Acta Numerica 2011: Volume 20

Acta Numerica (2011), pp. 1–119 doi:10.1017/S096249291100002X c Cambridge University Press, 2011 Printed in the Unite...

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Acta Numerica (2011), pp. 1–119 doi:10.1017/S096249291100002X

c Cambridge University Press, 2011 Printed in the United Kingdom

Topics in structure-preserving discretization∗ Snorre H. Christiansen Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway E-mail: [email protected]

Hans Z. Munthe-Kaas Department of Mathematics, University of Bergen, N-5008 Bergen, Norway E-mail: [email protected]

Brynjulf Owren Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway E-mail: [email protected] In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives. Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes. The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite ∗

Colour online available at journals.cambridge.org/anu.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

CONTENTS 1 2

Introduction Integration methods based on Lie group techniques 3 Schemes which preserve first integrals 4 Spatial symmetries, high-order discretizations and fast group-theoretic algorithms 5 Finite element systems of differential forms Appendix References

2 5 21 32 65 108 112

1. Introduction The solution of partial differential equations (PDEs) is a core topic of research within pure, applied and computational mathematics. Both measured in the volume of published work1 and also in terms of its practical influence on application areas such as computational science and engineering, PDEs rank above all other mathematical subjects. Historically, the field of numerical solutions of PDEs has its roots in seminal papers by Richard Courant and his students Kurt Friedrichs and Hans Lewy, in the early twentieth century. Among the nearly 4000 academic descendants of Courant we find a large fraction of the key contributors to the field, such as Friedrichs’ student Peter Lax, who received the Abel Prize in 2005 for his ground-breaking contributions to the theory and application of PDEs. Fundamental properties of PDE discretizations, which have been recognized as crucial ever since the days of the Old Masters, include accuracy, stability, good convergence properties and the existence of efficient computational algorithms. In more recent years various aspects of structure preservation have emerged as important in addition to these fundamental properties. 1

A count of all mathematical publications from 2001 to 2010, sorted according to the AMS Mathematics Subject Classification, reveals that number 35 PDEs ranks highest of all primary topics with 46 138 entries; in second place is 62 Statistics, with 39 176.

Topics in structure-preserving discretization

3

Within the topics of ordinary differential equations and time integration, a systematic study of structure-preserving discretizations was undertaken by Feng Kang in Beijing, starting in the 1980s. During the past decade a systematic study of the preservation of various geometric structures2 has evolved into a mature branch of numerical analysis, termed the geometric integration of differential equations (Hairer, Lubich and Wanner 2006, Leimkuhler and Reich 2004, Sanz-Serna and Calvo 1994). Experience has shown that the preservation of geometric properties can have a crucial influence on the quality of the simulations. In long-term simulations, structure preservation can have a dramatic effect on stability and global error growth. Examples of such structures are symplecticity, volume, symmetry, reversibility and first integrals. In short-term simulations it is frequently seen that discretization schemes designed with structure preservation in mind enjoy small errors per step, and hence become efficient for shorter time simulations too. An important class of problems is that of partial differential equations whose solutions may be subject to blow-up in finite time. We have seen that schemes which are designed to inherit certain symmetries of the continuous problem tend to perform well in capturing such finite-time singularities in the solution. For spatial PDEs, a parallel investigation of compatible discretizations has been undertaken (Arnold, Bochev, Lehoucq, Nicolaides and Shashkov 2006a). The equations of mathematical physics (describing fluids, electromagnetic waves or elastic bodies, for instance) have been presented in geometric language, easing the construction of discretizations which preserve important geometric features, such as topology, conservation laws, symmetries and positivity structures. When well-posedness of the continuous PDE depends on the conservation or monotonicity of certain quantities, such as energy, it seems equally important for the stability of numerical schemes that they enjoy similar properties. Rather than approximately satisfying the exact conservation law, as – it could be argued – any consistent scheme would do, it seems preferable, in order to obtain stable methods, to exactly satisfy a discrete conservation law. For PDEs written in terms of grad, curl and div acting on scalar and vector fields, one is led to construct operators acting on certain finite-dimensional spaces of scalar and vector fields, forming a complex which, in spatial domains with trivial topology, is an exact sequence. More generally, various discretizations of the de Rham sequence of differential forms have been introduced, and the successful ones are related to constructs of combinatorial topology such as simplicial cochain complexes (Arnold, Falk and Winther 2010).

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A geometric structure is understood as a structural property which can be defined independently of particular coordinate representations of the differential equations.

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S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren

This survey paper takes a view of PDEs analogous to looking at the moon through a telescope with very high magnification. In the vast lunar landscape we will focus on a small number of craters with particularly beautiful properties, and leave the rest of the lunar surface out of our main focus. One goal of the paper has been to tie together recent time integration techniques, in particular Lie group techniques and integral-preserving integration, and combine these with recent developments in structure-preserving spatial discretizations. The paper consists of four main parts: Sections 2–5. Section 2 covers integration methods based on Lie group techniques. We provide an introduction to the theory of Lie group integrators and describe how computational algorithms can be devised, given a group action on a manifold and coordinates on the acting group. Various choices of coordinates are discussed. Another type of Lie group integrator consists of those based on compositions of flows, and we discuss some applications of these methods to the time integration of PDEs. We focus in particular on applications outside the class of exponential integrators for semilinear problems. In Section 3 we discuss integral-preserving schemes. These methods apply to any PDE which has a known first integral, for instance an energy functional. We demonstrate how the method of discrete variational derivatives can be used to devise integral-preserving time integration schemes for PDEs. We consider, in particular, schemes that are linearly implicit, and therefore need the solution of only one linear system per time step. We also discuss methods which apply discrete variational derivatives, or discrete gradients, to conserve an arbitrary number of first integrals. In Section 4 we cover spatial symmetries, high-order discretizations and fast group-theoretic algorithms. The topic of this section is the exploitation of spatial symmetries. A recurring theme is that of linear differential operators commuting with groups of isometries acting on the domain. We investigate high-order spectral element discretization techniques based on simplicial subdivisions (into triangles, tetrahedra and simplexes in general) using high-order multivariate Chebyshev bases on the triangles. The efficient computation of matrix (and operator) exponentials is crucial for time integrators based on Lie group actions. We survey recent work on high-order discretization techniques, fast Fourier algorithms based on group-theoretic concepts and fundamental results from representation theory. In Section 5 we discuss finite element systems for differential forms. A framework for mixed finite elements is developed, general enough to allow for non-polynomial basis functions and a decomposition of space into noncanonical polytopes, but restrictive enough to yield spaces with local bases, interpolators commuting with the exterior derivative, and exact sequences where desirable. A smoothing technique gives Lq -stable commuting quasiinterpolation operators, from which various error estimates can be derived.


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2. Integration methods based on Lie group techniques The use of Lie group techniques for obtaining solutions to differential equations dates back to the Norwegian mathematician Sophus Lie in the second half of the nineteenth century. In Lie’s time these were used exclusively as analytical tools; however, more recently it has become increasingly popular to include Lie groups as an ingredient in numerical methods. The use of Lie groups in the numerical approximation of differential equations can be divided into two categories: one in which the aim is to preserve symmetries or invariance of the continuous model, and another in which Lie groups are used as building blocks for a time-stepping procedure. In this section we shall give a brief introduction to the mathematical machinery we use, and we will focus on the second category, that of using Lie groups as a fundamental component when designing numerical time integrators. We shall give a short introduction to the basics of Lie group integrators; for more details consult Iserles, Munthe-Kaas, Nørsett and Zanna (2000) and Hairer et al. (2006). The variety of studies related to Lie group integrators is now too large to cover in an exposition of this type. For this reason we shall focus on a selected part of the theory, and mostly consider integrators designed for nonlinear problems. Important classes of schemes that we shall not discuss here are methods based on the Magnus expansions and Fer expansions, usually applied to linear differential equations. This work developed in the 1990s, in large part due to Iserles and Nørsett: see, e.g., Iserles and Nørsett (1999). There are several excellent sources for a summary of these methods and their analysis, for instance the Acta Numerica article by Iserles et al. (2000), the monograph by Hairer et al. (2006) and the more recent survey by Blanes, Casas, Oteo and Ros (2009), which also contains many applications of these integrators. 2.1. Background and notation Let M be some differentiable manifold and let X (M ) be the set of smooth vector fields on M . We consider every X ∈ X (M ) as a differential operator on the set of smooth functions F(M ) on M . Thus, in local coordinates (x1 , . .∂. , xm ) in which X has components X1 , . . . , Xm , we write X = i Xi (x) ∂xi , so X[f ] = df (X), f ∈ F(M ) is the directional derivative of f along the vector field X. The flow of a vector field X ∈ X (M ) is a one-parameter family of maps exp(tX) : Dt → M , where Dt ⊂ M . For any x ∈ Dt we have exp(tX)x = γ(t), where γ(t) ˙ = X|γ(t) ,

γ(0) = x,

t ∈ (a(x), b(x)), a(x) < 0 < b(x).

The domain for exp(tX) is the set Dt = {x ∈ M : t ∈ (a(x), b(x))}; for further details see, e.g., Warner (1983, 1.48).

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The chain rule for the derivative of compositions of maps between manifolds is (ψ ◦ φ) = ψ ◦ φ , φ : M → N, ψ : N → P. From this, we easily obtain the useful formula X[f ◦ φ](x) = φ (X|x )[f ](y),

φ : M → N, X ∈ X (M ), f ∈ F(N ), (2.1) for x ∈ M, y = φ(x).

The Lie–Jacobi bracket on X (M ) is defined simply as the commutator of vector fields Z = [X, Y ] = XY − Y X. With respect to coordinates (x1 , . . . , xm ) it has the form Zi = [X, Y ]i =

m j=1

Xj

∂Yi ∂Xi − Yj . ∂xj ∂xj

This bracket makes X (M ) a Lie algebra; the important properties of the bracket is that it is bilinear, skew-symmetric and satisfies the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0,

∀X, Y, Z ∈ X (M ).

It is an easy consequence of (2.1) that φ ([X, Y ]) = [φ (X), φ (Y )],

X, Y ∈ X (M ), φ : M → N.

(2.2)

We say that Y is φ-related to X ∈ X (M ) if Y |φ(x) = φ (X|x ) for each x ∈ M . Note that Y is not generally a vector field on N : since φ does not have to be either injective or surjective, we must treat Y as a pullback section of φ∗ T N over M . In the particular case when φ is a diffeomorphism, there is a unique Y ∈ X (N ) which is φ-related to X; this vector field is then called the pushforward of X with respect to φ: Y = φ∗ X = φ (X) ◦ φ−1 . A Lie group G is a differentiable manifold which is furnished with a group structure such that the multiplication is a smooth map from G × G to G, and the map g → g −1 , g ∈ G is smooth as well. We briefly review some important concepts related to Lie groups. Every Lie group G has a Lie algebra associated to it, which can be defined as the linear subspace g ⊂ X (G), of right-invariant vector fields on G equipped with the Lie–Jacobi bracket. By right-invariance we mean invariance under right translation. Consider the diffeomorphism Rg : G → G defined as Rg (h) = h · g. A vector field X ∈ X (G) is right-invariant if it is Rg -related to itself, i.e., Rg∗ X = X. By (2.2) it follows that the Lie–Jacobi bracket between two right-invariant vector fields is again right-invariant. For every v ∈ Te G, where e is the identity element, there is a unique Xv ∈ g such that Xv |g = Rg (v), and for X ∈ X (G) there is an element vX ∈ Te G given as vX = X|e . This shows that g is isomorphic to Te G, and it may often be convenient to represent


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g as Te G rather than as right-invariant vector fields on G. We next let the group act locally on a manifold. A left group action on a differentiable manifold M by a Lie group G is a map G × M → M , which we denote as y = g · x ∈ M or simply y = gx for g ∈ G and x ∈ M . The group action must satisfy the conditions e · x = x, ∀x ∈ M,

g · (h · x) = (g · h) · x.

The group action is said to be free if g · x = x ⇒ g = e, so that the only group element which leaves x ∈ M fixed is the identity element. If it is true that for every pair of points x ∈ M, y ∈ M there exists a group element g such that y = g · x, the action is called transitive. We usually just need a local version of transitivity, requiring that for every x ∈ M , G · x contains some open neighbourhood of x. The orbit of the group action containing x is the set Ox = {g · x | g ∈ G}. It is sometimes useful to restrict the action to an orbit when transitivity is a desired property. Similarly, if one needs the group action to be free, it may be useful to extend the action to multispace M ×r M (r copies of M ) or to a suitable jet space whenever M takes the form of a fibred space. Let G be a Lie group with Lie algebra g. Assume that G acts locally on the manifold M , and set Λx (g) = g · x for g ∈ U ⊆ G, x ∈ M where U is some open neighbourhood of the identity element. For every fixed v ∈ g there is a vector field Xv ∈ X (M) defined by d Xv |x = λ∗ (v)|x = Λx (v) = γ(t) · x, dt t=0

where γ(t) is any smooth curve on G such that γ(0) = e, γ(0) ˙ = v. So λ∗ is a Lie algebra homomorphism of g into its image in X (M ). For Lie group integrators, it is important that the action is locally transitive; in this case it is true that for every x ∈ X λ∗ (g)|x = Tx M. This property means, in particular, that for any smooth vector field F ∈ X (M ) there exists a map f : M → g such that F |x = λ∗ (f (x))|x ,

(2.3)

a formulation called the generic presentation of ODEs on manifolds by Munthe-Kaas (1999). If the action is free then the map f (x) is unique for a given vector field F . Suppose further that the Lie algebra g is of dimension d and let e1 , . . . , ed be a basis for g. Let Ei = λ∗ (ei ), i = 1, . . . , d. We ¯ = span{E1 , . . . , Em }, that is, call the set {E1 , . . . , Ed } a frame,3 and let g 3

In the literature, a ‘frame’ is often used as a local object requiring that E1 |x , . . . , Ed |x is a basis for Tx M . We do not impose this condition here per se, as we find it useful to have a global representation of vector fields on M .

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the linear span of the frame fields over R or C. Now we can represent any smooth vector field X ∈ X (M ) by means of d functions fi : M → R: F |x =

d

fi (x) Ei |x .

(2.4)

i=1

The set of functions fi is uniquely given only if the action is free. There are two alternative ways to proceed, following either the terminology introduced by Crouch and Grossman (1993) or that of Munthe-Kaas (1995, 1998, 1999). In the former case, one introduces a freeze operator Fr : M × X (M ) → g¯ relative to the frame. It is defined by Xp |x := Fr(p, X)|x = fi (p)Ei |x . (2.5) i

The frozen vector field Xp has the property that it coincides with the unfrozen field X at the point x = p, i.e., Xp |x = X|x . A main assumption underlying many Lie group integrators is that flows of frozen vector fields can be calculated, or in some cases approximated with acceptable computational cost. 2.2. Integrators based on coordinates on the Lie group In what follows, we shall take g to be Te G. Consider a diffeomorphism defined from some open subset of U ⊆ g containing 0, i.e., Ψ : U → G and we require that Ψ(0) = e, Ψ0 (v) = v, ∀v ∈ g. For convenience, we shall work with a right-trivialized version of Ψu , setting ◦ dΨu , Ψu = RΨ(u)

dΨu : g → g.

Let the group act locally on the manifold M ; suppose that the action is defined on a subset Gp ⊆ G, where Ψ(U ) ⊂ Gp for every p ∈ M . For any p ∈ M and u ∈ U , set λp (u) = Ψ(u) · p, and let f : M → g represent the vector field F ∈ X (M ) through the form (2.3). We define the vector field F˜p on X (U ) by F˜p = dΨ−1 (2.6) u f (λp (u)). u

A simple calculation (Munthe-Kaas 1999, Owren and Marthinsen 2001) shows that F is λp -related to F˜ such that, for any u ∈ U , we have F |λp (u) = λp u (F˜p |u ). This local relatedness between vector fields F of the form (2.3) whose flows are to be approximated, and vector fields on the algebra of the acting group serves as the key underlying principle of many Lie group integrators. The idea is that near any point p ∈ M one may represent curves in the form y(t) = λp (σ(t)), and if the differential equation for y(t) is given by F , then a differential equation for σ(t) is that of F˜ . Most of the known integrators for ODEs have the property that when the solution belongs to


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some linear space, the numerical method will provide approximations for the solution belonging to the same linear space. So one may now approximate solutions to the ODE given by F˜ , and obtain numerical approximations in the linear space g. The map λp will then map these approximations onto M by definition. One may typically choose the initial value in each step to be p, so that when solving for σ in g one sets σ(tn ) = 0. Suppose that a one-step method map which preserves linear structure is denoted Φh,F˜p , where h is the time step. Algorithm 2.1. for n = 0, 1, . . . do p ← yn σn+1 ← Φh,F˜p (0) yn+1 ← λp (σn+1 ) end for We may be more specific and for instance insist that the scheme Φh,F˜ to be used in the Lie algebra is an explicit Runge–Kutta method, with weights bi and coupling coefficients aji , 1 ≤ j < i ≤ s. In this case we can phrase the scheme as follows for integrating a system in the form (2.3) from t0 to t0 + h, with initial value y(t0 ) = y0 ∈ M . Algorithm 2.2. (Runge–Kutta–Munthe-Kaas) for i = 1 → s do j ui ← h i−1 j=1 ai kj ki ← f (Ψ(ui ) · y0 ) ki ← dΨ−1 ui (ki ) end for v ← h sj=1 bj kj y1 = Ψ(v) · y0 An obvious benefit of schemes of this form is that they will preserve the manifold structure, a feature which cannot be expected when M is modelled as some embedded submanifold of Euclidean space. If the manifold happens to belong to a level set of one or more first integrals, then the Lie group integrators will automatically preserve these integrals. There are, however, other interesting situations, when the group action is used as a building block for obtaining a more accurate representation of the exact solution than is possible with other methods. The idea behind Lie group integrators can also be seen as a form of preconditioning. Coordinate maps. The computational cost of the Lie group integrators is an important issue, and the freedom one has in choosing the coordinate map Ψ may be used to optimize the computational cost. The generic choice for Ψ(u) is of course the exponential map Ψ(u) = exp u. This choice is called

10


canonical coordinates of the first kind. If the Lie group and its Lie algebra are realized as matrices, we have exp u =

∞ uk k=0

k!

.

Issues related to computing the matrix exponential have been thoroughly debated in the literature, going back to the seminal paper of Moler and Van Loan (1978), and its follow-up, Moler and van Loan (2003). For dense n × n matrices one would normally expect a computational complexity of approximately C n3 , where C will depend on several factors, as the tolerance for the accuracy, the size of the matrix elements and the conditioning of the matrix. The methods presented in these papers, however, do not usually respect the Lie group structure, such that an approximation g˜ ≈ exp u, u ∈ g will not belong to the group, i.e., g˜ ∈ G; this is a crucial issue as far as exact conservation is concerned. In practice, one is left with two alternatives. (1) Apply a standard method which yields g = exp u to machine accuracy. For relevant examples with Lie group integrators, the factor C typically lies in the range 20–30 (Owren and Marthinsen 2001). (2) Apply some approximation which is not exact, but which respects the Lie group structure, i.e., g˜ ≈ exp u with g˜ ∈ G. This approach has been pursued by Celledoni and Iserles (2000, 2001) for approximation by low-rank decomposition, as well as Zanna and Munthe-Kaas (2001/02) and Iserles and Zanna (2005) by means of the generalized polar decomposition. These approaches still have a computational cost of C n3 for Lie algebras whose matrix representation yields dense matrices, but the constant C may be smaller than for the general algorithms. It is interesting to explore other possible choices of analytic matrix functions than the exponential. One then replaces the exponential map with some local diffeomorphism from a neighbourhood of 0 ∈ g to G, usually required to map 0 → e. But it turns out that the exponential map is the only possible choice of analytic map that works for all Lie groups: this is asserted, for instance, using a result by Kang and Shang (1995). Lemma 2.3. Let sl(d) denote the set of all d × d real matrices with trace equal to zero and let SL(d) be the set of all d × d real matrices with determinant equal to one. Then, for any real analytic function R(z) defined in a neighbourhood of z = 0 in C satisfying the conditions: R(1) = 1 and R (0) = 1, we have that R(sl(d)) ⊆ SL(d) for some d ≥ 3 if and only if R(z) = exp(z). They prove this result by taking the Lie group SL(d) of unit determinant d × d matrices, d ≥ 3, whose Lie algebra sl(d) consists of d × d matrices with

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trace zero as an example: see also Hairer et al. (2006, p. 102). There are, however, exceptions when certain specific Lie groups are considered. For instance, if a matrix group GJ and its corresponding Lie algebra gJ can be characterized as GJ = {A ∈ GL(d) : A JA = J},

gJ = {a ∈ gl(d) : a J + Ja = 0}, (2.7)

for some fixed d × d matrix J, it turns out that every analytic function R(z) satisfying R(z)R(−z) = 1

(2.8)

will have the property a ∈ gJ ⇒ R(a) ∈ GJ . Such groups include the symplectic group Sp(d), the orthogonal groups O(d), SO(d), and the Lorenz groups SO(, d − ). One of the most popular choices of maps that satisfies (2.8) is the Cayley transformation Ψcay (z) =

1+z . 1−z

This transformation was used by Lewis and Simo (1994), and later by Diele, Lopez and Peluso (1998) for the group SO(d), Lopez and Politi (2001) for general groups of the form (2.7), and Marthinsen and Owren (2001) for linear equations. If one does not require the map to be realizable as an analytic function of the Lie algebra matrix, there are other choices, for instance the canonical coordinates of the second kind. This map requires the use of a basis for g, say v1 , . . . , vs where s = dim g. Then the map is defined as Ψccsk : v = α1 v1 + · · · + αs vs → exp(α1 v1 ) · exp(α2 v2 ) · · · exp(αs vs ). Here, the choice as well as the ordering of the basis is clearly an issue. In Owren and Marthinsen (2001) this coordinate map was, together with a generic basis of the Lie algebra, known as the Chevalley basis. The main advantage of this map is that when matrices are used, natural choices of bases are frequently realized as sparse matrices whose exponentials may be explicitly known. As an example, one may consider the special linear group SL(d), which can be realized as the set of d × d matrices with unit determinant. The Lie algebra is then the set of trace-free d × d matrices, and the Chevalley basis is obtained by choosing d − 1 trace-free diagonal matrices, say ei+1 e i+1 − ei ei , 1 ≤ i ≤ d − 1 together with all matrices ei ej , j = i. Also, hybrid variants are possible, where, for instance, the Lie algebra is decomposed into a direct sum of subspaces, say g = g1 ⊕ g2 ⊕ · · · ⊕ gs¯,

s¯ j=1

dim(gj ) = s,

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and a coordinate map can be realized as Ψhyb : v = v1 + · · · + vs¯ → exp(v1 ) · · · exp(vs¯),

vi ∈ gi .

In fact, such coordinate maps derived from the generalized polar decomposition were considered in Krogstad, Munthe-Kaas and Zanna (2009). Computing the inverse differential. In the transformation of the vector field to the Lie algebra, one needs to compute the inverse of the differential dΨ−1 u v. For the case of the exponential mapping, it was found by Baker (1905) and Hausdorff (1906) that it can be expanded in an infinite series of commutators. Defining the operator adu (v) = [u, v], for any u and v in g, we have ∞ Bk k −1 ad (v). (2.9) dexpu v = k! u k=0

Here, the constants Bk are the Bernoulli numbers, B2k+1 = 0, k ≥ 1, and the first non-zero ones are B0 = 1, B1 = −1/2, B2 = 1/12, B4 = −1/720. We will not dwell on the convergence properties of this series (discussed, for instance, in Varadarajan (1984)), the reason being an observation made by Munthe-Kaas (1999). The series (2.9) may be truncated at any point to yield an approximation in the Lie algebra g. Since one usually requires that the initial point of each time step in the Lie algebra, σ0 = 0, it follows that the term of index k will be of at least order hk as h → 0. It is therefore sufficient to truncate the series at an index that is consistent with the convergence order of the time-stepper Φh,F˜ . An extensive analysis of the number of terms that must be kept in such commutator expansions was presented in Munthe-Kaas and Owren (1999), but see also McLachlan (1995). Considering the coordinate maps Ψccsk and Ψhyb , their inverse differential maps are studied in Owren and Marthinsen (2001) and Krogstad et al. (2009) respectively. In these cases, the maps are computed exactly, but by carefully studying the structural properties of the respective Lie algebras it is possible to find computationally inexpensive algorithms: typically, if the 3/2 ). dimension of the Lie algebra is d, the cost of computing dΨ−1 u (v) is O(d Choosing the group and the group action. A particularly interesting case for a global group action is the homogeneous space. If the group G is acting transitively on M , then the manifold is called a homogeneous space. In this case it is well known (see, e.g., Bryant (1995)) that M is naturally diffeomorphic to G/Gp , where Gp is the isotropy or stabilizer subgroup of G, Gp = {g ∈ G : g · p = p}. The simplest case is when the action is free, so that Gp = e for all p ∈ M , meaning M ∼ = G. In this case the function f in (2.3) is unique, and there is no isotropy. In the case that there is a non-trivial isotropy group, the choice


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of f is not unique, in fact f may be replaced by f + w, where w(x) ∈ ker λ∗ for every x ∈ M . It turns out that the choice of w affects the numerical integrator. We proceed to give some examples frequently seen in applications: see Munthe-Kaas and Zanna (1997). Suppose that the Lie group G = SO(d) is a group of orthogonal d × d matrices. Its Lie algebra is so(d), the set of d × d matrices which are real and skew-symmetric. The manifold M is a connected open subset of the d × r matrices, and we give two examples. (1) Let the Lie group action be left multiplication g · m where · is now just matrix–matrix multiplication. We may now write the ODE vector field as: F |y = f (y) · y,

f : M → g.

(a) M is the set of vectors in Rd with unit length, identified with the (d − 1)-dimensional sphere S d−1 . (b) M is the set of d × r matrices with orthonormal columns, identified with the Stiefel manifold Vr (Rd ). (2) r = d, and the Lie group action is the conjugation g · m = gmg , the right-hand side is just products of matrices. We may choose M to be some isospectral set, such as the set of all matrices with a given prescribed Jordan form; one may also restrict to the set of symmetric matrices with a fixed set of eigenvalues. Another interesting choice is the affine group, first discussed in the context of Lie group integrators in Munthe-Kaas (1999). One may start with the group G = GL(d) Rd , the semidirect product between the general linear group and Rd . G is the group of all affine linear maps acting on Rd ; we have (A, a) · y = Ay + a. The group product is given as (A, a) · (B, b) = (AB, Ab + a). The corresponding Lie algebra is g = gl(d) Rd and the Lie bracket is given as [(A, a), (B, b)] = ([A, B], Ab − Ba). The exponential mapping can be expressed in terms of the matrix exponential exp(A, b) = (exp(A), dexpA (b)), where

∞

dexpA (b) =

1 exp(A) − I b= Aj b. A (j + 1)! j=0

Munthe-Kaas made the observation that this action has a large isotropy group: the dimension of the group is d(d + 1) whereas the manifold M = Rd

14


only has dimension d. Setting λp (A, a) = exp(A, a)·p = exp(A)p+dexpA (a), we find λ∗ (A, a)(y) = Ay + a, such that the isotropy algebra at y consists of all elements in G of the form (A, −Ay). Suppose that the ODE vector field is F . Then at one extreme one may choose f (y) = (0, F |y ), in which case one would recover the method Φh,F in the algorithm described above. However, as suggested in Munthe-Kaas (1999), one may instead add to this the element (J, −Jy), J ∈ GL(d) from ker λ∗ (.)(y), to obtain the function f (y) = (J, F |y − Jy), and a natural choice for J would be the Jacobian of the vector field F evaluated at some point near y. One may also replace the group GL(d) by a subgroup: the group SO(d) yields the special Euclidean group SE(d), whereas in applications to PDEs a popular choice is to select some oneparameter subgroup exp(tL), L ∈ gl(d). The Lie subalgebra elements are now of the form (αL, a) with fixed L; thus one may just write (α, a) ∈ R×Rd , and the Lie bracket of this subalgebra is simply [(α, a), (β, b)] = [0, L(αb − βa)].

(2.10)

The isotropy group Gp is now only one-dimensional; its Lie algebra is spanned by the element (1, −Lp). Time-symmetric Lie group integrators. For many problems it is of interest to apply integrators which are symmetric in time or self-adjoint. For onestep methods, say y1 = φh (y0 ), this means that if the integrator is applied backwards in time, i.e., with step size −h and y1 as input, the output y0 results, such that φh = φ−1 −h . For Lie group integrators, one may consider as an example the Runge–Kutta–Munthe-Kaas scheme based on the midpoint rule, using the coordinate map Ψ = exp. This scheme is an implicit counterpart to the method given in Algorithm 2.2: u=

h k, 2

k = f (exp(u) · y0 ),

y1 = exp(hk) · y0 .

Here we have omitted the correction dexp−1 u , since it is not going to alter the order of the method. Following Zanna, Engø andMunthe-Kaas (2001), we define the midpoint approximation y 1 = exp 12 hu · y0 ; the method may 2 be written in the form h h f y 1 .y 1 , y0 = exp − f y 1 .y 1 , y1 = exp 2 2 2 2 2 2


15

which is easily seen to be symmetric. Unfortunately, Lie group integrators based on coordinates on the Lie group are not generally symmetric even if the underlying scheme on the Lie algebra is a symmetric method. Zanna et al. (2001) considers how a recentering of the coordinate system for the Lie group can be used to obtain symmetric integrators. Rather than letting Ψ(0) = e be the base point of the coordinate chart, one may choose a θ ∈ g and define a new base point q := Ψ(θ)−1 · y0 . The curve y(t) ∈ G is now represented locally by a curve σ(t) ∈ g, as y(t) = Ψ(σ(t)) · q = Ψ(σ(t)) · Ψ(θ)−1 · y0 . This shift of coordinate chart results in the same differential equation for σ(t), but with initial value σ(0) = θ corresponding to y(0) = y0 . An implicit Runge–Kutta-type method generalizes Algorithm 2.2 as follows:   σi = θ + h sj=1 aji kj ,    −1 i = 1, . . . , s, (2.11) ki = f (Ψ(σi ) · Ψ(θ) · y0 ),     ki = dΨ−1 σi (ki ), v = θ + h si=1 bi ki , y1 = Ψ(v) · Ψ−1 (θ) · y0 . The idea is now to choose the centre θ depending on the stages ki and h in such a way that the resulting method is symmetric whenever the underlying Runge–Kutta scheme is symmetric. From Zanna et al. (2001) we find that this can be achieved by setting θ = θ(h; k1 , . . . , ks ) and requiring it to satisfy θ(−h; ks , . . . , k1 ) = θ(h; k1 , . . . , ks ) + h

s

bi ki .

(2.12)

i=1

There is of course no unique way of satisfying (2.12). An obvious choice which has been called the geodesic midpoint is h i b ki , 2 s

θ(h, k1 , . . . , ks ) = −

i=1

but in fact, one may choose more generally θ(h; k1 , . . . , ks ) = −h

s i=1

w i ki ,

where ws+1−i + wi = bi ,

i = 1, . . . , s.

16


A concrete example of a symmetric scheme in the format (2.11), using the geodesic midpoint for θ and the 2-stage Gauss method is h θ = − (k1 + k2 ), 4√ 3 σ1 = −h k2 , 6 k1 = f (Ψ(σ1 ) · Ψ(θ)−1 · y0 ),

√

3 k1 , 6 k2 = f (Ψ(σ2 ) · Ψ(θ)−1 · y0 ),

σ2 = h

k2 = dΨ−1 k1 = dΨ−1 σ1 (k1 ), σ2 (k2 ), h v = (k1 + k2 ), y1 = Ψ(v) · Ψ(θ)−1 · y0 . 4 McLachlan, Quispel and Tse (2009) recently studied symmetric and symplectic schemes which are linearization-preserving at fixed points, meaning that if y¯ is such that f (¯ y ) = 0 in the initial value problem y = f (y), y(0) = y0 then the integrator has the Taylor expansion about y¯:

y1 − y¯ = exp(hf (¯ y ))(y0 − y¯) + O(y0 − y¯2 ). Symmetric schemes for a class of exponential integrators applied to semilinear problems were proposed by Celledoni, Cohen and Owren (2008). 2.3. Lie group integrators based on compositions of flows A pioneering article by Crouch and Grossman (1993) suggests applying compositions of flows of frozen vector fields (2.5) to construct integration methods for vector fields which have been expressed in the form (2.4). They propose such generalizations both of multistep methods and of explicit Runge– Kutta methods. The latter takes a step from t0 to t0 + h as follows. Algorithm 2.4. (Crouch–Grossman) Y1 ← y0 F1 ← Fr(Y1 , F ) for r = 2 → s do 1 Yr ← exp(har−1 r Fr−1 ) · · · exp(har F1 )(y0 ) Fr ← Fr(Yr , F ) end for y1 ← exp(hbs Fs ) · · · exp(hb1 F1 )(y0 ) Here, br , akr are the weights and the coupling coefficients of the Runge– Kutta method. Explicit third-order with s = 3 were presented in Crouch and Grossman (1993), and Owren and Marthinsen (1999) presented a general theory of order conditions and examples of fourth-order methods with four stages. A potential drawback of the Crouch–Grossman methods is that they require the computation of a large number of flows of frozen vector fields in every time step. This led Celledoni, Marthinsen and Owren


17

(2003) to consider a version of these methods where flows of arbitrary linear combinations of frozen vector fields were considered. The methods take the following form. Algorithm 2.5. (Commutator-free methods) Y 1 ← y0 F1 ← Fr(Y1 , F ) for r = 2 → s do k k F ) · · · exp(h Yr ← exp(h k αr,J k k αr,1 Fk )(y0 ) Fr ← Fr(Yr , F ) end for y1 ← exp(h k βJk Fk ) · · · exp(h k β1k Fk )(y0 ) Note that when the vector fields commute, the expressions for Yr and y1 collapse to k k ar Fk (y0 ), y1 = exp b Fk (y0 ), Yr = exp k

k

where akr

=

J j=1

k αr,j ,

k

b =

J

βjk .

j=1

These coefficients correspond to the akr , bk in the Runge–Kutta–MuntheKaas methods, and they must in particular satisfy the classical order conditions for Runge–Kutta methods. The key idea of the commutator-free methods is to keep the number of exponentials or flows as low as possible in each stage. In Celledoni et al. (2003) the following fourth-order method was presented: Y 1 = y0 , Y2 = exp

1

F1 = Fr(Y1 , F ),

hF1 (y0 ), F2 = Fr(Y2 , F ), F3 = Fr(Y3 , F ), (2.13) Y3 = exp 2 hF2 (y0 ), 1 Y4 = exp − 2 hF1 + hF3 (Y2 ), F4 = Fr(Y4 , F ), h h (3F1 + 2F2 + 2F3 − F4 ) (y0 ). y1 = exp 12 (−F1 + 2F2 + 2F3 + 3F4 ) exp 12 21

Note that if the number of exponentials is counted according to the format of Algorithm 2.5, we would have two exponentials in the expression for Y4 , but since the rightmost exponential coincides with Y2 , we effectively compute only one exponential in each stage Yr , r = 2, 3, 4 and two exponentials in y1 , and thus the total number of flow calculations in this method is 4. In comparison, the methods of Crouch and Grossman would require 10 flow calculations. A complete description of the order theory for this type of method was given in Owren (2006). Wensch, Knoth and Galant (2009)

18


extend the idea of re-using flow calculations, as with Y2 in (2.13), to allow for the composition of exponentials to be applied to more general expressions involving y0 and the stage values Yi . For the case of linear differential equations, one can design commutator-free methods based on the Magnus series expansions: see Blanes and Moan (2006). 2.4. Applications of Lie group integrators to PDEs The Lie group integrators described in the preceding sections can be used as time-steppers in solving partial differential equations. It should come as no surprise that these integrators are intimately linked to a well-known class of methods known as ‘exponential integrators’, which go back to the early 1960s in pioneering work by Certaine (1960), Nørsett (1969) and Lawson (1967). An excellent account of these methods was presented in a recent Acta Numerica article by Hochbruck and Ostermann (2010), and we refer the reader to that article and the references therein for a detailed account of exponential integrators. There are several important papers which study analytical properties of exponential integrators, in particular for parabolic problems. An important issue is to describe the convergence order in the presence of unbounded operators: see, for instance, Hochbruck and Ostermann (2005) and Ostermann, Thalhammer and Wright (2006) and the references therein. In studying such time integrators one possibility is to apply the time-stepper without discretizing in space, thereby interpreting the method as a discrete trajectory in some abstract space of infinite dimension. Another option is to first do a spatial discretization to obtain a finite-dimensional system, and then apply the time-stepping scheme to this finite-dimensional problem. The most widespread application of exponential integrators is perhaps to the semilinear problem ut = Lu + N (t, u). (2.14) Here L is a linear operator, and N (t, u) is a nonlinear function or mapping. In many interesting cases, L may be a differential operator of higher order than those appearing in N (t, u); we refer to Minchev (2004) for an extensive treatment of this case. Such a problem lends itself to a formulation by the action of the affine group as described above; then any Lie group integrator can be applied. Cox and Matthews (2002) found an exponential integrator for (2.14) which turned out to be identical to the general commutatorfree Lie group integrator in Celledoni et al. (2003) applied with the affine action adapted to (2.14), when the Lie algebra g is a subalgebra of the affine Lie algebra consisting of elements (αL, b) as described above. Also, Krogstad (2005) derived exponential integrators with excellent properties for semilinear problems. In what follows we shall consider the application of Lie group integrators to some non-standard cases, differing from (2.14).


19

Celledoni–Kometa methods. Whereas the most common application of exponential integrators is to problems of the form (2.14), there are other examples in the literature. One is a class of schemes proposed by Celledoni and Kometa (Celledoni 2005, Celledoni and Kometa 2009). They consider convection–diffusion problems of the form ∂u + V(u) · ∇u = ν∇2 u + f, (2.15) ∂t where the convection term V(u) · ∇u is dominating, and 0 < ν 1. A semidiscretized version of (2.15) is obtained as yt − C(y)y = Ay + f,

y(0) = y0 .

(2.16)

Here the nonlinear convection term is assumed to take the form C(y)y for some matrix C(y) and the diffusion operator is similarly replaced by a constant matrix A. A typical strategy is now to treat the convection term and the diffusion term separately, using an explicit and possibly high-order approximation for the convection term, and an implicit integrator for the comparably stiff diffusion term. This is the idea behind the IMEX methods, for instance: see Kennedy and Carpenter (2003) for a general theory. Note that the matrix or linear operator A will usually be symmetric and negative definite: therefore implicit schemes can be implemented efficiently in such a splitting. The approach of Celledoni and Kometa (2009) is to use a non-trivial flow calculation for the convection part. A model example of a method in the class they consider for (2.16) is y1 = exp(h C(y0 ))y0 + h Ay1 + h f. This scheme can be generalized by adding stages in a Runge–Kutta fashion and by introducing compositions of exponentials in a similar way to that of commutator-free Lie group integrators. This leads to a semi-Lagrangian approach in which linearized versions of the convection part of the problem are solved exactly by the method of characteristics, for example. The general form of the method applied to (2.16) is   Yi = φi y0 + h sj=1 ai,j φi,j AYj ,   k k  i = 1, . . . , s, φi = exp h k αi,J C(Yk ) · · · exp h k αi,1 C(Yk ) ,     φi,j = φi φ−1 j , ¯ 0 + h s bi φ¯i AYj , y1 = φy i=1 k φ¯ = exp h β k C(Yk ) · · · exp h β C(Yk ) , k

¯ −1 . φ¯i = φφ i

J

k

1

Examples of schemes of order two and three are provided in Celledoni and Kometa (2009).

20


Commutator-free methods. Wensch et al. (2009) introduce the multirate infinitesimal step method for differential equations formulated on Euclidean space of the form y = f (y) + g(y). (2.17) The method is defined by the authors as follows. (1) Define the initial points of the stage flows as Zni (0) = yn + αij (Ynj − yn ). j

(2) Compute the stages by solving the following equations exactly: 1 Zni (τ ) = γij (Ynj − yn ) + βij f (Ynj ) + di g(Zni (τ )), h j

j

Yni = Zni (h), and the update stage, yn+1 , can be interpreted as an additional stage Yn,s+1 . It is thus assumed that one can calculate exactly flows of vector fields of the form y = c + f (y), (2.18) and the explicitness of the scheme is ensured by requiring that αij = βij = γij = 0 for j ≥ i. Furthermore, one assumes that the method is balanced, meaning that di = j βij for every i. The order analysis can be treated by the theory developed in Owren (2006), and third-order methods are derived under some simplifying assumptions, setting the γij = 0 and demanding that precisely one αij = 0 for each i. When g(y) ≡ 0, the scheme reduces to a standard Runge–Kutta method. In practical situations one may relax the requirement that the flow of (2.18) is solved exactly, but one may instead approximate it with high order of accuracy, for instance by using much smaller time steps than the h which features in the method format above – hence the name multirate methods. The main application of these methods in Wensch et al. (2009) is the 2D Euler equations for atmosphere processes, in which the two main scales are the fast acoustic waves and the slower advection terms: ∂ρ = −∇ · (ρu), ∂t ∂ρu ∂p = −∇ · (u ⊗ ρu) − ∇Θ − f (ρ), ∂t ∂Θ ∂Θ = −∇ · (Θu), ∂t p = ΘR(p/p0 )κ .


21

Here u = (u, w) denotes the horizontal and vertical velocity, ρ is the density, Θ = θρ where θ is the potential temperature, p is the pressure and R, p0 , κ are physical parameters. The spatial discretizations used are third-order upwind differences for the advection terms and centred differences for the pressure terms.

3. Schemes which preserve first integrals The development of energy-preserving finite difference schemes go all the way back to the seminal work of Courant, Friedrichs and Lewy (1928) where the celebrated energy method was presented for the first time. Energy is an example of a first integral, a functional which is preserved along exact solutions of the PDE. In the early days of integral-preserving methods, the focus was mostly on stability issues, the preservation of energy, for instance, is an invaluable tool in proving stability of numerical methods for evolutionary problems. More recently some of the interest has shifted to the conservation property itself, considering not only stability issues, but also the ability of the numerical solution to inherit structural properties of the continuous system. In this section we shall discuss some rather general procedures that can be used to construct numerical schemes which exactly preserve one or more first integrals of the partial differential equations. The framework we present includes, but is not limited to, Hamiltonian PDEs. That is, we shall consider systems which can be written in the form δH , (3.1) δu where D is a skew-adjoint operator, which is applied to the variational derivative of a first integral H. In general, the operator D may depend on the solution itself as well as its partial derivatives with respect to spatial variables. In the literature one can also find several examples in which ‘dissipative versions’ of (3.1) are studied, where the operator D is replaced by an operator which causes the functional H to decrease monotonically along exact solutions, the aim then being to design a numerical method which has the same dissipation property. Here we shall consider only the conservative case, and our aim is to derive numerical schemes which, at each time level t = tn , produce a numerical approximation U n to the exact solution u(tn , ·) of (3.1), satisfying H(U n ) = H(U 0 ) for all n ≥ 1, or Hd (un ) = Hd (u0 ) where Hd is a suitable spatially discretized version of H and un is the corresponding space discretization of U n . We shall also assume that the spatial domain X and boundary conditions are of such a type that (possibly repeated) integration by parts does not cause boundary terms to appear. In the rest of this section we define M to be a fixed connected open subset M ⊂ X × U , where X is the space of independent variables x1 , . . . , xp , and ut = D

22


U is the space of dependent variables u1 , . . . , uq . The spaces of prolongations M (n) ⊂ U (n) are then open subsets of the corresponding jet spaces. Following Olver (1993), we define the C-algebra of smooth differential functions A from M (n) to C. The space Aq is the set of q-plets of differential functions. Usually, the order n of the prolongation is of minor importance to the arguments presented here, and thus we just write F [u] or F ((uαJ )) for F ∈ Aq . We shall make use of operators D = D[u], whose adjoint D∗ [u] is defined via P D∗ Q

DP Q = X

X

for all compactly supported differential functions P and Q. The operator D is skew-adjoint if D∗ = −D. We consider the subclass of evolution equations which can be written in the form ∂u δH = F [u] = D[u] [u], ∂t δu

(3.2)

where D[u] is skew-adjoint. The functional H[u] is usually of the form H[u] = G[u]. (3.3) Equations of this form include the set of Hamiltonian PDEs, but here we do not need to assume that the operator D[u] defines a Poisson bracket. The general formula for the variational derivative expressed in terms of the Euler operator is included here for later use: δH δu is an m-vector depending on uαJ for |J| ≤ ν where ν ≥ ν, defined via (Olver 1993, p. 245) δH ∂ · ϕ dx = H[u + ϕ], (3.4) ∂ =0 Ω δu for any sufficiently smooth m-vector of functions ϕ(x). One may calculate δH δu by applying the Euler operator to G[u]; the α-component is given by δH α = Eα G[u], (3.5) δu where Eα =

|J|≤ν

(−1)|J| DJ

∂ , ∂uαJ

(3.6)

so that the sum ranges over all J corresponding to derivatives uαJ featuring in G. We have used total derivative operators: ∂uα ∂ J . D J = D j1 . . . D jk , D i = ∂xi ∂uαJ α,J

23


Another example of a Hamiltonian PDE is a generalized version of the Korteweg–de Vries equation ut + uxxx + (up−1 )x = 0, which can be cast in the form (3.2), with 1 2 1 p u − u dx, H[u] = 2 x p

D=

(3.7) ∂ . ∂x

(3.8)

Clearly, the functional H[u] is conserved along solutions of (3.2), since d δH δH δH H[u] = [u] ut = [u]D[u] [u] = 0, dt δu δu δu because D is skew-adjoint. The idea behind conservative integration schemes is to construct numerical methods which exactly reproduce this property as the numerical method advances from time tn to tn+1 , i.e., assuming that the solution to (3.2) at time t = tn is approximated by U n , we ask the scheme to satisfy H[U n ] = H[U 0 ], ∀n ≥ 1. (3.9) Alternatively, (3.9) may be imposed for some approximation Hd to H. The methodology we present here has its source in the ODE literature. One of the most prevalent approaches is that of discrete gradients, examples of which were proposed by LaBudde and Greenspan (1974), and treated systematically by Gonzalez (1996) and McLachlan, Quispel and Robidoux (1999). A finite-dimensional counterpart to (3.2) is the system y˙ = S(y)∇H(y) = f (y),

y ∈ Rm ,

(3.10)

where S(y) is an antisymmetric matrix. The idea is to introduce a discrete approximation to the gradient, letting ∇H : Rm ×Rm → Rm be a continuous map satisfying H(u) − H(v) = ∇H(v, u) (u − v), ∇H(u, u) = ∇H(u). The existence of such discrete gradients is well established in the literature, and their construction is not unique: see, for instance, the monograph by Hairer et al. (2006). The averaged vector field (AVF) gradient, for example, is defined by 1 ∇AVF (v, u) = ∇H(ξu + (1 − ξ)v) dξ. (3.11) 0

Once a discrete gradient has been found, one immediately obtains an integral-preserving method, simply letting y n+1 − y n = S(y n , y n+1 ) ∇(y n , y n+1 ). ∆t

24


Here ∆t is the time step, and S(y n , y n+1 ) is some approximation to the matrix S in (3.10); one would normally require that S(y) = S(y, y). Discrete gradient methods are usually implicit. The generalization of this approach to PDEs is immediate, and has been developed independently of the ODE literature by Furihata, Matsuo and co-authors (Furihata 1999, Furihata 2001a, Furihata 2001b, Furihata and Matsuo 2003, Matsuo 2007, Matsuo 2008, Matsuo and Furihata 2001, Matsuo, Sugihara, Furihata and Mori 2000, Matsuo, Sugihara, Furihata and Mori 2002, Yaguchi, Matsuo and Sugihara 2010), and recently presented in its full generality in Dahlby and Owren (2010). One may define a discrete variational derivative of the Hamiltonian H[u] to be a continuous (differential) function of two arguments δH u and v δ(v,u) satisfying the relations δH H[u] − H[v] = (u − v) dx, (3.12) δ(v, u) Ω δH δH = . δ(u, u) δu

(3.13)

To obtain a conservative scheme we simply replace the skew-adjoint operator D[u] in (3.2) by some approximation D[v, u] satisfying D[u, u] = D[u], and define the method as U n+1 − U n δH = D[U n , U n+1 ] . (3.14) n ∆t δ(U , U n+1 ) The AVF scheme can of course also be interpreted as a discrete variational derivative method, where 1 δH δHAVF = [ξu + (1 − ξ)v] dξ. (3.15) δ(v, u) 0 δu The fact that (3.15) verifies the condition (3.12) is seen from the elementary identity 1 d H[u] − H[v] = H[ξu + (1 − ξ)v] dξ. (3.16) 0 dξ The derivative under the integral is written d d H[v + (ξ + ε)(u − v)] H[ξu + (1 − ξ)v] = dξ dε ε=0 δH = [ξu + (1 − ξ)v](u − v) dx. Ω δu Now substitute this into (3.16) and interchange the integrals to obtain (3.12). We illustrate the discrete variational derivative method to the Korteweg– de Vries equaton (3.7)–(3.8) using (3.15). The variational derivative is

Topics in structure-preserving discretization δH δu

25

= −uxx − up−1 such that 1 δHAVF =− (ξu + (1 − ξ)v)xx + (ξu + (1 − ξ)v)p−1 dξ δ(v, u) 0 uxx + vxx 1 up − v p =− − . 2 p u−v

Thus, taking D = D = U n+1 − U n + ∆t

∂ ∂x , the conservative method for (3.7) reads n+1 + U n Uxxx 1 ∂ (U n+1 )p − (U n )p xxx = 0. + 2 p ∂x U n+1 − U n

3.1. Linearly implicit methods for polynomial Hamiltonians A major principle behind the conservative schemes is that they preserve the (discretized) first integral exactly. This means that for an implicit scheme, the nonlinear equation to be solved in each time step must be solved exactly or to machine accuracy, and this may sometimes cause the overall method to be expensive compared to other approaches. A possible remedy would be to look for an iteration method having the property that the first integral is preserved in every iteration; another possibility is to design schemes which are linearly implicit by construction, thereby requiring only one linear solve for every time step. In the second of these two approaches, discussed in Dahlby and Owren (2010), one may apply a polarization technique if the Hamiltonian is of polynomial type, meaning that the differential function G in (3.3) is a multivariate polynomial in the jet space coordinates u(n) . The idea is to substitute G[u] by a function of k ≥ 2 indeterminates G[w1 , . . . , wk ], satisfying G[u, . . . , u] = G[u] G[w1 , w2 , . . . , wk ] = G[w2 , . . . , wk , w1 ]

(consistency),

(3.17)

(cyclicity).

(3.18)

In particular, whenever G[u] = G((uαJ )) is a multivariate polynomial of degree p, one may construct G[w1 , . . . , wk ] satisfying (3.17)–(3.18), which is also multi-quadratic as long as k ≥ (p + 1)/2. We next replace the skewadjoint operator D[u] by some skew-adjoint approximation D[w1 , . . . , wk−1 ] of k − 1 arguments, satisfying D[u, . . . , u] = D[u] D[w1 , w2 , . . . , wk−1 ] = D[w2 , . . . , wk−1 , w1 ] Defining

(consistency), (cyclicity).

(3.19) (3.20)

H[w1 , . . . , wk ] =

Ω

G[w1 , . . . , wk ],

we may now define a polarized version of the discrete variational derivative in a similar way to (3.12)–(3.13), but it is now a function of k +1 arguments.

26


We refer to Dahlby and Owren (2010) for the general definition and give only the particular expression for the AVF case: 1 δH δH = [ξwk+1 + (1 − ξ)w1 , w2 , . . . , wk ] dξ. (3.21) δ(w1 , . . . , wk+1 ) 0 δw1 δH , is defined as Here the variational derivative on the right-hand side, δw 1 before, considering H as a function of its first argument only, leaving the others fixed. The following method was proposed in Dahlby and Owren (2010):

U n+k − U n δH = kD , k∆t δ(U n , . . . , U n+k )

n ≥ 0.

(3.22)

Remark 3.1. Note that the method (3.22) is a multistep method, and thus approximations U 1 , . . . , U k−1 to the first time steps are required. Remark 3.2. The procedure is linear, in the sense that if the Hamiltonian can be split into a sum of two terms H1 + H2 , the method can be applied to each term separately. We summarize some important properties of the scheme (3.22), the proof of which can be found in Dahlby and Owren (2010). Theorem 3.3. • The scheme (3.22) is conservative in the sense that H[U n , . . . , U n+k−1 ] = H[U 0 , . . . , U k−1 ],

∀n ≥ 1,

for any polarized Hamiltonian function H satisfying (3.17)–(3.18). • The polarized AVF scheme (3.22) using (3.21) has formal order of consistency two for any polarized Hamiltonian satisfying (3.17)–(3.18), and skew-symmetric operator D satisfying (3.19)–(3.20). • If G[w1 , . . . , wk ] is multi-quadratic in all its k arguments, (wJα )j , j = 1, . . . , k, then the polarized AVF method (3.22) is linearly implicit. To illustrate the linearly implicit scheme just presented, consider (3.7) with p = 6, the mass critical generalized Korteweg–de Vries equation. We then have 1 1 G[u] = u2x − u6 . 2 6 As indicated in Remark 3.2, we can treat the two terms separately. For the second term, one needs to use k ≥ 3: choosing k = 3 leads to the unique multi-quadratic choice G2 [w, v, u] = G2 (w, v, u) = − 16 u2 v 2 w2 . In the first term we have several options, but for reasons to be explained later, ¯ := D = ∂/∂x. we choose G1 [w, v, u] = 16 (u2x + vx2 + wx2 ). As before, we let D

27


This now leads to the scheme

n + U n+3 U n+3 − U n Uxxx ∂ U n + U n+3 n+1 2 n+2 2 xxx ) (U ) = 0. + + (U 3∆t 2 ∂x 2

Thus the scheme is linear in U n+3 , and preserves exactly the time-averaged Hamiltonian 1 n 2 n n+1 n+2 (Ux ) + (Uxn+1 )2 + (Uxn+2 )2 H[U , U ,U ]= 6 2 1 + U n U n+1 U n+2 dx. 6 We remark that some care should be taken when choosing the polarization. Since the resulting method is a multistep scheme, it has spurious solutions which might be unstable. Considering as a test case the term H[u] = u2x , letting D be a constant operator having the exponentials as eigenfunctions, such as in differentiation operators with constant coefficients, one may choose the one-parameter family of polarizations 2 ux + vx2 1 θ H[v, u] = + (1 − θ)ux vx dx. 2 2 Dahlby and Owren (2010) remark that this leads to a stable scheme only if θ ≥ 12 . 3.2. Preserving multiple first integrals The preservation of more than one first integral may often be of interest. For instance, in PDEs which have soliton solutions, a well-known technique for proving stability of the exact solution is based on using the preservation of two specific invariants: see the articles by Benjamin and co-authors (Benjamin 1972, Benjamin, Bona and Mahony 1972). A procedure for preserving multiple first integrals by a numerical integrator for ODEs was described in McLachlan et al. (1999). Their method was based on finding a presentation of the ODE vector field f (y) by means of a completely skewsymmetric rank q + 1 tensor field S, writing y˙ = f (y) = S(·, ∇H1 , . . . , ∇Hq ),

(3.23)

where H1 , . . . , Hq are q known first integrals. This form effectively encodes the preservation of all the q first integrals, since, for 1 ≤ i ≤ q, d Hi (y) = ∇Hi (y) · f (y) = S(y)(∇Hi (y), ∇H1 (y), . . . , ∇Hq (y)) = 0. dt Proof of the existence of the formulation (3.23), along with constructive examples, is provided in McLachlan et al. (1999). The discrete gradient method is now easily implemented. Given a discrete gradient, ∇Hi (y n , y n+1 )

28


for each of the first integrals, one may, for instance, define the method via y n+1 − y n ¯ n , y n+1 )( · , ∇H1 , . . . , ∇Hq ), = S(y ∆t for a suitable approximation S¯ to S. Another method for preserving multiple first integrals was presented in Minesaki and Nakamura (2006) for a subclass of Hamiltonian ODEs called Stäckel systems. In a recent paper by Dahlby, Owren and Yaguchi (2010), the problem of retaining several first integrals simultaneously has been considered. Again we explain this approach for ODEs, or in the case when our PDE has already been discretized in space to yield a system y˙ = f (y), y ∈ Rm , with first integrals H1 , . . . , Hq . Using the Euclidean structure on Rm one may consider the immersed submanifold Mc ⊂ Rm , a leaf of the foliation induced by the integrals M = Mc = {y ∈ Rm : H1 (y) = c1 , H2 (y) = c2 , . . . , Hq (y) = cq }. The tangent space to this manifold may be characterized as the orthogonal complement to span(∇H1 , . . . , ∇Hq ). In order to adapt this setting to the situation with discrete tangents, the discrete tangent space was defined relative to two points, sufficiently close to each other in M × M . For all y in a neighbourhood of a point p ∈ Mc , let T(p,y) M = {η ∈ Rm : ∇Hj (p, y), η = 0, 1 ≤ j ≤ q} for the Euclidean inner product ·, · on Rm . A vector η = η(p,y) ∈ T(p,y) M is called a discrete tangent vector. Note that this definition causes T(y,y) M = Ty M . The following observation is immediate. Any integrator satisfying y n+1 − y n ∈ T(yn+1 ,yn ) M

(3.24)

preserves all integrals, in the sense that Hi (y n+1 ) = Hi (y n ), 1 ≤ i ≤ q. The obvious approach to adopt in order to obtain a method satisfying (3.24) is to combine a non-conservative scheme with projection as follows. For simplicity, let φ∆t denote a one-step method of order p, φ∆t : Rm → Rm . For any y ∈ M , let y also denote its injection into Rm . Then consider un+1 = φ∆t (yn ),

yn+1 = yn + P(yn+1 , yn )(un+1 − yn ),

(3.25)

where P(yn , yn+1 ) is a smooth projection operator onto the discrete tangent space T(yn ,yn+1 ) M . An alternative scheme is yn+1 = yn + ∆tP(yn , yn+1 )ψ∆t (yn+1 , yn ), where ψ∆t is an integrator which can be written in the form yn+1 = yn + ∆t ψ∆t (yn+1 , yn ).

(3.26)


29

This method is itself assumed to be of order p, that is, y(t + ∆t) − y(t) − h ψ∆t (y(t + ∆t), y(t)) = O(∆tp+1 ).

(3.27)

The following result was proved in Dahlby et al. (2010). Theorem 3.4. If the non-conservative methods φ∆t , ψ∆t are of order p, then so are the schemes (3.25) and (3.26): y(t + h) − y(t) − P(y(t), y(t + ∆t))(un+1 − y(t)) = O(∆tp+1 ),

(3.28)

where un+1 = φ∆ t(y(t)), and y(t + h) − y(t) − hP(y(t + h), y(t))ψh (y(t + h), y(t)) = O(hp+1 ).

(3.29)

An alternative approach to projection is to derive methods based on a local coordinate representation of the manifold. Based on the assumption that the q first integrals are all independent, we can define a coordinate chart centred at a point p ∈ M as follows. Suppose y ∈ M is sufficiently close to p such that all discrete gradients ∇H1 (p, y), . . . , ∇Hq (p, y) form a linearly independent set of vectors in Rm . Let Tp (y) be a smooth map from the manifold M into the set of orthogonal m × m matrices T : M → O(m) such that its last q columns form a basis for the linear span of the discrete gradients. We now define the coordinate map implicitly to be η m−q χp : R → M : χp (η) = y : y − p = Tp (y) . (3.30) 0q That these maps induce an atlas on M is proved in Dahlby et al. (2010). One may now express the original differential equation in terms of η: η˙ = −T (χp ◦ η)DTp (f (χp ◦ η))η + T (χp ◦ η)f (χp ◦ η). (3.31) χp ◦η

The method proposed takes one step as follows. (1) Let η0 = 0. (2) Take a step with any pth-order method applied to the ODE (3.31), using p = y n in (3.30). The result is η1 . (3) Compute y n+1 = χ(η1 ). The method defined in this way will clearly be of order at least p. Using this approach, one clearly needs to compute DTp because it is needed explicitly in (3.31), and this map may also be useful in computing the implicitly defined coordinate map. It is shown in Dahlby et al. (2010) how DTp |y (v)η can be computed with a complexity of order O(mq 2 + q 3 ). 3.3. Discretizing in space The adaptation of the approaches just presented to PDEs which have already been discretized in space is of course straightforward: the integral in

30

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren Continuous calculus

Discrete calculus

First integral H[u] = G[u]

-

Discrete first integral Hd [u] = (Gd [u])k ∆x k

?

? Discrete variational δHd derivative δ(v, u)

Variational derivative δH δu

?

?

PDE ut = D[u]

-

δH δu

Finite difference scheme δHd u − un ¯ =D ∆t (δun , δun+1 ) n+1

Figure 3.1. The Furihata formalism. This illustration is taken from Furihata (1999).

(3.3) may be replaced by a quadrature rule, resulting in a discrete version, for instance, Hd : Rm → R. The skew-adjoint operator D must be replaced by a skew-symmetric m × m matrix, Dd . We now consider finite difference approximations. The function space to which the solution u belongs is replaced by a finite-dimensional space with functions on a grid indexed by Ig ⊂ Zd . We use boldface symbols for these functions. Let there be Nr grid points in the space direction r so that N = N1 · · · Nd is the total number of grid points. We denote by uα the approximation to uα on such a grid, and by u the vector consisting of (u1 , . . . , um ). We will replace each derivative uαJ by a finite difference approximation δJ uα , and replace the integral by a quadrature rule. We then let bi (Gd ((δJ u)))i ∆x. (3.32) Hd (u) = i∈Ig

Here ∆x is the volume (length, area) of a grid cell and b = (bi )i∈Ig are the


31

weights in the quadrature rule. The discretized Gd has the same number of arguments as G, and each input argument as well as the output are vectors in RN . We have here approximated the function uαJ by a difference approximation δJ uα , where δJ : RN → RN is a linear map. As in the continuous case, we use square brackets, say F [u], as shorthand for a list of arguments involving difference operators F [u] = F (u, δJ1 u, . . . , δJq u). We compute 1 ∂Gd bi [ξu + (1 − ξ)v]dξ(δJ (uα − vα )) ∆x Hd [u] − Hd [v] = ∂δJ uα i 0 J,α i∈Ig δHd = ,u − v , (3.33) δ(v, u) where

δHd = δJ B δ(v, u) J,α

1 0

∂Gd α α [ξu + (1 − ξ)v ] dξ , ∂uJ

B is the diagonal linear map B = diag(bi ), i ∈ Ig , and the discrete inner product used in (3.33) is u, v = uαi viα . α,i∈Ig

Notice the resemblance between the operator acting on Gd in (3.33) and the continuous Euler operator in (3.6). We make the following assumptions. (1) The spatially continuous method (3.14) (using (3.15)) is discretized in space, using a skew-symmetric Dd and a selected set of difference quotients δJ for each derivative ∂J . (2) Considering (3.5) and (3.6), the choice of discretization operators δJ used in ∂G/∂uαJ [u] is arbitrary, but the corresponding DJ is replaced by the transpose δJ . In this case, using the same Dd , an identical set of difference operators in discretizing H (3.32), and choosing all the quadrature weights bi = 1, the resulting scheme is the same. Letting er denote the rth canonical unit vector in Rd , we define the most common first-order difference operators ui+er − ui , (δr+ u)i = ∆xr ui − ui−er , (δr− u)i = ∆xr ui+er − ui−er . (δr1 u)i = 2∆xr

32


These difference operators are all commuting, but only the last one is skewsymmetric. However, for the first two we have the useful identities (δr+ ) = −δr− ,

(δr− ) = −δr+ .

Higher-order difference operators δJ can generally be defined by taking compositions of these operators: in particular, we shall consider examples in the next section using the second and third derivative approximations δr2 = δr+ ◦ δr− ,

δ 3 = δ 1 ◦ δ 2 .

We may now introduce numerical approximations Un representing the fully discretized system: the scheme is δHd Un+1 − Un = Dd . n ∆t δ(U , Un+1 ) The conservative schemes based on polarization are adapted in a straightforward manner, introducing a function Hd [w1 , . . . , wk ] which is consistent and cyclic, and a skew-symmetric map Dd depending on at most k − 1 arguments. The scheme is then Un+k − Un δHd = kDd . k∆t δ(Un , . . . , Un+k )

(3.34)

This scheme conserves the function Hd in the sense that Hd [Un+1 , . . . , Un+k ] = Hd [U0 , . . . , Uk−1 ],

n ≥ 0.

4. Spatial symmetries, high-order discretizations and fast group-theoretic algorithms In this chapter we survey some group-theoretic techniques applied to the discretization and solution of PDEs. Inspired by recent active research in Lie group and exponential time integrators for differential equations, we will in Section 4.1 present algorithms for computing matrix and operator exponentials based on Fourier transforms on finite groups. As a final example, we consider spherically symmetric PDEs, where the discretization preserves the 120 symmetries of the icosahedral group. This motivates the study of spectral element discretizations based on triangular subdivisions. In Section 4.2 we introduce novel applications of multivariate non-separable Chebyshev polynomials in the construction of spectral element bases on triangular and simplicial subdomains. These generalized Chebyshev polynomials are intimately connected to kaleidoscopes of mirrors acting on a vector space, i.e., groups of isometries generated by mirrors placed on each face of a triangle (2D), a tetrahedron (3D) or in general simplexes. Mathematically this is described by the theory of root systems and Weyl groups,


33

which we will review. Group-theoretic techniques lead to FFT-based fast computational algorithms. This is well known in the case of commutative groups, but it turns out to be important to understanding such ideas in the setting of non-commutative groups as well. Lie group integrators enjoy a number of nice geometrical properties, of which we will here focus on their symmetry and equivariance properties. Fundamental to the theory of differential equations is the equivariance of the solution curves with respect to any diffeomorphism φ : M → M acting on the domain. Let φ∗ F denote the pushforward of the vector field F , i.e., (φ∗ F )(z) = T φ · F (φ−1 (z)) for all z ∈ M, where T φ denotes the tangent map (in coordinates, the Jacobian matrix). Then the two differential equations y (t) = F (y(t)), y(0) = y0 and z (t) = (φ∗ F )(z(t)), z(0) = φ(y0 ) have analytical solution curves related by z(t) = φ(y(t)). In particular, if φ∗ F = F , we say that φ is a symmetry of the vector field, and in that case φ maps solution curves to other solution curves of the same equation. For numerical integrators it is in general impossible to satisfy equivariance with respect to arbitrary diffeomorphisms, since this would imply an analytically correct solution. (There always exists a local diffeomorphism that straightens the flow to a constant flow in the x1 direction, and this is integrated exactly by any reasonable numerical method.) The equivariance group of a numerical scheme is the largest group of diffeomorphisms under which the numerical solutions transform equivariantly. It is known that the equivariance group of classical Runge–Kutta methods is the group of all affine linear transformations of Rn . Lie group integrators based on exact computation of exponentials have equivariance groups that include the Lie group G on which the method is based; hence, if some elements g ∈ G are symmetries of the differential equation, then G-equivariant Lie group integrators will exactly preserve these symmetries. However, if the exact exponential is replaced with approximations, care must be taken not to destroy G-equivariance and symmetry preservation of the numerical scheme. In the case of PDEs, symmetry preservation also depends on symmetry-preserving spatial discretizations. Equivariance is the foundation of Fourier analysis. Classical Fourier analysis can be defined as the study of linear operators L which are equivariant with respect to an (abelian) group of translations acting on a domain, i.e., L◦τ = τ ◦L for all translations τ (t) : t → t + τ . The fact that exponential functions are the eigenfunctions of translation operators, exp(2πiλ(t + τ )) = exp(2πiλτ ) exp(2πiλt), and hence also eigenfunctions for L, is the explanation for the omnipresence of Fourier bases in computational science. The more specialized cos and sin bases appear naturally when boundary conditions are taken into account, as symmetrization and skewsymmetrization of the exponentials with respect to the reflection t → −t. Chebyshev polynomials again occur naturally through a change of variables

34


cos(t) → y. The availability of fast computational algorithms (FFT-based) for all these bases again relies in a crucial manner upon the underlying group structures provided by the equivariance. Products of equivariant operators lead naturally to the convolution product, which is diagonalized by Fourier transforms. Many important operators are equivariant with respect to larger noncommutative groups of transformations, such as the important Laplacian ∇2 which is equivariant with respect to every isometry of the domain.4 The generalized Fourier theory of linear operators equivariant with respect to non-commutative groups of transformations is not as widely known as the abelian theory within computational science. Convolution products also occur naturally in this setting. However, due to non-commutativity, exponential bases are no longer sufficient to diagonalize convolutions. Instead one must also use the higher-dimensional representations of the group, which can be understood as homomorphisms of the group into a space of (unitary) matrices. Exponentials are one-dimensional representations of abelian groups. In the natural bases obtained from representation theory, equivariant operators are block-diagonalized. This is a starting point for constructing fast algorithms for matrix exponentials, eigenvalue computations and linear equations in cases where the operator is equivariant. We start our presentation with a discussion of matrices equivariant with respect to a finite non-abelian group acting upon the index set. This is a good example for understanding some basic concepts of non-commutative harmonic analysis, without having to consider more technical analysis questions which occur in the case of infinite groups. 4.1. Introduction to discrete symmetries and equivariance The topic of this subsection is the application of Fourier analysis on groups to the computation of matrix exponentials. Assuming that the domain is discretized with a symmetry-respecting discretization, we will show that by a change of basis derived from the irreducible representations of the group, the operator is block-diagonalized. This simplifies the computation of matrix exponentials. The basic mathematics here is the representation theory of finite groups (James and Liebeck 2001, Lomont 1959, Serre 1977). Applications of this theory in scientific computing have been discussed by a number of authors: see, e.g., Allgower, Böhmer, Georg and Miranda (1992), Allgower, Georg, Miranda and Tausch (1998), Bossavit (1986), Douglas and Mandel (1992) and Georg and Miranda (1992). Our exposition, based on the group algebra, is explained in detail in ˚ Ahlander and Munthe-Kaas (2005). 4

The Laplacian can indeed be defined as the unique (up to constant) second-order linear differential operator which commutes with all isometries.


35

G-equivariant matrices. A group is a set G with a binary operation g, h → gh, inverse g → g −1 and identity element e, such that g(ht) = (gh)t, eg = ge = g and gg −1 = g −1 g = e for all g, h, t ∈ G. We let |G| denote the number of elements in the group. Let I denote the set of indexes used to enumerate the nodes in the discretization of a computational domain. We say that a group G acts on a set I (from the right) if there exists a product (i, g) → ig : I × G → I such that ie = i for all i ∈ I, i(gh) = (ig)h for all g, h ∈ G and i ∈ I.

(4.1) (4.2)

The map i → ig is a permutation of the set I, with the inverse permutation being i → ig −1 . An action partitions I into disjoint orbits Oi = {j ∈ I | j = ig for some g ∈ G},

i ∈ I.

We let S ⊂ I denote a selection of orbit representatives, i.e., one element from each orbit. The action is called transitive if I consists of just a single orbit, |S| = 1. For any i ∈ I we let the isotropy subgroup at i, Gi be defined as Gi = {g ∈ G | ig = i}. The action is free if Gi = {e} for every i ∈ I, i.e., there are no fixed points under the action of G. Definition 4.1.

A matrix A ∈ CI×I , is G-equivariant if

Ai,j = Aig,jg

for all i, j ∈ I and all g ∈ G.

(4.3)

The definition is motivated by the result that if L is a linear differential operator commuting with a group of domain symmetries G, and if we can find a set of discretization nodes I such that every g ∈ G acts on I as a permutation i → ig, then L can be discretized as a G-equivariant matrix A: see Allgower et al. (1998) and Bossavit (1986). The group algebra. We will establish that G-equivariant matrices are associated with (scalar or block) convolutional operators in the group algebra. Definition 4.2. The group algebra CG is the complex vector space CG where each g ∈ G corresponds to a basis vector g ∈ CG. A vector a ∈ CG can be written as a= a(g)g where a(g) ∈ C. g∈G

The convolution product ∗ : CG × CG → CG is induced from the product in G as follows. For basis vectors g, h, we set g ∗ h ≡ gh, and in general,

36


a(g)g and b = h∈G b(h)h, then a(g)g ∗ b(h)h = a(g)b(h)(gh) = (a ∗ b)(g)g, a∗b=

if a =

g∈G

g∈G

where

h∈G

(a ∗ b)(g) =

g∈G

g,h∈G

a(gh−1 )b(h) =

h∈G

a(h)b(h−1 g).

(4.4)

h∈G

Consider a G-equivariant A ∈ Cn×n in the case where G acts freely and transitively on I. In this case there is only one orbit of size |G| and hence I may be identified with G. Corresponding to A there is a unique A ∈ CG, given as A = g∈G A(g)g, where A is the first column of A, i.e., A(gh−1 ) = Agh−1 ,e = Ag,h .

Cn

(4.5)

Similarly, any vector x ∈ corresponds uniquely to x = g∈G x(g)g ∈ CG, where x(g) = xg for all g ∈ G. Consider the matrix vector product Ag,h xh = A(gh−1 )x(h) = (A ∗ x)(g). (Ax)g = h∈G

h∈G

If A and B are two equivariant matrices, then AB is the equivariant matrix, where the first column is given as (AB)g,e = Ag,h B h,e = A(gh−1 )B(h) = (A ∗ B)(g). h∈G

h∈G

We have shown that if G acts freely and transitively, then the algebra of G-equivariant matrices acting on Cn is isomorphic to the group algebra CG acting on itself by convolutions from the left. In the case where A is G-equivariant with respect to a free, but not transitive, action of G on I, we need a block version of the above theory. Let Cm× G ≡ Cm× ⊗CG denote the space of vectors consisting of |G| matrix blocks, each block of size m × . Thus A ∈ Cm× G can be written as A= A(g) ⊗ g where A(g) ∈ Cm× . (4.6) g∈G

The convolution product (4.4) generalizes to a block convolution ∗ : Cm× G × C×k G → Cm×k G given as A(g) ⊗ g ∗ B(h) ⊗ h = (A ∗ B)(g) ⊗ g, A∗B = g∈G

where

(A ∗ B)(g) =

A(gh−1 )B(h) =

h∈G

and

A(h)B(h−1 g)

g∈G

h∈G

denotes a matrix product.

h∈G

A(h)B(h−1 g),

(4.7)


37

If the action of G on I is free, but not transitive, then I splits into m orbits, each of size |G|. We let S denote a selection of one representative from each orbit. We will establish an isomorphism between the algebra of G-equivariant matrices acting on Cn and the block-convolution algebra Cm×m G acting on Cm G. We define the mappings µ : Cn → Cm G, ν : Cn×n → Cm×m G by µ(y)i (g) = yi (g) = y ig ∀i ∈ S, g ∈ G, ν(A)i,j (g) = Ai,j (g) = Aig,j ∀i, j ∈ S g ∈ G.

(4.8) (4.9)

In ˚ Ahlander and Munthe-Kaas (2005) we show the following. Proposition 4.3. Let G act freely on I. Then µ is invertible and ν is invertible on the subspace of G-equivariant matrices. Furthermore, if A, B ∈ Cn×n are G-equivariant, and y ∈ Cn , then µ(Ay) = ν(A) ∗ µ(y), ν(AB) = ν(A) ∗ ν(B).

(4.10) (4.11)

To complete the connection between G-equivariance and block convolutions, we need to address the general case where the action is not free, and hence some of the orbits in I have reduced size. One way to treat this case is to duplicate the nodes with non-trivial isotropy subgroups; thus a point j ∈ I is considered to be |Gj | identical points, and the action is extended to a free action on this extended space. Equivariant matrices on the original space is extended by duplicating the matrix entries, and scaled according to the size of the isotropy. We define µ(x)i (g) = xi (g) = xig ∀i ∈ S, g ∈ G, 1 Aig,j ∀i, j ∈ S g ∈ G. ν(A)i,j (g) = Ai,j (g) = |Gj |

(4.12) (4.13)

With these definitions it can be shown that (4.10)–(4.11) still hold. It should be noted that µ and ν are no longer invertible, and the extended block convolutional operator ν(A) becomes singular. This poses no problems for the computation of exponentials since this is a forward computation. Thus we just exponentiate the block convolutional operator and restrict the result back to the original space. However, for inverse computations such as solving linear systems, the characterization of the image of µ and ν as subspaces of Cm G and Cm×m G is an important issue for finding the correct solution (˚ Ahlander and Munthe-Kaas 2005, Allgower, Georg and Miranda 1993). The generalized Fourier transform (GFT). So far we have argued that a symmetric differential operator becomes a G-equivariant matrix under discretization, which again can be represented as a block convolutional operator. In this subsection we will show how convolutional operators are

38


block-diagonalized by a Fourier transform on G. This is the central part of Frobenius’s theory of group representations from 1897–1899. We recommend the monographs by F¨ assler and Stiefel (1992), James and Liebeck (2001), Lomont (1959) and Serre (1977) as introductions to representation theory with applications. Definition 4.4. Cd×d such that

A d-dimensional group representation is a map R : G → R(gh) = R(g)R(h)

for all g, h ∈ G.

(4.14)

Generalizing the definition of Fourier coefficients we define for any A ∈ ˆ ∈ Cm×k ⊗ Cm×k G and any d-dimensional representation R a matrix A(R) d×d C as ˆ A(R) = A(g) ⊗ R(g). (4.15) g∈G

Proposition 4.5. (Convolution theorem) For any A ∈ Cm×k G, B ∈ Ck× G and any representation R we have ˆ ˆ (A ∗ B)(R) = A(R) B(R).

(4.16)

Proof.

The statement follows from ˆ ˆ A(R)B(R) = A(g) ⊗ R(g) B(h) ⊗ R(h) g∈G

=

h∈G

A(g)B(h) ⊗ R(g)R(h) =

g,h∈G

=

A(g)B(h) ⊗ R(gh)

g,h∈G

A(gh−1 )B(h) ⊗ R(g) = (A ∗ B)(R).

g,h∈G

Let dR denote the dimension of the representation. For use in practical computations, it is important that A ∗ B can be recovered by knowing (A ∗ B)(R) for a suitable selection of representations, and furthermore that their dimensions dR are as small as possible. Note that if R is a represen˜ tation and X ∈ CdR ×dR is non-singular, then also R(g) = XR(g)X −1 is ˜ a representation. We say that R and R are equivalent representations. If ˜ ˜ there exists a similarity transform R(g) = XR(g)X −1 such that R(g) has a block diagonal structure, independent of g ∈ G, then R is called reducible, otherwise it is irreducible. Theorem 4.6. (Frobenius) For any finite group G there exists a complete list R of non-equivalent irreducible representations such that d2R = |G|. R∈R

39


Defining the GFT for a ∈ CG as a(g)R(g) a ˆ(R) =

for every R ∈ R,

(4.17)

g∈G

we may recover a by the inverse GFT (IGFT): 1 a(g) = dR trace (R(g −1 )ˆ a(R)). |G|

(4.18)

R∈R

For the block transform of A ∈ Cm×k G given in (4.15), the GFT and the IGFT are given componentwise as Ai,j (g)R(g) ∈ CdR ×dR , (4.19) Aî,j (R) = g∈G

Ai,j (g) =

1 dR trace (R(g −1 )Aî,j (R)). |G|

(4.20)

R∈R

Complete lists of irreducible representations for several common groups are to be found in Lomont (1959). Applications to the matrix exponential. We have seen that via the GFT, any G-equivariant matrix is block-diagonalized. Corresponding to an irreducible ˆ representation R, we obtain a matrix block A(R) of size mdR × mdR , where m is the number of orbits in I and dR is the size of the representation. Let Wdirect denote the computational work, in terms of floating-point operations, for computing the matrix exponential on the original data A, and let Wfspace be the cost of doing the same algorithm on the corresponding block diagonal ˆ Thus GFT-based data A. 3 3 3 d2R , Wfspace = cm3 d3R Wdirect = c(m|G|) = cm R∈R

and the ratio becomes O(n ) : 3

Wdirect /Wfspace =

R∈R

R∈R

d2R

3 / d3R . R∈R

Table 4.1 lists this factor for the symmetries of the triangle, the tetrahedron, the 3D cube and the maximally symmetric discretization of a 3D sphere (icosahedral symmetry with reflections). The cost of computing the GFT is not taken into account inthis estimate. There exist fast GFT algorithms of complexity O |G| log (|G|) for a number 2 | , the of groups, but even if we use a slow transform of complexity O |G total cost of the GFT becomes just O m2 |G|2 , which is much less than Wfspace .

40


Table 4.1. Gain in computational complexity for matrix exponential via GFT. Domain Triangle Tetrahedron Cube Icosahedron

G

|G|

{dR }R∈R

D3 S4 S4 × C 2 A5 × C 2

6 24 48 120

{1, 1, 2} {1, 1, 2, 3, 3} {1, 1, 1, 1, 2, 2, 3, 3, 3, 3} {1, 1, 3, 3, 3, 3, 4, 4, 5, 5}

Wdirect /Wfspace 21.6 216 864 3541

Example 4.7. (Equilateral triangle) The non-commutative group of smallest order is D3 , the symmetries of an equilateral triangle. There are six linear transformations that map the triangle onto itself: three pure rotations and three rotations combined with reflections. In Figure 4.1(a) we indicate the two generators α (rotation 120◦ clockwise) and β (right–left reflection). These satisfy the algebraic relations α3 = β 2 = e, βαβ = α−1 , where e denotes the identity transform. The whole group is D3 = {e, α, α2 , β, αβ, α2 β}. Given an elliptic operator L on the triangle such that L(u◦α) = L(u)◦α and L(u◦β) = L(u)◦β for any u satisfying the appropriate boundary conditions on the triangle, let the domain be discretized with a symmetryrespecting discretization: see Figure 4.1(b). In this example we consider a finite difference discretization represented by the nodes I = {1, 2, . . . , 10}, such that both α and β map nodes to nodes. In finite element discretizations one would use basis functions mapped to other basis functions by the symmetries. We define the action of D3 on I as (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)α = (5, 6, 1, 2, 3, 4, 9, 7, 8, 10), (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)β = (2, 1, 6, 5, 4, 3, 7, 9, 8, 10), and extend to all of D3 using (4.2). As orbit representatives, we may pick S = {1, 7, 10}. The action of the symmetry group is free on the orbit O1 = {1, 2, 3, 4, 5, 6}, while the points in the orbit O7 = {7, 8, 9} have isotropy subgroups of size 2, and finally O10 = {10} has isotropy of size 6. The operator L is discretized as a matrix A ∈ C10×10 satisfying the equivariances Aig,jg = Ai,j for g ∈ {α, β} and i, j ∈ S. Thus we have, e.g., A1,6 = A3,2 = A5,4 = A4,5 = A2,3 = A6,1 . D3 has three irreducible representations given in Table 4.2 (extended to the whole group using (4.14)). To compute exp(A), we find A = ν(A) ∈ C3×3 G from (4.13) and find Aˆ = GFT(A) from (4.19). The transformed maˆ 1 ) ∈ Cm×m and A(ρ ˆ 2 ) ∈ Cm×m ⊗C2×2 ˆ 0 ), A(ρ trix Aˆ has three blocks, A(ρ 2m×2m C , where m = 3 is the number of orbits. We exponentiate each of these blocks, and find the components of exp(A) using the inverse GFT (4.20).

41

Topics in structure-preserving discretization Table 4.2. A complete list of irreducible representations for D3 .

ρ0 ρ1 ρ2

α

β

1 1 √ −1/2 − 3/2 √ 3/2 −1/2

1 −1 1 0 0 −1 7

4

5

α 3

6

10

β 9

(a)

2

1

8

(b)

Figure 4.1. Equilateral triangle with a symmetry-preserving set of 10 nodes.

We should remark that in Lie group integrators, it is usually more important to compute y = exp(A) · x for some vector x. In this case, we ˆ i ))·ˆ x(ρi ), and recover y by inverse GFT. Note that compute yˆ(ρi ) = exp(A(ρ x ˆ(ρ2 ), yˆ(ρ2 ) ∈ Cm ⊗C2×2 C2m×2 . Example 4.8. (Icosahedral symmetry) As a second example illustrating the general theory, we solve the simple heat equation u t = ∇2 u on the surface of a unit sphere. See Trønnes (2005) for details of the implementation. The sphere is divided into 20 equilateral triangles, and each triangle is subdivided into a finite difference mesh respecting all the 120 symmetries of the full icosahedral symmetry group (including reflections). To understand this group, it is useful to realize that five tetrahedra can be simultaneously embedded in the icosahedron, so that the 20 triangles correspond to the (in total) 20 corners of these five tetrahedra. From this one sees that the

42


icosahedral rotation group is isomorphic to A5 , the group of all 60 even permutations of the five tetrahedra. The 3D reflection matrix −I obviously commutes with any 3D rotation, and hence we realize that the full icosahedral group is isomorphic to the direct product C2 × A5 , where C2 = {1, −1}. The irreducible representations of A5 have dimensions {1, 3, 3, 4, 5}, and the representations of the full icosahedral group are found by taking tensor products of these with the two one-dimensional representations of C2 . The fact that the full icosahedral group is a direct product is also utilized in faster computation of the GFT. This, however, is not of major importance, since the cost of the GFT in any case is much less than the cost of the matrix exponential. 4.2. Spatial symmetries and high-order discretizations In this subsection we will discuss novel techniques for spatial discretizations based on reflection groups and multivariate Chebyshev polynomials. Given high-order time integration methods, such as Lie group methods, it is desirable to also use high-order discretizations in space. We will see that it is important to employ discretizations respecting spatial symmetries, both for the quality of the discretization error and also for efficiency of linear algebra computations such as matrix exponentials, eigenvalue computations and solution of linear systems. Spectral methods are important discretization techniques, where special function systems are used to expand the solution on simple (e.g., boxshaped) domains. For PDEs with analytic solutions spectral discretization techniques enjoy exponentially fast convergence (Canuto, Hussaini, Quarteroni and Zang 2006). On more general domains spectral element methods are constructed by patching together simple subdomains. Typically the basis functions on the subdomains arise as eigenfunctions for a self-adjoint linear operator, such as the exponential Fourier basis on periodic domains and orthogonal polynomials (Legendre or Chebyshev) on bounded domains with homogeneous boundary conditions. Most of the popular orthogonal polynomial systems used in current spectral (and spectral-element) methods are special cases of Jacobi polynomials, which are the solutions of singular Sturm–Liouville problems on a bounded domain, i.e., they are eigenfunctions of a second-order linear differential operator which is self-adjoint with respect to a weighted inner product, where the weight function is singular on the boundary. A problem with standard polynomial-based spectral element methods is a lack of flexibility with respect to geometries. On box-shaped domains it is easy to construct polynomial bases by tensor products of univariate polynomials. Box-shaped domains are, however, difficult to patch together in more general geometries, and difficult to match with domain symmetries,


43

e.g., for problems on spherical surfaces. For these reasons, it is desirable to explore high-order function systems based on triangles (2D), tetrahedra (3D) and in general simplexes for higher dimensions. 4.2.1. Root systems and Laplacian eigenfunctions on triangles and simplexes For certain triangular or simplicial domains with homogeneous Dirichlet or Neumann boundary conditions, the eigenfunctions of the Laplacian are known explicitly. We will see later that these systems are closely related to reflection groups (crystallographic groups and Coxeter groups) and root systems. In order to motivate a discussion of reflection groups, we start with an important particular example, the eigenfunctions of the Laplacian on an equilateral triangle . What is the sound of an equilateral drum? A detailed discussion of this example is also found in Huybrechs, Iserles and Nørsett (2010). Without loss of generality, the triangle has corners in the origin [0; 0], √ √ we assume that √ √ λ1 = [1/ 2; 1/ 6] and λ2 = [0; 2/ 3], shown as the shaded domain in Figure 4.4(a), labelled A2 (see p. 50). Let {sj }3j=1 denote reflections of R2 ˜ denote the full group of isometries of about the edges of the triangle. Let W 2 3 R generated by {sj }j=1 . This is an example of a crystallographic group,5 a group of isometries of Rd where the subgroup of translations form a lattice in Rd . From this fact we will derive the Laplacian eigenfunctions on . Recall that a lattice L in Rd is the Z-linear span of d linearly independent vectors in Rd , L = spanZ {α1 , . . . , αd }; thus L is an abelian group isomorphic to Zd . The reciprocal lattice L∗ = {λ ∈ Rd : (λ, α) ∈ Z for all λ ∈ Λ and α ∈ L}, where (·, ·) is the standard inner product on Rd . We have L∗ = spanZ {λ1 , . . . , λd }, where {λ1 , . . . , λd } is the dual basis of {α1 , . . . , αd }, i.e., (λj , αk ) = δj,k . The reciprocal lattice serves as the index set for the Fourier basis for L-periodic functions on Rd . Consider the L-periodic domain (a d-torus) Td = Rd /L and the space of square-integrable periodic functions L2 (Td ). The Fourier basis for L2 (Td ) is the L2 -orthogonal family of exponential functions indexed by the dual lattice, {exp(2πi(λ, y))}λ∈L∗ . ˜ is the Back to our equilateral triangle, where the translation lattice of W lattice L = spanZ {α1 , α2 } < R2 √ √ √ √ generated by the vectors α1 = ( 2, 0) and α2 = (−1 2, 3/ 2). The unit 5

More specifically it is an affine Weyl group, to be defined below.

44


cell of L can be taken as either the rhombus spanned by α1 and α2 or the hexagon indicated in Figure 4.4. The hexagon is the Voronoi cell of the origin in the lattice L, i.e., its interior consists of the points in R2 that are closer to the origin than to any other lattice points. As a first step in our construction, we consider the L-periodic eigenfunctions of the Laplacian ∇2 . Since (λj , αk ) = δj,k , the dual lattice is L∗ = spanZ {λ1 , λ2 } and the periodic eigenfunctions are ∇2t e2πi(λ,t) = −(2π)2 λ2 e2πi(λ,t)

for all λ ∈ L∗ , t ∈ R2 /L.

We continue to find the Laplacian eigenfunctions on by folding the expo˜ be the linear subgroup which leaves the origin fixed nentials. Let W < W (the symmetries of ): W = {e, s1 , s2 , s1 s2 , s2 s1 , s1 s2 s1 }, where e is the identity and si , i ∈ {1, 2} act on v ∈ R2 as si v = v − 2(αi , v)/(αi , αi ). We define even and odd (cosine- and sine-type) foldings of the exponentials 1 2πi(λ,wt) 1 2πi(w λ,t) e = e , (4.21) cλ (t) = |W | |W | w∈W

sλ (t) =

1 |W |

w∈W

w∈W

det(w)e2πi(λ,wt) =

1 det(w)e2πi(w λ,t) , |W |

(4.22)

w∈W

where, in our example, |W | = 6. The Laplacian commutes with any isometry, in particular ∇2 (f ◦ w) = (∇2 f ) ◦ w for w ∈ W . Hence, reflected exponentials and the functions cλ (t) and sλ (t) are also eigenfunctions with the same eigenvalue. Lemma 4.9. The eigenfunctions of ∇2 on the equilateral triangle with Dirichlet (f = 0) and Neumann boundary conditions (∇f · n = 0) are given by ∇2t cλ (t) = −(2π)2 λ2 cλ (t), ∇2t sλ (t) = −(2π)2 λ2 sλ (t), respectively, for λ ∈ spanN {λ1 , λ2 }, the set of all non-negative integer combinations of λ1 = [1/2; 1/6] and λ2 = [0; 2/3]. The set spanN {λ1 , λ2 } ⊂ L∗ contains exactly one point from each W -orbit: see the discussion of Weyl chambers below. An important question is how good (or bad!) these eigenfunctions are as bases for approximating analytic functions on . It is well known from the univariate case that a similar construction, yielding the eigenfunctions cos(kθ) and sin(kθ) on [0, π], does not give fast convergence. For an analytic function f (θ) where f (0) = f (π) = 0, the even 2π-periodic extension is piecewise smooth, with only C 0


45

continuity at θ ∈ {0, π}. Hence the Fourier cosine series converges only as O(k −2 ). There are several different ways to achieve spectral convergence O(exp(−ck)). One possible solution is to approximate f (θ) in a frame (not linearly independent) consisting of both {cos(kθ)}k∈Z and {sin(kθ)}k∈Z+ . Another possibility is to employ a change of variables x = cos(θ), yielding (univariate) Chebyshev polynomials of the first and second kind: Tk (x) = cos(kθ), Uk (x) = sin((k + 1)θ)/ sin(θ). These polynomials (in particular the first kind) are ubiquitous in approximation theory and famous for their excellent approximation properties. We will develop the corresponding multivariate theory and see that we have similar possibilities for constructing spectrally converging frames and bases. Root systems and affine Weyl groups. Let V be a finite-dimensional real Euclidean vector space with standard inner product (·, ·). The construction above can be generalized to all those simplexes ⊂ V with the property ˜ generated by reflecting about its faces is a that the group of isometries W crystallography group. All such simplexes are determined by a root system, a set of vectors in V which are perpendicular to the reflection planes of W passing through the origin. We review some basic definitions and results about root systems. For more details we refer to Bump (2004). Definition 4.10. A root system in V is a finite set Φ of non-zero vectors, called roots, that satisfy the following conditions. (1) The roots span V. (2) The only scalar multiples of a root α ∈ Φ that belong to Φ are ±α. (3) For every root α ∈ Φ, the set Φ is invariant under reflection in the hyperplane perpendicular to α, i.e., for any two roots α and β, the set Φ contains the reflection of β, sα (β) := β − 2

(α, β) α ∈ Φ. (α, α)

(4) (Crystallographic restriction.) For any α, β ∈ Φ, we have 2

(α, β) ∈ Z. (α, α)

Condition (4) implies that the obtuse angle between two different reflection planes must be either 90◦ , 120◦ , 135◦ or 150◦ . This is a fundamental fact in crystallography, implying that rotational symmetries of a crystal must be either 2-fold, 3-fold, 4-fold or 6-fold. The rank of the root system is the dimension d of the space V. Any root system contains a subset (not uniquely defined) Σ ⊂ Φ of so-called simple

46


An g

g

g

g

g

g> g

Cn g

g

g

g< g g

Dn g

g

g @g

E8 g

g

G2 g< g

Bn

g

g

g E6 g

g

g

g

g

g E7 g

g

g

g

g

g

g

g

g

g

g> g

g

g

F4

g

g

g

Figure 4.2. Dynkin diagrams for irreducible root systems.

positive roots. This is a set of d linearly independent roots Σ = {α1 , . . . , αd } such that any root β ∈ Φ can be written either as a linear combination of αj with non-negative integer coefficients, or as a linear combination with nonpositive integer coefficients. We call Σ a basis of the root system Φ. A root system is conveniently represented by its Dynkin diagram. This is a graph with d nodes corresponding to the simple positive roots. Between two nodes j and k, no line is drawn if the angle between αj and αk is 90◦ , a single line is drawn if it is 120◦ , a double line for 135◦ and a triple line for 150◦ . It is only necessary to understand the geometry of irreducible root systems when the Dynkin diagram is connected. Disconnected Dynkin diagrams (reducible root systems) are trivially understood in terms of products of irreducible root systems. For irreducible root systems, the roots are either all of the same length, or have just two different lengths. In the latter case a marker < or > on an edge indicates the separation of long and short roots (short < long). Since the work of W. Killing and E. Cartan in the late nineteenth century it has been known that Dynkin diagrams of irreducible root systems must belong to one of four possible infinite cases, An (n > 0), Bn (n > 1), Cn (n > 2), Dn (n > 3), or five special cases, E6 , E7 , E8 , F4 , G2 , shown in Figure 4.2. We say that two root systems are equivalent if they differ only by a scaling or an isometry. Up to equivalence, there corresponds a unique root system to each Dynkin diagram. We will see that the root system and also the Weyl group can be explicitly computed (as real vectors and matrices) using from the Cartan matrix defined below. A root system Φ is associated with a dual root system Φ∨ defined such that a root α ∈ Φ corresponds to a co-root α∨ := 2α/(α, α) ∈ Φ∨ . If all roots have equal lengths then Φ∨ = Φ (up to equivalence), i.e., the root system is self-dual. For the cases with two root lengths we have Bn∨ = Cn , F4∨ = F4 and G∨ 2 = G2 . The Cartan matrix is defined as Cj,k = (αj , αk∨ ) = 2(αj , αk )/(αk , αk ). Its diagonal is Cj,j = 2. Off-diagonal entries Cj,k are found from the number of edges between αj and αk and their

47


relative lengths:  0 if no edge,     − 1 if single edge, or multiple edge where α < α , j k Cj,k =  − 2 if double edge and αj > αk ,    − 3 if triple edge and αj > αk . The ratio of long and short roots is as follows. If αj > αk then αj = √ > = is a double edge and αj = √2αk in the cases where the division = 3αk when it is triple ≡ > ≡. It is convenient such √ to choose normalization ∨ that the longest roots have length αj = 2. This leads to αj = αj for the long roots. For short roots this implies αk∨ = 2αk in the double-line cases = > = and αk∨ = 3αk in the triple-edge case ≡ > ≡. Let D be the diagonal matrix Dk,k = αk . Then S = D−1 CD/2 is a symmetric matrix, and we compute bases for the root system Φ and the dual root system Φ∨ as the columns of the matrices6 Σ and Σ∨ given as Σ = RD, ∨

Σ = 2RD

(4.23) −1

,

(4.24)

where R is the Cholesky factorization of S, i.e., R is upper-triangular, with positive diagonal elements, such that R R = D−1 CD/2. A root system leads to the definition of Weyl groups and affine Weyl groups. Let Φ be a d-dimensional root system with dual root system Φ∨ . For any root α ∈ Φ consider the reflection sα : V → V given by sα (t) = t −

2(t, α) α = t − (t, α∨ )α. (α, α)

For the dual roots α∨ ∈ Φ∨ we define translations τα∨ : V → V by τα∨ (t) = t + α∨ . The Weyl group of Φ is the finite group of isometries on V generated by the reflections sα for α ∈ Σ: W = {sα }α∈Σ . The dual root lattice L∨ is the lattice spanned by the translations τα∨ for α∨ ∈ Σ∨ . We identify this with the abelian group of translations on V generated by the dual roots L∨ = {τα∨ }α∨ ∈Σ∨ . ˜ is the infinite crystallographic symmetry group The affine Weyl group W of V generated by the reflections sα for α ∈ Σ and the translations τα∨ for 6

By abuse of notation we use Σ to denote both the basis for the root system and the matrix whose columns form the basis of the root system.

48


α∨ ∈ Σ∨ , thus it is the semidirect product of the Weyl group W with the dual7 root lattice L∨ ˜ = {sα }α∈Σ , {τα∨ }α∨ ∈Σ∨ = W L∨ . W Let Λ denote the reciprocal lattice of L∨ , i.e., for all λ ∈ Λ and all α∨ ∈ Φ∨ we have (λ, α∨ ) ∈ Z. The lattice Λ is spanned by vectors {λj }dj=1 such that (λj , αk∨ ) = δj,k for all αk∨ ∈ Σ∨ . The vectors λj are called the fundamental dominant weights of the root system Φ and Λ is called the weights lattice. We use the weights λj and dual roots αk∨ to represent the Weyl group in matrix form as follows: wj,k := (λj , wαk∨ )

for all w ∈ W .

(4.25)

In particular, the reflections sαr for αr ∈ Σ are represented as (sαr )j,k = δj,k − Cr,k δr,j ⇒ sαr = I − er e r C, where C is the Cartan matrix and {er } is the standard basis on Rd . This shows that the Weyl group can be represented as integer matrices with respect to the basis Σ∨ and the dual basis {λ1 , . . . , λd } for V. The positive Weyl chamber C+ is defined as the closed conic subset of V containing the points with non-negative coordinates with respect to the dual basis {λ1 , . . . , λd }, in other words C+ = {t ∈ V : (t, αj ) ≥ 0}. This is a fundamental domain for the Weyl group acting on V. The boundary of C+ consists of the hyperplanes perpendicular to {α1 , . . . , αd }. The affine Weyl group contains reflection symmetries about affine planes perpendicular to the roots, shifted a half-integer multiple of the length of a co-root away from the origin, i.e., for each α∨ ∈ Φ∨ and each k ∈ Z there is an affine plane consisting of the points Pk,α∨ = {t ∈ V : 2(t, α∨ ) = k(α∨ , α∨ )} = {t ∈ V : (t, α) = k}, and this affine plane is invariant under the affine reflection τkα∨ · sα . A connected closed subset of V limited by such affine planes is ˜. called an alcove, and is a fundamental domain for the affine Weyl group W The situation is particularly simple for irreducible root systems, where the alcoves are always d-simplexes. Recall that any root α ∈ Φ can be expressed as α = dk=1 nk αk , where the nk = 2(α, λk )/(αk , αk ) are either all non-negative or all non-positive integers. A root α ˜ strictly dominates another root α, written α ˜ α, if n ˜ k ≥ nk for all k, with strict inequality for at least one k. Irreducible root systems have a unique dominant root 7

Since the Weyl groups of Φ and of Φ∨ are identical, it is no problem to instead define ˜ = W L as the semidirect product of the Weyl group with the primal root lattice. W We have, however, chosen here to follow the most common definition, which leads to a slightly simpler notation for the Fourier analysis.


49

A × A root system 1 1 α 2

λ

2

λ1

α1

Figure 4.3. Fundamental domains of reducible root system A1 × A1 .

α ˜ ∈ Φ such that α ˜ α for all α = α ˜ . The dominant root α ˜ is the unique long root in the Weyl chamber (possibly on the boundary). Basic geometric properties of affine Weyl groups are summarized by the following lemma. Lemma 4.11. (i) If Φ is irreducible with dominant root α ˜ then a fundamental domain ˜ is the simplex ⊂ V given by for W = {t ∈ V : (t, α) ˜ ≤ 1 and (t, αj ) ≥ 0 for all αj ∈ Σ}, where has corners in the origin and in the points λj /(λj , α ˜ ) for j = 1, . . . , d. (ii) The affine Weyl group is generated by the affine reflections8 about the boundary faces of the fundamental domain . For irreducible Φ these are ˜ = {sα }dj=1 , τα˜ · sα˜ . W j (iii) If Φ is reducible then a fundamental domain for the affine Weyl group is given as the Cartesian product of the fundamental domains for each of its irreducible components. Example 4.12. The simplest rank d root system is the reducible system A1 × · · · × A1 , where the Dynkin diagram consists of d non-connected dots. Figure 4.3 shows A1 × A1 . The square outlined in black is the fundamental domain of the root lattice, and the small shaded square is the fundamental ˜ . Reducible root systems can be easily domain of the affine Weyl group W understood as products of irreducible root systems. For instance, the multivariate Chebyshev polynomials corresponding to A1 ×· · ·×A1 are the tensor products of d univariate (classical) Chebyshev polynomials on a d-cube. 8

˜ is generated by reflections it is also a special case of a Coxeter group. Since W

50

S. H. Christiansen, H. Z. Munthe-Kaas and B. Owren α2 A2 root system α

2

λ

λ

2

1

2

λ2

α2

λ1

λ1 α1

α

α1

(a)

G root system

B2 root system

λ2

1

(b)

(c)

Figure 4.4. The 2D irreducible root systems.

Figure 4.5. Fundamental domains of A3 root system.

Example 4.13. The 2D and 3D irreducible root systems are A2 , B2 , G2 , A3 , B3 and C3 . The Cartan matrix, the matrix representation of generators for the Weyl group, and the size of the Weyl group are shown in Table 4.3. Figure 4.4 shows the 2D cases with roots α (large dots), dominant root α ˜ (larger dot), simple positive roots (α1 , α2 ), and their fundamental dominant weights√(λ1 , λ2 ). The roots are normalized such that the longest roots have length 2, thus for long roots α∨ = α. For short roots we have for B2 that α∨ = 2α2 and for G2 that α∨ = 3α. The fundamental domain of the dual root lattice (Voronoi region of L∨ ) is indicated by , and , and the fundamental domain for the affine Weyl group is indicated by triangles. Figure 4.5 shows the A3 case (self-dual), where the fundamental domain (Voronoi region) of the root lattice is a rhombic dodecahedron, a convex polyhedron with 12 rhombic faces. Each of these faces (composed of two triangles) is part of a plane perpendicular to one of the 12 roots, half-way out to the root (roots are not drawn). The fundamental domain of the affine Weyl group is the tetrahedron with an inscribed octahedron. The

51

Topics in structure-preserving discretization Table 4.3. The Cartan matrix, the matrix representation of generators for the Weyl group, and the size of the Weyl group. C A2 B2 G2  A3

B3

C3

2 −1 −1 2 2 −2 −1 2 2 −1 −3 2

2 −2 0  2 −1 0  2 −1 0

s1

 −1 0 2 −1 −1 2  −1 0 2 −2 −1 2  −1 0 2 −1 −2 2



−1 1 0 1 −1 2 0 1 −1 1 0 1

−1 0 0  −1 0 0  −1 0 0

s2

 1 0 1 0 0 1  1 0 1 0 0 1  1 0 1 0 0 1

1 1 1 1 1 3

|W |

s3

0 −1

6

0 −1

8

0 −1

12

   1 0 0 −1 0 1 −1 1  0 1 0 0 1 0 1    1 0 0 −1 0 1 −1 2  0 1 0 0 1 0 1    1 0 0 −1 0 1 −1 1  0 1 0 0 1 0 2

 0 0 −1  0 0 −1  0 0 −1

24

48

48

corners of this tetrahedron constitute the fundamental dominant weights λ1 , λ2 , λ3 . The regular octahedron drawn inside the Weyl chamber is important for application of multivariate Chebyshev polynomials, discussed in Section 4.2.2. Laplacian eigenfunctions on triangles, tetrahedra and simplexes. In this subsection we will consider real- or complex-valued functions on V-respecting symmetries of an affine Weyl group. Consider first L2 (T), the space of complex-valued L2 -integrable periodic functions on the torus T = V/L∨ i.e., functions f such that f (y + α∨ ) = f (y) for all y ∈ V and α∨ ∈ L∨ . Since the weights lattice Λ is reciprocal to L∨ , the Fourier basis for periodic functions is given as {exp(2πi(λ, t))}λ∈Λ and Fourier transforms are defined for f ∈ L2 (T): 1 f (λ) = F(f )(λ) = f (t)e−2πi(λ,t) dt, (4.26) vol(T) T f (t) = F −1 (f)(t) = (4.27) f(λ)e2πi(λ,t) . λ∈Λ

We are interested in functions which are periodic under translations in L∨ ˜ , e.g., functions and which in addition respect the other symmetries in W

52


˜ (we will with odd or even symmetry with respect to the reflections in W see below that there are also other possibilities). Due to the semi-direct ˜ = W L∨ it follows that any w ˜ can be written product structure W ˜∈W ∨ ∨ ∨ w ˜ = w · τα , where w ∈ W and α ∈ L . Thus, on the space of periodic ˜ and the finite group W are identical. We functions L2 (T) the action of W define subspaces of symmetric and skew-symmetric periodic functions as follows: L2∨ (T) = {f ∈ L2 (T) : f (wt) = f (t) for all w ∈ W , t ∈ T }, L2∧ (T) = {f ∈ L2 (T) : f (wt) = (−1)|w| f (t) for all w ∈ W , t ∈ T }, where |w| denotes the length, defined as |w| = , where w = sαj1 · · · sαj is written in the shortest possible way as a product of reflections about the simple positive roots αj ∈ Σ. Thus (−1)|w| = det(w) = ±1 depending on whether w is a product of an even or odd number of reflections. We define L2 -orthogonal projections π∨W and π∧W on these subspaces as 1 π∨W f (t) = f (wt), (4.28) |W | w∈W 1 W π∧ f (t) = (−1)|w| f (wt). (4.29) |W | w∈W

Orthogonal bases for these subspaces are obtained by projecting the exponentials, yielding the cosine- and sine-type basis functions cλ (t) := π∨W exp 2πi(λ, t), sλ (t) := π∧W exp 2πi(λ, t) as in (4.21)–(4.22). Note that these are not all distinct functions. Since they possess symmetries cλ (t) = cwλ (t), sλ (t) = (−1)|w| swλ (t), for every w ∈ W , we need only one λ from each orbit of W . The weights in the Weyl chamber, Λ+ = C+ ∩ Λ, form a natural index set of orbit representatives, and we find L2 -orthogonal bases by taking the corresponding cλ (t) and sλ (t). Lemma 4.9 also holds in the following more general case. Lemma 4.14. Let φ be a rank d root system with weights lattice Λ. Let W be the Weyl group and let denote the fundamental domain of the affine ˜ = W L∨ . The functions {cλ (t)}λ∈Λ and {sλ (t)}λ∈Λ Weyl group W + + form two distinct L2 -orthogonal bases for L2 (). These basis functions are eigenfunctions of the Laplacian ∇2 on satisfying homogeneous Neumann and Dirichlet boundary conditions, as in Lemma 4.9.


53

Unfortunately, truncations of these bases do not form excellent approximation spaces for analytic functions on . Approximation of a function f , defined on , in terms of {cλ (t)} is equivalent to Fourier approximation of the even extension of f in L2∨ (T), and in general we only observe quadratic convergence due to discontinuity of the gradient across the boundary of . A route to spectral convergence is by a change of variables, which turns the trigonometric polynomials {cλ (t)} and {sλ (t)} into multivariate Chebyshev polynomials of the first and second kind. Before we dive into this topic, we remark on an alternative approach. Trigonometric polynomials with generalized symmetries. In 1D it is known that spectral (exponentially converging) approximations for analytic f defined on can be obtained by employing the frame (i.e., a spanning set of vectors which is not linearly independent) generated by taking both the cosine and sine functions. In the frame we compute an approximating function by solving the following periodic extension problem. Find a smooth extension of f to L2 (T) and approximate this by the frame on . A detailed discussion of such techniques is found in Huybrechs (2010). One question is whether a frame consisting of only cλ (t) and sλ (t) might also be sufficient to obtain spectral convergence for higher-dimensional domains, or whether we have to include basis functions with other kinds of symmetries. Consider L2 (T, W), the space of functions periodic on T returning values in a Hilbert space W with inner product (·, ·)W . The inner product on L2 (T, W) is (f, g) = (f (t), g(t))W dt. T

Recall that a representation of the group W on W is a map ρ : W → Gl(W), defining a linear action of W on W, i.e., for w ∈ W , ρ(w) is an invertible linear map on W such that ρ(ww) ˜ = ρ(w)ρ(w). ˜ The integer dρ = dim(W) is called the dimension of the representation ρ. We will (without loss of generality) always assume representations are unitary with respect to (·, ·)W . We say that f ∈ L2 (T, W) is ρ-symmetric if f (wt) = ρ(w)f (t)

for all w ∈ W , t ∈ T.

The space of such symmetric functions is denoted L2ρ (T, W). As an example, the functions in L2∨ (T) and L2∧ (T) are ρ-symmetric with respect to the trivial representation ρs (w) = 1 and the alternating representation ρa (w) = (−1)|w| acting on W = R. Since the Weyl group W is constructed as a group of linear transformations on V, we always also have the faithful identity representation ρI (w) = w, acting on W = V. Seen through a kaleidoscope,

54


a vector field on appears to be ρI -symmetric on T, where the vectors are reflected in the mirrors. We define a projection πρW : L2 (T, W) → L2ρ (T, W) as πρW f (t) =

1 ρ(w−1 )f (wt). |W |

(4.30)

w∈W

Since ρ is unitary, it can be shown that πρW is an L2 -orthogonal projection. Note that scalar functions f : T → C can be symmetrized with respect to representations on any vector space W in a similar manner, i.e., 1 πρW f (t) = ρ(w−1 )f (wt), |W | w∈W

which yields a function πρW f (t) returning values in the space of all matrices over W. Using Theorem 4.6, we can recover f on T from its symmetrizations on . Proposition 4.15. Let f be a periodic scalar function f ∈ L2 (T). Given the symmetrizations fρ = πρW f ∈ L2ρ (T, W) for all ρ ∈ R, where R is a complete list of irreducible representations of W , we recover f on all of T as f (wt) = dρ trace (ρ(w)fρ (t)), for all t ∈ and w ∈ W . ρ∈R

It should be remarked that for the root system An the Weyl group is W Sn+1 , the symmetric group consisting of all permutations of n + 1 items. The representation theory of Sn+1 is well known, and there also exist fast algorithms for computing the corresponding transforms. Similarly to the symmetrization of f we can symmetrize the Fourier basis with respect to ρ and obtain matrix-valued symmetrized exponentials defined as the dρ × dρ matrices expρ,λ (t) := πρW e2πi(λ,t) ,

(4.31)

where dρ = dim(W). Restricting to λ ∈ Λ+ and ρ ∈ R, where R is a complete list of irreducible representations of W , the components of {expρ,λ }λ∈Λ+ ,ρ∈R form an L2 -orthogonal basis for L2 (T). Restricted to they form a tight frame with frame bound |W |: see Huybrechs (2010) for a treatment of the classical case A1 . These generalized symmetric exponential functions are (almost) equivalent to the irreducible representations of the ˜ obtained by the technique of small groups (Serre 1977). affine Weyl group W In Section 4.2.2 we will see that the odd and even symmetrized exponentials cλ (t) and sλ (t) transform to multivariate Chebyshev polynomials of the first and second kind under a particular change of variables. Also, the


55

more general ρ-symmetrized exponentials expρ,λ (t) are related to multivariate Chebyshev polynomials of ‘other kinds’. Approximation theory on the simplex using the frame spanned by {expρ,λ }, and the theory of multivariate Chebyshev polynomials of other kinds, is currently under investigation.9 It is known in the A2 case that the frame consisting only of cλ (t) and sλ (t) is not sufficient to guarantee spectral convergence of arbitrary analytic functions on . By including the last four basis functions arising from the irreducible two-dimensional representation ρI (w) = w we do obtain spectral convergence. However, practical use of these techniques in discretizations of PDEs has not yet been completed. 4.2.2. Multivariate Chebyshev polynomials Bivariate Chebyshev polynomials were constructed independently by Koornwinder (1974) and Lidl (1975) by folding exponential functions. Multidimensional generalizations (the A2 family) appeared first in Eier and Lidl (1982). Hoffman and Withers (1988) presented a general folding construction. Characterization of such polynomials as eigenfunctions of differential operators is found in Beerends (1991) and Koornwinder (1974). Applications to the solution of differential equations are found in Munthe-Kaas (2006) and to triangle-based spectral element Clenshaw–Curtis-type quadratures in Ryland and Munthe-Kaas (2011). Recall that classical Chebyshev polynomials of the first and second kind, Tk (x) and Uk (x), are obtained from cos(kθ) and sin(kθ) by a change of variable x = cos(θ), as Tk (x) = cos(kθ), sin((k + 1)θ) Uk (x) = . sin(θ) We want to understand this construction in the context of affine Weyl groups. We recognize cos(kθ) and sin(kθ) as the symmetrized and skewsymmetrized exponentials. The cos(θ) used in the change of variables is the 2π-periodic function, which is symmetric, non-constant and has the longest wavelength (as such, uniquely defined up to constant). In other words cos(θ) = π∨ exp(λ1 θ), where λ1 = 1 is the generator of the weights lattice. Any periodic band-limited even function f has a symmetric Fourier series of finite support on the weights lattice, and must hence be a polynomial in the variable x = cos(θ). The denominator sin(θ) is similarly the odd function of longest possible wavelength. Any periodic band-limited odd function f has a skew-symmetric Fourier series on the weights lattice. Dividing out sin(θ) results in a band-limited even function which again must 9

H. Munthe-Kaas and D. Huybrechs, work in progress.

56


map to a polynomial under our change of variables. The denominator, which in our special case is sin(θ), we call the Weyl denominator. It plays an important role in the representation theory of compact Lie groups as the denominator in Weyl’s character formula, the jewel in the crown of representation theory. We will detail these constructions below. Notation. As in Section 4.2.1, we let Φ be a rank d root system on V = Rd , ˜ = W L∨ . with Weyl group W , co-root lattice L∨ and affine Weyl group W ∨ Let T = V/L be the torus of periodicity and Λ = spanZ {λ1 , . . . , λd } the reciprocal lattice of L∨ . The lattice Λ is an abelian group known as the Pontryagin dual of the abelian group T. Specifically, each λ ∈ Λ corresponds uniquely to an exponential function exp(2πi(λ, t)), which is one of the irreducible one-dimensional representations on T. These form an abelian group under multiplication. In the following, it is convenient to write the group Λ in multiplicative form as follows. Let {eλ }λ∈Λ denote the elements of the multiplicative group,10 understood as formal symbols such that for λ, µ ∈ Λ we have eλ · eµ = eλ+µ . Let E = E(C) denote the free complex over the symbols eλ . vector space λ This consists of all formal sums a = λ∈Λ a(λ)e where the coefficients a(λ) ∈ C and all but a finite number of these are non-zero. This is the complex group algebra over the multiplicative group Λ, with a commutative product induced from the group product a(λ) b(µ) = a(λ)b(µ) · eν . λ∈Λ

We define conjugation

µ∈Λ

ν∈Λ

λ+µ=ν

: E → E as a= a(λ)e−λ . λ∈Λ

An element a ∈ E is identified with a trigonometric polynomial f (t) = F −1 (a)(t) on the torus T (i.e., a band-limited periodic function) through the Fourier transforms given in (4.26)–(4.27). Multiplication and conjugation in E corresponds to pointwise multiplication and complex conjugation of the functions on T. Let E∨W ⊂ E denote the symmetric subalgebra of those elements that are invariant under the action of the Weyl group W on E. This consists of those a ∈ E where a(λ) = a(wλ) for all λ and all w ∈ W . Similarly, E∧W ⊂ E denotes those a ∈ E whose sign alternates under reflections sα , i.e., where the coefficients satisfy a(λ) = (−1)|w| a(wλ). Here |w| denotes the length 10

Those who prefer can simply consider eλ to be shorthand for exp(2πi(λ, t)), as a function of t ∈ T.


57

of w, the length of the shortest factorization of w into a product of simple reflections. Thus (−1)|w| = det(w). The alternating elements do not form a subalgebra, since the product of two alternating functions is symmetric. The subspaces E∨W and E∧W correspond to W -symmetric and W -alternating trigonometric polynomials on T. As a digression we remark that a parallel theory is the representation theory of compact connected Lie groups G: see Bump (2004) for an excellent exposition. In this setting T is the maximal torus of G, i.e., a maximal abelian subgroup. Any g ∈ G is conjugate to a point in T, i.e., there exist a k ∈ G such that kgk−1 ∈ T. The normalizer N (T) is composed of those elements of G which fix T under conjugation, N (T) = {k ∈ GkTk −1 = T}. The Weyl group is W = N (T)/T, which is always finite. It can be shown that the action of W on T through conjugation can be represented in terms of a root system. This implies that class functions on G, i.e., functions such that f (g) = f (kgk −1 ) for all k, can be represented as W invariant functions on T, and that any function in E∨W can be interpreted as a class function on G. Of particular interest are the class functions which arise as traces of the irreducible representations on G. These are called the irreducible characters, and are known explicitly through the celebrated Weyl character formula. We will see that the irreducible characters and multivariate Chebyshev polynomials of the second kind are equivalent: see Beerends (1991). Multivariate Chebyshev polynomials. Define a projection π∧W : E → E∧W by its action on the coefficients a → π∧W a: (π∧W a)(λ) =

1 (−1)|w| a(wλ). |W | w∈W

The projection π∨W : E → E∨W is defined similarly, omitting (−1)|w| . The algebra E is generated by {eλj }dj=1 ∪ {e−λj }dj=1 where λj are the fundamental dominant weights. E∨W is the subalgebra generated by the symmetric generators {zj }dj=1 defined by zj = π∨W eλj =

1 wλj 2 wλj e = e , |W | |W | + w∈W

w∈W

where W + denotes the even subgroup of W containing those w such that |w| is even. The latter identity follows from sαj λj = λj , so it is enough to consider w of even length. The action of W + on λj is free and effective. It can be shown that E∨W is a unique factorization domain over the generators {zj }, i.e., any a ∈ E∨W can be expressed uniquely as a polynomial in {zj }dj=1 .

58


The skew subspace E∧W does not form an algebra, but this can be corrected by dividing out the Weyl denominator. Define the Weyl vector ρ ∈ Λ as ρ=

d

λj =

j=1

1 α. 2 + α∈Φ

We define the Weyl denominator D ∈ E∧W as (−1)|w| ewρ . D= w∈W

Proposition 4.16. Any a ∈ E∧W is divisible by D, i.e., there exists a unique b ∈ E∨W such that a = bD. Proof.

See Bump (2004, Proposition 25.2).

Any a ∈ E∨W can be written as a polynomial in z1 , . . . , zd , and hence the following polynomials are well-defined. Definition 4.17. For λ ∈ Λ we define multivariate Chebyshev polynomials of the first and second kind, Tλ and Uλ , as the unique polynomials that satisfy 1 wλ Tλ (z1 , . . . , zd ) = π∨W eλ = e , (4.32) |W | w∈W |w| w(ρ+λ) |W |π∧W eλ+ρ w∈W (−1) e Uλ (z1 , . . . , zd ) = . (4.33) = |w| wρ D w∈W (−1) e By a slight abuse of notation we will also consider zj as W -invariant functions on T as 2 2πi(λj ,t) e . (4.34) zj (t) = F −1 (zj )(t) = |W | + w∈W

We will study coordinates on T obtained from these. The functions zj (t) may be real or complex. If there exists an w ∈ W + such that wλj = λj , then zj = zj is real. Otherwise there must exist an index j = j and a w ∈ W + such that wλj = λj and we have zj = zj . In the latter case we can 1 (zj − zj ). replace these with d real coordinates xj = 12 (zj + zj ), xj = 2i ∨ d Using the basis {αj }, we identify T (R/Z) , and let tj ∈ [0, 1) denote standard coordinates on this unit-periodic torus. Let Jk,l (t) =

∂zk (t) ≡ (αl∨ · ∇t )zj (t) ∂αl∨

k, l ∈ {1, . . . , d}

be the Jacobian matrix of the map (t1 , . . . , td ) → (z1 , . . . , zd ).

59


Proposition 4.18. The Fourier transform of the Jacobian determinant, J := F(det(J)(t)), is the alternating function J = cD ∈ E∧W , where D is the Weyl denominator and the constant c =

4πi d |W |

.

Proof. Let sα be any reflection in W . Since z(t) = z(sα t) we have J(t) = J(sα t)sα . Hence det(J(t)) = − det(J(sα t)) and we conclude that J ∈ E∧W . By Proposition 4.16, D divides J. We need to confirm that J/D is constant. We compute 4πi J α , wλk ewλk . k, = |W | + w∈W

+ Thus, J k, is supported on W λk and J is supported on the set w j λj . w1 ,...,wd ∈W +

The highest weight in this set is the Weyl vector ρ = dj=1 λj , which is reached if and only if w1 = w2 = · · · wd = I. Since the highest weight is ρ we conclude that J/D = c. The constant c is computed as follows: d c = J(ρ) = sign(σ) Jj,σj (ρ) j=1

σ∈Sd

=

sign(σ)

σ∈Sd

=

d j=1

4πi d

|W |

Jj,σj (λj ) =

σ∈Sd

sign(σ)

σ∈Sd

d j=1

δσj ,j =

d 4πi ασj , λj sign(σ) |W |

j=1

4πi |W |

d

.

Finally, in this paragraph, we want to remark that (4.33) is exactly the same formula as Weyl’s character formula, giving the trace of all the irreducible characters on a semi-simple Lie group (Bump 2004). These characters form an L2 -orthogonal basis for the space of class functions on the Lie group. Thus, expansions in terms of multivariate Chebyshev polynomials of the second kind is equivalent to expansions in terms of irreducible characters on a Lie group. Just as the basis given by the irreducible representations block-diagonalize equivariant linear operators on a Lie group, we can also use irreducible characters to obtain block diagonalizations: see James and Liebeck (2001). Thus, our software, which is primarily constructed to deal with spectral element discretizations of PDEs, may also have important applications to computations on Lie groups. This opens up a whole area

60


of possible applications of these approximations, such as in random matrix theory. We will not pursue these topics here. We will focus below on applications of Chebyshev polynomials of the first kind in spectral element discretizations. Analytical properties. The polynomials Tλ (z) of the first kind are by construction W-symmetric in λ, Tλ = Twλ for all w ∈ W . Thus the full space of multivariate polynomials is spanned by {Tλ }λ∈Λ+ , where Λ+ are the weights within the positive Weyl chamber. We will first show that this is a family of multivariate orthogonal polynomials. ˜ . The change of variLet ⊂ T denote the fundamental domain of W ables t → z in (4.34) has a Jacobian determinant proportional to the Weyl denominator D. It is known from representation theory that D is always zero on the boundary of and non-zero in the interior. Thus the coordinate map t → z is invertible on , regular in the interior of and singular on the boundary. Let δ = z(). This is a domain which in the A2 case is a deltoid or three-cusp Steiner hypocycloid: see Figure 4.6(b). Note that D ∈ E∧W is an alternating function, thus DD ∈ E∨W is a symmetric function, hence it is a real multivariate polynomial in z. Proposition 4.19. The coordinate map t → z is invertible on δ = z(). The absolute value of the Jacobian determinant is = |c| DD, |J| is a real polynomial in z. The boundary of δ is given as an where DD algebraic ideal, DD = 0. In the A1 case we have J = sin(t), hence DD = sin2 (t) = 1 − z 2 . In the A2 case we have DD =

1 8 3 + (x − 3xy 2 ) − (x2 + y 2 )2 − 2(x2 + y 2 ), 3 3

where x = 12 (z + z) and y = 12 (z − z). Proposition 4.20. The polynomials {Tλ }λ∈Λ+ are orthogonal on δ with respect to the weighted inner product 1 dz. (f, g) = f (z)g(z) √ δ DD Proof. Under the change of variable z → t, we have that Tλ maps to π∨W exp(2πi(λ, t)), and these are L2 -orthogonal on . The weight follows from the formula for the Jacobian determinant.


(a)

(b)

61

(c)

Figure 4.6. The equilateral domain ∆ in (a) maps to the deltoid δ in (b) under coordinate map t → z, and to a triangle in (c) under a straightening map.

From the convolution product in E∨W one finds that the Tλ satisfy the recurrence relations T0 = 1,

(4.35)

Tλj = zj ,

(4.36)

Tλ = Twλ for w ∈ W, T−λ = Tλ , Tλ T µ =

1 |W |

(4.37) (4.38)

Tλ+wµ .

(4.39)

w∈W

These reduce to classical three-term recurrences for A1 and four-term recurrences for A2 : see Munthe-Kaas (2006). The recurrences provide a practical way of computing values of Tk (z) at arbitrary points z. For special collocation points we will see below that FFT-based computations are much more efficient. Discrete properties. Pulled back to t-coordinates, polynomials in z become band-limited symmetric functions f ∈ L2∨ (T). Due to the Shannon sampling theorem, any band-limited function f ∈ L2 (T) can be exactly reconstructed from sampling on a sufficiently fine lattice S ⊂ T. Due to periodicity, the reciprocal lattice S ∗ must be a sublattice of Λ. The condition for perfect reconstruction is that the only point of the reciprocal lattice S ∗ within the support of f in Λ is the origin. In addition to this condition, we also require the sampling lattice to be W -invariant, W S = S.

62


There are several ways to construct such a lattice. A convenient way is to let S be a down-scaled version of the co-root lattice, S = L∨ /M,

(4.40)

where M is an integer sufficiently large to ensure the Shannon condition for perfect reconstruction. On the lattice S the sampled function fS becomes a symmetric function on a finite abelian group. Thus we can employ FFTs to find f ∈ E∨W . Special group-theoretic versions of symmetrized FFTs can also be used, which are more efficient, but harder to program than the standard FFT: see Munthe-Kaas (1989). The lattice S maps to collocation points z(S) for the multivariate polynomials, as shown in Figure 4.6 for the A2 case, M = 12. The nodal points z(S) ⊂ δ form the two-dimensional analogue of Chebyshev extremal points. The straight lines in Figure 4.6(a), which are in the direction of a root, map to straight lines in Figure 4.6(b). Along these lines inside δ, the nodal points distribute like 1D classical Chebyshev collocation points (either as Chebyshev zeros, or as Chebyshev extremal points). Similarly, the multivariate Chebyshev points in higher dimensions contain flat hyperplanes of lower dimension, and under restriction to these one restricts to lower-dimensional Chebyshev nodes. This is an important feature of these polynomials, which simplifies restriction of polynomials to (certain specific) linear subspaces. In particular, we note that the hexagon in Figure 4.6(a) maps to an equilateral triangle embedded nicely inside δ (Figure 4.6(b)). The fact that the collocation nodes distribute as 1D Chebyshev nodes on the boundary of this triangle allows us to express in closed form conditions for continuity of polynomials patched together along these triangle boundaries. For high-order polynomial collocation it is extremely important that the Lebesgue number of the collocation points grows slowly in M . It is well known that for classical Chebyshev polynomials the Lebesgue number grows at the optimal rate O(log(M )). From properties of the Dirichlet kernel, one can show a similar result for multivariate Chebyshev polynomials. Proposition 4.21. For the collocation points z(S), where S = L∨ /M ⊂ Rd , the Lebesgue number grows as O(logd (M )). 4.2.3. Spectral element methods based on triangles and simplexes Triangle-based spectral element methods were first considered in Dubiner (1991), where a basis was constructed by a ‘warping’ of a 2D tensor product to a triangle. More recent work is Giraldo and Warburton (2005), Hesthaven (1998) and Warburton (2006), based on interpolation of polynomials in good interpolation nodes on a triangle. Our approach differs from these by basing the construction on the multivariate Chebyshev polynomials, which allows for fast FFT-based computations of all the basic operations on the polynomials. It is well known


63

that a logarithmic growth of the Lebesgue number implies exponentially fast (spectral) convergence of polynomial approximations to analytic functions (Canuto et al. 2006). This result, together with the availability of FFT-based algorithms is, just as in the univariate case, among the most important properties of the multivariate Chebyshev polynomials as a practical tool in computational approximation theory and solution of PDEs. Thus, as a conclusion, the coordinate map t → z has a dramatic positive effect on the convergence rate of finite approximations using multivariate Chebyshev polynomials, compared to expansions in the symmetric exponentials π∧W exp(2πi(λ, t). A practical difficulty is, however, the fact that under this change of coordinates the simplex becomes a significantly more complicated domain with cusps in the corners. This has to be dealt with in an efficient manner if we want to construct practical spectral element methods. We have been working with three possible solutions to this problem. • Work with the domain ⊂ T. Use the tight frame provided by the ρ-symmetrized exponentials for ρ ∈ R, and solve the periodic extension problem. This approach has not been developed in detail yet. • Straighten δ to a nearby simplex by a straightening map (MuntheKaas 2006). We have had reasonably good experiences with this approach, but the straightening maps must be singular in the cusps. In numerical experiments with this approach we have seen quite good Lebesgue numbers, and reasonably good convergence rates: see MuntheKaas (2006). However, we have not been able to obtain spectral convergence, probably due to the corner singularities of the straightening map. • Patch together deltoids with overlap, so that the equilateral triangles that are inscribed in the deltoid form a simplicial subdivision of the total space. This approach is working successfully, and will be discussed below. One important question is whether or not this inscribed triangle is a particular feature for the A2 case, or if the higher-dimensional cases also have a similar structure with a nicely inscribed simplex in δ. This is indeed the case. We can prove this for the An family (which is the most important case), and we conjecture that a similar property also holds for the other infinite families of Dynkin diagrams. In Figure 4.7 we illustrate the δ domain in the A3 case. The 3D deltoid-shaped domain contains an inscribed tetrahedron. This tetrahedron is the image of the regular octahedron inscribed in ⊂ T as seen in Figure 4.5. On the faces of this tetrahedron, the nodal points distribute as the Chebyshev points for the A2 case.

64


Figure 4.7. The deltoid domain δ in the A3 case. The inscribed tetrahedron is the image of the regular octahedron inscribed in the Weyl chamber of : see Figure 4.5.

Numerical experiments. A library of routines has been implemented for the A2 case, both in MATLAB and in C++. All basic algorithms such as products of functions, derivation and integration is implemented using FFTbased techniques; see Ryland and Munthe-Kaas (2011) for more details. Algorithms exist for exactly integrating the polynomials both over δ and also over the inscribed triangle, with respect to weight function 1 or the 1 weight function (DD)− 2 . In Ryland and Munthe-Kaas (2011) we demonstrate the use of these bases in spectrally converging Clenshaw–Curtis-type cubatures based on triangular subdivisions. The software allows for triangles mapped from the equilateral reference triangle both with linear and with non-linear non-singular coordinate maps. The Jacobian of these maps can be supplied analytically, or computed numerically from the sampling values. One problem with this type of quadrature is that, since the deltoid stretches outside its inscribed triangle, one has to evaluate functions at points outside the domain. If this is not possible, one could use the collocation points of the straightened triangle, Figure 4.6(c), for the triangles on the boundary. This, however, reduces the convergence rate. We are currently exploring spectral element discretizations for PDEs based on similar subdivisions. In the experiments, we have explicitly imposed continuity across triangle boundaries. The numerical experiments confirm the exponential convergence rate for these approximations.

65

Topics in structure-preserving discretization solution N=48

100 80 60 40 20 0 −20 50 40

50 30

40 30

20

20

10

10 0

0

Figure 4.8. The numerical experiment shows spectral element solution of ∇2 u = −1 on an L-shaped domain composed of three squares, with Dirichlet boundary conditions. Each square is divided into two isosceles triangles, and all these 6 subdomains are patched together by explicitly imposing conditions for continuity across common boundaries of two triangles. Each triangle has a Chebyshev A2 family of Chebyshev polynomials, in this figure using the root lattice down-scaled with the factor 48 as collocation lattice.

5. Finite element systems of differential forms Many partial differential equations (PDEs) can be naturally thought of as expressing that a certain field, say a scalar field or a vector field, is a critical point of a certain functional. Often this functional will be of the form S(u) = L(x, u(x), ∇u(x), . . .) dx. S

u is a section of a vector bundle over S and L is Here S is a domain in a Lagrangian involving x ∈ S as well as values of u and its derivatives. In this context the functional S is called the action. Criticality of the action can be written as Rd ,

∀u

DS(u)u = 0.

More generally, a PDE written in the form F(u) = 0 can be given a variational formulation: ∀u

F(u), u = 0.

(5.1)

The operator F is continuous from a Banach space X to another one Y , and ·, · denotes a duality product on Y × Y .The unknown u is sought in

66


X and the equation is tested with u in Y . The duality product is most often a generalization of the L2 (S) scalar product, such as a duality between Sobolev spaces. In the case of least action principles it is of interest to find X such that one can take Y = X (and Y = X). This point of view leads quite naturally to discretizations. We construct a trial space Xn ⊆ X and a test space Yn ⊆ Y , and solve u ∈ Xn , ∀u ∈ Yn

F(u), u = 0.

(5.2)

In theory we have sequences indexed by n ∈ N and establish convergence properties of the method as n → ∞. This discretization technique is called the Galerkin method when Xn = Yn and the Petrov–Galerkin method when Xn and Yn are different. The finite element (FE) method consists in constructing finite-dimensional trial and test spaces adapted to classes of PDEs. Mixed FE spaces have been constructed to behave well for the differential operators grad, curl and div. They provide a versatile tool-box for discretizing PDEs expressed in these terms. Thus Raviart–Thomas div-conforming finite elements (Raviart and Thomas 1977) are popular for PDEs in fluid dynamics and Nédélec’s curl-conforming finite elements (Nédélec 1980) have imposed themselves in electromagnetics. For reviews and references see Brezzi and Fortin (1991), Roberts and Thomas (1991), Hiptmair (2002) and Monk (2003). These spaces fit the definition of a finite element given by Ciarlet (1978). In particular they are equipped with unisolvent degrees of freedom, which determine inter-element continuity and provide interpolation operators: projections In onto Xn which are defined at least for smooth fields. In applications one often needs a pair of spaces Xna × Xnb , and these spaces should be compatible in the sense of satisfying a Brezzi inf-sup condition (Brezzi 1974). The spaces are linked by a differential operator d : Xna → Xnb (one among grad, curl, div), and the proof of compatibility would follow from a commuting diagram: Xa

d

Ina

Xna

d

/ Xb

Inb

/ Xb

n

A technical problem is that one would like the interpolators to be defined and bounded on the Banach spaces X a , X b , whereas the natural degrees of freedom are not bounded on these. For instance X a is usually of the form: X a = {u ∈ L2 (S) ⊗ E a : du ∈ L2 (S) ⊗ E b }, for some finite-dimensional fibres E a , E b , but this continuity is usually insufficient for degrees of freedom, such as line integrals, to be well-defined. The convergence of eigenvalue problems requires even more boundedness prop-


67

erties of the interpolation operators, something more akin to boundedness in L2 (S) (see Christiansen and Winther (2010) for precise statements). Mixed finite elements have been constructed to contain polynomials of arbitrarily high degree. Applying the Bramble–Hilbert lemma to the interpolators gives high orders of approximation, and under stability conditions this gives a high order of convergence for numerical methods. But here too the intention is hampered by the lack of continuity of interpolators. As remarked by Bossavit (1988), lowest-order mixed finite elements, when translated into the language of differential forms, correspond to constructs from differential topology called Whitney forms (Weil 1952, Whitney 1957). The above-mentioned differential operators can all be interpreted as instances of the exterior derivative. From this point of view it becomes natural to arrange the spaces in full sequences linked by operators forming commuting diagrams: X0

d

In0

Xn0

d

/ X1

d

/ ...

d

In1

/ X1

n

d

/ . . .

d

/ Xd

Ind

/ X d. n

An important property is that the interpolators should induce isomorphisms on cohomology groups. This is essentially de Rham’s theorem, when the top row consists of spaces of smooth differential forms and the bottom row consists of Whitney forms. In particular, on domains homeomorphic to balls, the sequences of FE spaces should be exact. This applies in particular to single elements such as cubes and simplexes. High-order FE spaces of differential forms naturally generalizing Raviart– Thomas–Nédélec elements were presented by Hiptmair (1999). For a comprehensive review, encompassing Brezzi–Douglas–Marini elements (Brezzi, Douglas and Marini 1985), see Arnold, Falk and Winther (2006b). Elements are constructed using the Koszul operator, or equivalently the Poincaré homotopy operator, to ensure local sequence exactness, that is, exactness of the discrete sequence attached to a single element. Mixed FE spaces have thus been defined for simplexes. Tensor product constructions yield spaces on Cartesian products. Given spaces defined on some mesh, it would sometimes be useful to have spaces constructed on the dual mesh, matching the spaces on the primal mesh in some sense. Since the dual mesh of a simplicial mesh is not simplicial, this motivates the construction of FE spaces on meshes consisting of general polytopes. On these, it seems unlikely that good FE spaces can be constructed with only polynomials: one should at least allow for piecewise polynomials. In some situations, such as convection-dominated fluid flow, stability requires some form of upwinding. This could be achieved by a

68


Petrov–Galerkin method, by including upwinded basis functions in either the trial space Xn or the test space Yn . For a recent review of this topic see Morton (2010). This provides another motivation for constructing a framework for finite elements that includes non-polynomial functions. Discretization of PDEs expressed in terms of grad, curl and div on polyhedral meshes has long been pursued in the framework of mimetic finite differences, reviewed in Bochev and Hyman (2006). The convergence of such methods has been analysed in terms of related mixed finite elements (Brezzi, Lipnikov and Shashkov 2005). For connections with finite volume methods see Droniou, Eymard, Gallouët and Herbin (2010). While recent developments tend to exhibit similarities and even equivalences between all these methods, the FE method, as it is understood here, could be singled out by its emphasis on defining fields inside cells, and ensuring continuity properties between them, in such a way that the main discrete differential operators acting on discrete fields are obtained simply by restriction of the continuous ones. Ciarlet’s definition of a finite element does not capture the fact that mixed finite elements behave well with respect to restriction to faces of elements. From the opposite point of view, once this is realized, it becomes natural to prove properties of mixed finite elements by induction on the dimension of the cell. The goal is then to construct a framework for FE spaces of differential forms on cellular complexes accommodating arbitrary functions. One requires stability of the ansatz spaces under restriction to subcells and under the exterior derivative. For the good properties of standard spaces to be preserved, one must impose additional conditions on the ansatz spaces, essentially surjectivity of the restriction from the cell to the boundary, and sequence exactness under the exterior derivative on each cell. As it turns out, these conditions, which we refer to as compatibility, imply the existence of interpolation operators commuting with the exterior derivative. Degrees of freedom are not part of the definition of compatible finite element systems but are rather deduced, and it becomes more natural to compare various degrees of freedom for a given system. The local properties of surjectivity and sequence exactness also ensure global topological properties by the general methods of algebraic topology. Interpolation operators still lack desired continuity properties. But combining them with a smoothing technique yields commuting projection operators that are stable in L2 (S). Stable commuting projections were first proposed in Schöberl (2008). Smoothing was achieved by taking averages over perturbations of the grid. Another construction using cut-off and smoothing by convolution on reference macro-elements was introduced in Christiansen (2007). While commutativity was lost, the lack of it was controlled by an auxiliary parameter. As observed in Arnold et al. (2006b), for quasi-uniform


69

meshes one can simplify these constructions and use smoothing by convolution on the physical domain. If, in the method of Sch¨ oberl (2008), one can say that the nodes of the mesh are shaken independently, smoothing by convolution consists in shaking them in parallel. Much earlier, in Dodziuk and Patodi (1976), convergence for the eigenvalue problem for the Hodge– Laplacian, discretized with Whitney forms, was proved using smoothing by the heat kernel. For scalar functions, smoothing by convolution in the FE method has been used at least as far back as Strang (1972) and Hilbert (1973), but Clément interpolation (Clément 1975) seems to have supplanted it. Christiansen and Winther (2008) introduced a space-dependent smoothing operator, commuting with the exterior derivative, allowing for general shape-regular meshes. These constructions also require a commuting extension operator, extending differential forms outside the physical domain. This section is organized as follows. In Section 5.2, cellular complexes and the associated framework of finite element systems are introduced. Basic examples are included, as well as some constructions like tensor products. Section 5.2 serves to introduce degrees of freedom and interpolation operators on FE systems. In Section 5.3, we construct smoothers and extensions which commute with the exterior derivative and preserve polynomials locally. When combined with interpolators they yield Lq (S)-stable commuting projections for scale-invariant FE systems. In Section 5.4 we apply these constructions to prove discrete Sobolev injections and a translation estimate. The framework of FE systems was implicit in Christiansen (2008a) and made explicit in Christiansen (2009), but we have improved some of the proofs. Upwinding in this context is new, as well as the discussion of interpolation and degrees of freedom. The construction of smoothers and extensions improves that of Christiansen and Winther (2008) by having the additional property of preserving polynomials locally, up to any given maximal degree. The analysis is also extended from L2 to Lq estimates, for all finite q. This is used in the proof of the Sobolev injection and translation estimate, which are also new (improving Christiansen and Scheid (2011) and Karlsen and Karper (2010)). 5.1. Finite element systems Cellular complexes. For any natural number k ≥ 1, let Bk be the closed unit ball of Rk and Sk−1 its boundary. For instance S0 = {−1, 1}. We also put B0 = {0}. Let S denote a compact metric space. A k-dimensional cell in S is a closed subset T of M for which there is a Lipschitz bijection Bk → T with a Lipschitz inverse. If a cell T is both k- and l-dimensional then k = l. For k ≥ 1, we denote by ∂T its boundary, the image of Sk−1 by the chosen

70


bi-Lipschitz map. Different such maps give the same boundary. The interior of T is by definition T \ ∂T (it is open in T but not necessarily in S). Definition 5.1. A cellular complex is a pair (S, T ) where S is a compact metric space and T is a finite set of cells in S, such that the following conditions hold. • Distinct cells in T have disjoint interiors. • The boundary of any cell in T is a union of cells in T . • The union of all cells in T is S. In this situation we also say that T is a cellular complex on S. We first make the following remarks. Proposition 5.2. The intersection of two cells in T is a union of cells in T . Proof. Let T, U be two cells in T and suppose x ∈ T ∩ U . Choose a cell T included in T of minimal dimension such that x ∈ T . Choose also a cell U included in U of minimal dimension such that x ∈ U . Suppose neither of the cells T and U are points. Then x belongs to the interiors of both T and U , so that T = U . Therefore there exists a cell included in both T and U to which x belongs. This conclusion also trivially holds if T or U is a point. In fact, if (S, T ) is a cellular complex, S can be recovered from T as follows. Proposition 5.3.

Let (S, T ) be a cellular complex.

• S is recovered as a set from T as its union. • The topology of S is determined by the property that a subset U of S is closed if and only if, for any cell T in T , of dimension k, the image of U ∩ T under the chosen bi-Lipschitz map is closed in Bk . • A compatible metric on S can be recovered from metrics dT on each cell T in T inherited from Bk , for instance by (max ∅ = +∞): d(x, y) = min{1, max{dT (x, y) : x, y ∈ T and T ∈ T }}. A simplicial complex is a cellular complex in which the intersection of any two cells is a cell (not just a union of cells) and such that the boundary of any cell is split into subcells in the same way as the boundary of a reference simplex is split into sub-simplexes. The reference simplex ∆k of dimension k is k+1 (x0 , . . . , xk ) ∈ R : xi = 1 and ∀i xi ≥ 0 . i


71

Its sub-simplexes are parametrized by subsets J ⊆ {0, · · · , k} as the solution spaces {x ∈ ∆k : ∀i ∈ J xi = 0}. For each k ∈ N the set of k-dimensional subsets of T is denoted by T k = {T ∈ T : dim T = k}. The k-skeleton of T is the cellular complex consisting of cells of dimension at most k: T (k) = T 0 ∪ · · · ∪ T k . The boundary ∂T of any cell T of T can be naturally equipped with a cellular complex, namely {T ∈ T : T ⊆ T and T = T }. We use the same notation for the boundary of a cell and the cellular complex it carries. A refinement of a cellular complex T on S is a cellular complex T on S such that each element of T is the union of elements of T . We will be particularly interested in simplicial refinements of cellular complexes. A cellular subcomplex of a cellular complex T on S is a cellular complex T on some closed part S of S such that T ⊆ T . For instance, if T ∈ T is a cell, its subcells form a subcomplex of T , which we denote by T˜. We have seen that the boundary of any cell T ∈ T can be equipped with a cellular complex which is a subcomplex of T˜. Fix a cellular complex (S, T ). In the following we suppose that for each T ∈ T of dimension ≥ 1, the manifold T has been oriented. The relative orientation of two cells T and T in T , also called the incidence number , is denoted o(T, T ) and defined as follows. For any edge e ∈ T 1 its vertices are ordered, from say e˙ to e¨. Define o(e, e) ˙ = −1 and o(e, e¨) = 1. As for higher-dimensional cells, fix k ≥ 1. Given T ∈ T k+1 and T ∈ T k such that T ⊆ T , we define o(T, T ) = 1 if T is outward-oriented compared with T and o(T, T ) = −1 if it is inward-oriented. For all T, T ∈ T not covered by these definitions we put o(T, T ) = 0. For each k, let C k (T ) denote the set of maps c : T k → R. Such maps associate a real number with each k-dimensional cell and are called k-cochains. The coboundary operator δ : C k (T ) → C k+1 (T ) is defined by (δc)T = o(T, T )cT . T ∈T k

The space of k-cochains has a canonical basis indexed by T k . The coboundary operator is the operator whose canonical matrix is the incidence matrix o, indexed by T k+1 × T k . We remark that the coefficients in the sum can be non-zero only when T ∈ ∂T ∩ T k .

72


Lemma 5.4. Proof.

We have that δδ = 0 as a map C k (T ) → C k+2 (T ).

See, e.g., Christiansen (2009, Lemma 3.6).

In other words the family C • (T ) is a complex, called the cochain complex and represented by 0 → C 0 (T ) → C 1 (T ) → C 2 (T ) → · · · . When S is a smooth manifold we denote by Ωk (S) the space of smooth differential k-forms on S. Differential forms can be mapped to cochains as follows. Let S be a manifold and T a cellular complex on S. For each k we denote by ρk : Ωk (S) → C k (T ) the de Rham map, which is defined by k u . ρ : u → T ∈T k

T

Proposition 5.5.

For each k the following diagram commutes:

ρk

ρk+1

/ C k+1 (T )

δ

C k (T ) Proof.

/ Ωk+1 (S)

d

Ωk (S)

This is an application of Stokes’ theorem.

Suppose T is a cellular complex equipped with an orientation (of the cells) and T is a cellular refinement also equipped with an orientation (for instance the same complex but with different orientations). For each cell T ∈ T k and each T ∈ T k define ι(T, T ) = ±1 if T ⊆ T and they have the same/different orientation, and ι(T, T ) = 0 in all other cases. Proposition 5.6.

The map ι : C • (T ) → C • (T ), defined by ι(T, T )uT , (ιu)T = T ∈T

is a morphism of complexes, meaning that ι and δ commute. Proof.

See, e.g., Christiansen (2009, Proposition 3.5).

We also remark that for a manifold S we have the following. Proposition 5.7.

The following diagram commutes: Ω(S)

w ww ww w w {w w ρ

C(T )

ι

GG GG ρ GG GG # / C(T )


73

Element systems. For any cell T , we denote by Ωks,q (T ) the space of differential k-forms on T with Ws,q (T ) Sobolev regularity, and put Ωkq (T ) = Ωk0,q (T ). Fix q ∈ [1, +∞[ and define X k (T ) = {u ∈ Ωkq (T ) : du ∈ Ωkq (T )}. When i : T → T is an inclusion of cells and u is a smooth enough form on T we denote by u|T = i u the pullback of u to T . Thus we restrict to the subcell and forget about the action of u on vectors not tangent to it. In the topology (5.1), restrictions to subcells T of codimension one are well-defined, for instance as elements of Ωk−1,q (T ). When T is a cell in a given cellular complex T we may therefore set ˆ k (T ) = {u ∈ X k (T ) : ∀T ∈ T X

T ⊆ T ⇒ u|T ∈ X k (T )}.

Definition 5.8. Suppose T is a cellular complex. For each k ∈ N and each ˆ k (T ) called a differential T ∈ T we suppose we are given a space Ak (T ) ⊆ X k-element on T . We suppose that the exterior derivative induces maps d : Ak (T ) → Ak+1 (T ) and that if i : T ⊆ T is an inclusion of cells, pullback induces a map i : Ak (T ) → Ak (T ). Such a family of elements is called an element system. A differential element is said to be finite if it is finite-dimensional. A finite element system is an element system in which all the elements are finite. ˆ • (•) themselves define an element system. It Example 5.9. The spaces X is far from finite. Example 5.10. Let U be an open subset of a vector space V . We denote by Pp (U ) the space of real polynomials of degree at most p on U . For k ≥ 1 the space of alternating maps V k → R is denoted Ak (V ). The space of differential k-forms on U , which are polynomial of degree at most p, is denoted PAkp (U ). We identify PAkp (U ) = Pp (U ) ⊗ Ak (V )

and

PA0p (U ) = Pp (U ).

Choose a cellular complex where all cells are flat. Choose a function π : T × N → N and define Ak (T ) = PAkπ(T,k) (T ). One gets a finite element system when the following conditions are satisfied: T ⊆ T ⇒ π(T , k) ≥ π(T, k)

and

π(T, k + 1) ≥ π(T, k) − 1.

Example 5.11. Denote the Koszul operator on vector spaces by κ. It is the contraction of differential forms by the identity, considered as a vector field: (κu)x (ξ1 , . . . , ξk ) = ux (x, ξ1 , . . . , ξk ).

74


Alternatively one can use the Poincaré operator associated with the canonical homotopy from the identity to the null-map. Let T be a simplicial complex. Define, for non-zero p ∈ N, k+1 k Λkp (T ) = {u ∈ PAkp (T ) : κu ∈ PAk−1 p (T )} = PAp−1 (T ) + κPAp−1 (T ).

For fixed p we call this the trimmed polynomial finite element system of order p. The case p = 1 corresponds to constructs in Weil (1952) and Whitney (1957). Arbitrary order elements were introduced in Nédélec (1980) for vector fields in R3 . In Hiptmair (1999) these spaces were extended to differential forms. The correspondence between lowest-order mixed finite elements and Whitney forms was pointed out in Bossavit (1988). See Arnold et al. (2006b) for a comprehensive review. It was usual to start the indexing at p = 0 but, as remarked in the preprint of Christiansen (2007), the advantage of letting the lowest order be p = 1 is that the wedge product induces maps: +k1 ∧ : Λkp00 (T ) × Λkp11 (T ) → Λkp00 +p (T ). 1

See also Arnold et al. (2006b, p. 34). In words, the wedge product respects the grading in k and the filtering in p. This observation was useful in the implementation of a scheme for the Yang–Mills equation (Christiansen and Winther 2006). The first example of a finite element system yields quite useless Galerkin spaces in general, whereas the second one yields good ones. We shall elaborate on this in what follows, starting by defining what the Galerkin space associated with a finite element system is. For any subcomplex T of T we define Ak (T ) as follows: k k A (T ) = u ∈ A (T ) : ∀T, T ∈ T T ⊆ T ⇒ uT |T = uT . T ∈T

Elements of Ak (T ) may be regarded as differential forms defined piecewise, which are continuous across interfaces between cells, in the sense of equal pullbacks. For a cell T its collection of subcells is the cellular complex T˜. Applied to this case, the above definition gives a space canonically isomorphic to Ak (T ). We can identify Ak (T˜) = Ak (T ). An FE system over a cellular complex is an inverse system of complexes: for an inclusion of cells there is a corresponding restriction operator. The space A• (T ) defined above is an inverse limit of this system and is determined by this property up to unique isomorphism. We also point out that this kind of construction, involving glueing of polynomial differential forms on cellular complexes, has been used to study homotopy theory (Griffiths and Morgan 1981).

75


Of particular importance is the application of the above construction to the boundary ∂T of a cell T , considered as a cellular complex consisting of all subcells of T except T itself. Considering ∂T as a cellular complex (not only a subset of T ), we denote the constructed space by Ak (∂T ). If i : ∂T → T denotes the inclusion map, the pullback by i defines a map i : Ak (T ) → Ak (∂T ), which we denote by ∂ and call restriction. Conventions. In the following, the arrows starting or ending at 0 are the only possible ones. Arrows starting at R are, unless otherwise specified, the maps taking a value to the corresponding constant function. Arrows ending at R are integration of forms of maximal degree. Other unspecified arrows are instances of the exterior derivative. Consider now the following conditions on an element system A on a cellular complex T . • Extensions. For each T ∈ T and k ∈ N, restriction ∂ : Ak (T ) → Ak (∂T ) is onto. • Exactness. The following sequence is exact for each T : 0 → R → A0 (T ) → A1 (T ) → · · · → Adim T (T ) → 0. The first condition can be written symbolically as

∂Ak (T )

=

(5.3)

Ak (∂T ).

Definition 5.12. We will say that an element system admits extensions if the first property holds, is locally exact if the second condition holds and is compatible if both hold. Given a finite element system A, we say that its points carry reals if, for each point T ∈ T 0 , A0 (T ) contains the constant maps T → R (so that A0 (T ) = RT ≈ R). Proposition 5.13. If A admits extensions and its points carry reals, then for each cell T , Adim T (T ) contains a form with non-zero integral. Proof.

By induction on the dimension of the cell, using Stokes’ theorem.

Notation. We let Ak0 (T ) denote the kernel of ∂ : Ak (T ) → Ak (∂T ). Proposition 5.14.

We have dim Ak (T ) ≤

dim Ak0 (T ),

T ∈T

with equality when the finite element system admits extensions. Proof. For a given m ≥ 0, let T (m) be the m-skeleton of T . We have a sequence: 0→ Ak0 (T ) → Ak (T (m) ) → Ak (T (m−1) ) → 0. T ∈T m

76


The second arrow is bijective onto the kernel of the third. If all cells of dimension m admit extensions, the whole sequence is exact. The proposition follows from applying these remarks for all m. Combining Propositions 5.13 and 5.14, we get the following. Corollary 5.15. If A admits extensions and its points carry reals, then Ak (T ) has dimension at least the number of k-dimensional subcells of T . Proposition 5.16. When the element system is compatible, the de Rham map ρ• : A• (T ) → C • (T ) induces isomorphisms in cohomology. Proof. We use induction on the dimension of T . In dimension 0 it is clear. Suppose now that m ≥ 1, and that we have proved the theorem when dim T < m. Suppose that T has dimension m. Note that the de Rham map gives isomorphisms in cohomology on cells, A• (T ) → C • (T˜), since both complexes are acyclic. Denote by U the (m − 1)-skeleton of T . Consider the diagram: • • A (U ) A (T ) A• (∂T ) • / / / A (T ) 0 T ∈T m

0

/ C (T ) •

• / C (U )

T ∈T m

T ∈T m

C • (T˜)

C • (∂T ) / T ∈T m

/0

/0

The vertical maps are de Rham maps. The second horizontal arrow consists of restricting to the summands whereas the third one consists of restricting and comparing, as in the Mayer–Vietoris sequence. Both rows are exact sequences of complexes, the diagram commutes, and the last two vertical arrows induce isomorphisms in cohomology by the induction hypothesis. Write the long exact sequences of cohomology groups associated with both rows (e.g., Christiansen (2009, Theorem 3.1)) and connect them with the induced morphisms. Applying the five lemma (e.g., Christiansen (2009, Lemma 3.2)) gives the result for T . Proposition 5.17. For an element system with extensions the exactness of (5.3) on each T ∈ T is equivalent to the combination of the following two conditions. • For each T ∈ T , A0 (T ) contains the constant functions. • For each T ∈ T , the following sequence (with boundary condition) is exact: 0 → A00 (T ) → A10 (T ) → · · · → Adim T (T ) → R → 0. (5.4)

77


Proof. When the extension property is satisfied on all cells, both versions (with and without boundary condition) of the cohomological condition guarantee that for each T , Adim T (T ) contains a form with non-zero integral, by Proposition 5.13. When the spaces A0 (T ) all contain the constant functions, we may consider the following diagrams, where T is a cell of dimension m and Am − (T ) denotes the space of forms with zero integral: 0

/R O

/ A0 (∂T ) O

/ ··· O

/ Am−1 (∂T ) O

/R O

/0

0

/R O

/ A0 (T ) O

/ ··· O

/ Am−1 (T ) O

/ Am (T ) O

/0

0

/ A0 (T )

/ ···

/ Am−1 (T )

/ Am (T ) −

/0

0

0

(5.5) The columns, extended by 0, are exact if and only if the extension property holds on T and Am (T ) contains a form of non-zero integral. In this case, if one row is exact, the two other rows are either both exact or both inexact. We now prove the stated equivalence. (i) If compatibility holds then in (5.5) the extended columns are exact, as well as the first and second row, so also the third. Hence (5.4) is exact. (ii) Suppose now exactness of (5.4) holds for each T . Choose m ≥ 1 and suppose that we have proved exactness of (5.3) for cells of dimension up to m − 1. Let T be a cell of dimension m. In (5.5) the extended columns are exact. Apply Proposition 5.16 to the boundary of T to get exactness of the first row. The third row is exact by hypothesis, and we deduce exactness of the second. The induction, whose initialization is trivial, is complete. ˆ is a compatible element system. Corollary 5.18. X Tensor products. Suppose we have two manifolds M and N , equipped with cellular complexes U and V. We suppose we have differential elements Ak (U ) for U ∈ U and B k (V ) for V ∈ V, both forming systems as defined above. Let U × V denote the product cellular complex on M × N , whose cells are all those of the form U × V for U ∈ U and V ∈ V. Recall that the tensor product of differential forms u on U and v on V , is the form on U × V defined as the wedge product of their pullbacks by the respective canonical projections pU : U × V → U and pV : U × V → V . In symbols we can write u ⊗ v = (pU u) ∧ (pV v). We equip U × V with elements C • (U × V ) = A• (U ) ⊗ B • (V ).

78


Explicitly we put C k (U × V ) =

Al (U ) ⊗ B k−l (V ).

l

This defines an element system C, called the tensor product of A and B. Proposition 5.19.

We have C0• (U × V ) = A•0 (U ) ⊗ B0• (V ).

Proof.

See Christiansen (2009, Lemma 3.10).

Proposition 5.20.

When A and B admit extensions we have C • (U × V) = A• (U) ⊗ B • (V).

Proof. The right-hand side is included in the left-hand side. Moreover, by Proposition 5.14, dim C(U × V) ≤ U,V dim C0 (U × V ). On the other hand Proposition 5.19 gives U,V dim C0 (U × V ) = U,V dim A0 (U ) ⊗ B0 (V ) = U,V dim A0 (U ) dim B0 (V ) = U dim A0 (U ) V dim B0 (V ) = dim A(U ) dim B(V). This completes the proof. Proposition 5.21. product. Proof.

If A and B admit extensions, so does their tensor

Consider cells U ∈ U and V ∈ V. Note that (∂U × V ) ∪ (U × ∂V ) = ∂(U × V ), (∂U × V ) ∩ (U × ∂V ) = ∂U × ∂V.

The Mayer–Vietoris principle gives an exact sequence, 0 → C(∂(U × V )) → C(∂U × V ) ⊕ C(U × ∂V ) → C(∂U × ∂V ) → 0, where the second and third mappings are w → w|∂U ×V ⊕ w|U ×∂V , u ⊕ v → u|∂U ×∂V − v|∂U ×∂V . It follows that dim C(∂(U × V )) = dim C(∂U × V ) + dim C(U × ∂V ) − dim C(∂U × ∂V ).


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Applying Proposition 5.20 three times, we get dim C(∂(U × V )) = dim A(∂U ) dim B(V ) + dim A(U ) dim B(∂V ) − dim A(∂U ) dim B(∂V ) = dim A(U ) dim B(V ) − dim A0 (U ) dim B0 (V ) = dim C(U × V ) − dim C0 (U × V ) = dim ∂C(U × V ). Therefore we obtain ∂C(U × V ) = C(∂(U × V )), as claimed. Proposition 5.22. product.

If A and B are locally exact, then so is their tensor

Proof. This follows from the Kunneth theorem (e.g., Christiansen (2009, Theorem 3.2)). Nesting. Suppose T is a cellular complex and that (Ξ, #) is an ordered set. Suppose that, for each parameter ξ ∈ Ξ, an FE system A[ξ] on T has been chosen. We assume that if ξ # ξ then A[ξ]k (T ) ⊆ A[ξ ]k (T ). Choose now a parameter function π : T → Ξ, which is order-preserving in the sense that if T ⊆ T then π(T ) # π(T ). Define an FE system A[π] by A[π]k (T ) = {u ∈ A[π(T )]k (T ) : ∀T ∈ T

T ⊆ T ⇒ u|T ∈ A[π(T )]k (T )}.

Proposition 5.23. If, for each ξ ∈ Ξ, the system A[ξ] is compatible, then the constructed system A[π] is too. Proof. Note that if u ∈ A[π]k (∂T ) then u ∈ A[ξ(T )]k (∂T ) so it can be extended to an element u ∈ A[ξ(T )]k (T ). This element is in A[π]k (T ). Thus A[π]k (T ) admits extensions. We also have A[π]k0 (T ) = A[π(T )]k0 (T ), which gives local exactness thanks to Proposition 5.17. Example 5.24. For a simplicial complex T , let A[p] denote the trimmed finite element system of order p. We model variable orders of approximation by a function π : T → N∗ such that when T ⊆ T we have π(T ) ≤ π(T ). The above construction defines an FE system A[π], of the type used for hp-methods (Demkowicz, Kurtz, Pardo, Paszy´ nski, Rachowicz and Zdunek 2008). We will check later that the trimmed systems for fixed order p are compatible, and then the above result gives compatibility of the variableorder system defined by π. Locally harmonic forms. Let T be a cell where, for each k, Ak (T ) is equipped with a scalar product a. Orthogonality with respect to a will be denoted

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by ⊥. We say that a k-form u on T is A-harmonic if du ⊥ dAk0 (T )

and

u ⊥ dAk−1 0 (T ).

(5.6)

One can, for instance, take a to be the L2 scalar product on differential forms, associated with some Riemannian metric. Denote by d the formal adjoint of d with respect to this scalar product. The continuous analogue of the above condition (5.6) is d du = 0

and

d u = 0.

(5.7)

From the other point of view, (5.6) is the Galerkin variant of (5.7). Proposition 5.25. Let A be a finite element system where each Ak (T ) is equipped with a scalar product a. Suppose T is a cell such that (5.4) is exact. Put m = dim T . m • For

each α ∈ R, there is a unique A-harmonic u ∈ A (T ) such that T u = α.

• For k < m, any u ∈ Ak (∂T ) admitting an extension in Ak (T ), has a unique A-harmonic extension in Ak (T ). Let A be a finite element system on T . Define a finite element system ˚ A by ˚k (T ) = {u ∈ Ak (T ) : ∀T ∈ T A

T ⊆ T ⇒ u|T is A-harmonic}.

˚ is the subsystem of locally harmonic forms. We say that A ˚ is a compatProposition 5.26. If A is a compatible FE system then A k k ˚ ible FE system such that the de Rham map ρ : A (T ) → C k (T ) is an isomorphism. Proof.

This was essentially proved in Christiansen (2008a).

This construction generalizes Kuznetsov and Repin (2005), in which divconforming finite elements are defined on polyhedra in R3 . Since C k (T ) has a canonical basis, the de Rham map determines a corresponding canonical ˚k (T ). Its elements can be constructed by recursive harmonic basis of A extension. Example 5.27. On simplexes, lowest-order trimmed polynomial differential forms are locally harmonic in the sense of (5.7), with respect to the L2 -product associated with any piecewise constant Euclidean metric (Christiansen 2008a). Example 5.28. Locally harmonic forms can be used to define finite element spaces on the dual cellular complex of a given simplicial one (Buffa and Christiansen 2007). Choose a simplicial refinement of the dual mesh,


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for instance the barycentric refinement of the primal mesh. Consider Whitney forms on this refinement as a compatible finite element system on the dual mesh. Then take the subsystem of locally harmonic forms. In space dimension d, this provides a space of k-forms with the same dimension as the space of (d−k)-forms on the primal mesh. Duality in the sense of an inf-sup condition was proved in Buffa and Christiansen (2007) with applications to the preconditioning of integral operators appearing in electromagnetics. Duality methods are quite common in finite volume settings; for a recent development see Andreianov, Bendahmane and Karlsen (2010). Example 5.29. For a given fine mesh one can agglomerate elements into a coarser cellular mesh. The finite element system on the fine mesh then provides a finite element system on the coarse one, with identical function spaces. If the former system is compatible so is the latter. Associated with the latter, we can consider the subsystem of locally harmonic forms. This procedure can be applied recursively: at each level one can consider the locally harmonic forms of the finer level. This yields a multilevel analysis which can be used for multigrid preconditioning (Pasciak and Vassilevski 2008). Here is a third application, again involving the dual finite elements. Example 5.30. written as

Recall that the incompressible Euler equation can be

u˙ + div(u ⊗ u) + grad p = 0

and

div u = 0.

Here, u is a vector field with time derivative u. ˙ One uses div-conforming Raviart–Thomas elements for u: let Xh denote this space. One uses a weak formulation with locally harmonic curl-conforming elements on the dual grid as test functions: let Yh denote this space. The L2 duality and its extensions to Sobolev spaces are denoted ·, ·. It is invertible on Xh × Yh , at least in dimension 2, as proved in Buffa and Christiansen (2007). The semi-discrete problem reads as follows. Find a time-dependent uh ∈ Xh such that, for all vh ∈ Yh which are orthogonal to the subspace of Yn of curl-free elements, we have u˙h , vh + div(uh ⊗ uh ), vh = 0. To see that the second bracket is well-defined, note that div(uh ⊗ uh ) and div div(uh ⊗ uh ) both have W−1,q Sobolev regularity for all q < 2. On the other hand vh and curl vh have W0,q -regularity, for q > 2. Analogous spaces, with regularity expressed by a second-order operator, have been used for Regge calculus (Christiansen 2008b) and elasticity (Schöberl and Sinwel 2007).

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In computational fluid dynamics it is often important to include some form of upwinding in the numerical method to obtain stability. A model problem for this situation is the equation ∆u + V · grad u = f,

(5.8)

where the field u ∈ H1 (S) satisfies homogeneous Dirichlet boundary condition, f is a given forcing term, whereas V is a given flow field, which we take to be divergence-free. One is interested in the asymptotic behaviour for small positive (viscosity). Such problems can for instance be solved with Petrov–Galerkin methods. Consider a cellular complex and a large compatible finite element system on it, obtained for instance by refinement. We let one space (say the trial space) consist of locally harmonic forms for the standard L2 -product. For the other space (say the test space) we use locally harmonic forms for a weighted L2 -product. To motivate our choice of weight we introduce some more notions of differential geometry. Given a 1-form α we define the covariant exterior derivative: dα : u → du + α ∧ u, These operators do not form a complex, but we have dα dα u = (dα) ∧ u. In gauge theory the term dα is called curvature. Supposing that α = dβ for a function β, we have dα = exp(−β) d exp(β)u. One says that u → exp(β)u is a gauge transformation. We suppose the domain S is equipped with a Riemannian metric. It provides in particular an L2 scalar product on differential forms. We denote by dα the formal adjoint of d. When α = exp(−β), we have dα u = exp(β)d exp(−β). A natural generalization of (5.7) is dα du = 0

and

dα u = 0,

d exp(−β)du = 0

and

d exp(−β)u = 0.

which can be written as

Example 5.31. To address (5.8) define the 1-form α by α(ξ) = −−1 V · ξ, and note that (5.8) can be rewritten as dα du = −−1 f.

(5.9)


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Given a cellular complex T on S, with flat cells, and a large compatible FE system A, we construct two spaces of locally harmonic forms, distinguished by the choice of scalar product a. For one (the trial space) take a to be the L2 scalar product. For the other (the test space), we choose for each T a constant approximation αT of the pullback of α to T . Let βT be the affine function with zero mean on T such that dβT = αT . For the trial spaces, use the scalar product defined on a cell T by a(u, v) = exp(−βT )u · v (5.10) T

to define the locally harmonic functions. If v is a constant differential form, u = exp(βT )v satisfies the equations (5.9). The canonical basis of the test space will then be upwinded or downwinded (depending on the sign in front of βT in (5.10)), compared with the canonical basis of the trial space. Example 5.32. equation:

Similar notions can be used to address the Helmholtz ∆u + k 2 u = 0.

(5.11)

We wish to construct a compatible FE system over C, which contains a certain number of plane waves: uξ : x → exp(iξ · x). To contain just one of them we remark that for any (flat) cell T , if ξT is the tangent component of ξ on T we have, on T , ∆T uξ |T − iξT · gradT uξ |T = 0. Connecting with the previous example, on a cell T we let βT be the affine real function with zero mean and gradient ξT on T . Define a(u, v) = exp(−iβT )u · v. T

Here, extra care must be taken because this bilinear form is not positive definite and will in fact be degenerate at interior resonances of the cell. Away from them, the locally harmonic forms for the infinite-dimensional ˆ (with q = 2) are well behaved, in the sense of satisfying element system X Proposition 5.26, and contain the plane wave uξ . More generally, suppose one wants to construct an FE system containing a good approximation to a particular solution of (5.11). To the extent that the solution can be locally approximated by a plane wave, the finite element system will contain a good approximation of it, for a choice of a family of functions βT , one for each cell T , to be determined (maybe adaptively).

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5.2. Interpolators Mirrors and interpolators. The notion of a mirror system formalizes that of degrees of freedom, with particular emphasis on their geometric location. Definition 5.33. A mirror system is a choice, for each k and T , of a ˆ k (T ) , called a k-mirror on T . subspace Z k (T ) of X ˆ k (T ) then gives a linear form ·, u on Z k (T ), which Any k-form u in X we call the mirror image of u. For a global k-form u the mirror images can be collected into a single object. We define Φk u = ·, u|T T ∈T ∈ Z k (T ) , where, for any subcomplex T of T , we define a (global) mirror Z k (T ) by Z k (T ) = Z k (T ). T ∈T

We say that a mirror system is faithful to an element system A if, for any subcomplex T , restricting Φ to T determines an isomorphism: Φk (T ) : Ak (T ) → Z k (T ) . Example 5.34. The canonical mirror system for the trimmed polynomial FE system of order p is the following, where dim T = m:

Z k (T ) = {u → T v ∧ u : v ∈ PAm−k p−m+k−1 (T )}. It follows from results in Arnold et al. (2006b) that it is faithful. Proposition 5.35. When A admits extensions, a given mirror system Z is faithful if and only if the duality product on Z k (T ) × Ak0 (T ) is invertible for each k and T . Proof. (i) Suppose Z is faithful. By induction on dimension, dim Z k (T ) = dim Ak0 (T ). Moreover, Φk (T ) induces an injection Ak0 (T ) → Z k (T ) . Thus duality on Z k (T ) × Ak0 (T ) is invertible. (ii) Suppose duality on Z k (T ) × Ak0 (T ) is invertible for all k and T . Then dim Ak (T ) = T ∈T dim Ak0 (T ) = dim Z k (T ) . Moreover, Φk (T ) is injective. Indeed, if Φk u = 0 then for T ∈ T , u|T is proved to be 0 by starting with cells T of dimension k, and incrementing cell dimension inductively using u|∂T = 0. Definition 5.36. For a finite element system A, an interpolator is a colˆ k (T ) → Ak (T ), one for each k ∈ N lection of projection operators I k (T ) : X and T ∈ T , which commute with restrictions to subcells.


85

One can then denote it simply with I • and extend it unambiguously to any subcomplex T of T . Any faithful mirror system defines an interpolator by Φk I k u = Φk u. We call this the interpolator associated with the mirror system. Proposition 5.37.

The following are equivalent.

• A admits extensions. • A has a faithful mirror system. • A can be equipped with an interpolator. Proof. (i) Suppose A admits extensions. For each k and T , choose a closed ˆ k (T ) and let Z k (T ) be its annihilator. The supplementary of Ak0 (T ) in X duality product on Z k (T ) × Ak0 (T ) is then invertible for each k and T , so that Z is faithful to A. (ii) As already stated, any faithful mirror system defines an interpolator. ˆ k (T ) and (iii) Suppose A has an interpolator. If u ∈ Ak (∂T ) extend it in X k interpolate it to get an extension in A (T ). Example 5.38. In particular, the trimmed polynomial FE system of order p is compatible. As remarked in Christiansen (2010), it is minimal among compatible finite element systems containing polynomial differential forms of order p − 1. For a given element system A, admitting extensions, it will be useful to construct extension operators Ak (∂T ) → Ak (T ), i.e., linear left inverses of the restriction. One also remarks that a faithful mirror system determines a particular extension. Namely, to u ∈ Ak (∂T ) one associates the unique v ∈ Ak (T ) extending u and such that, for all l ∈ Z k (T ), l(v) = 0. Proposition 5.39. Let A be an element system on T admitting extensions. Let M denote the set of mirror systems that are faithful to A, let I be the set of interpolators onto A and let E be the set of extensions in A. The natural map M → I × E is bijective. Proof. For a given interpolator I ∈ I and E ∈ M, define a mirror system z(I, E) as follows. For k ∈ N and T ∈ T consider the map ˆ k (T ) → X ˆ k (T ). QkT = (id −E∂) ◦ I : X It is a projector with range Ak0 (T ). Let z(I, E)k (T ) denote the annihilator of its kernel. In other words l ∈ z(I, E)k (T ) if and only if ˆ k (T ) ∀u ∈ X

Iu = EI∂u ⇒ l(u) = 0.

86


We claim that z inverts the given map a : M → I × E. (i) Given (I, E), we check that the interpolator and extension defined by z(I, E) are I and E. ˆ k (T ). We have I(u − Iu) = 0 and EI∂(u − Iu) = 0, hence • Pick u ∈ X l(u − Iu) = 0. The interpolator deduced from z(I, E) is thus I. • Pick u ∈ Ak (∂T ). We have IEu = EI∂Eu, hence for all l ∈ z(I, E)k (T ) we have l(u) = 0. The extension deduced from z(I, E) is thus E. (ii) Given a faithful mirror system Z with associated interpolators I and E, we check that z(I, E) = Z. ˆ k (T ) is such that Iu = EI∂u then l(u) = Pick l ∈ Z k (T ). If u ∈ X l(Iu) = l(EI∂u) = 0. Hence l ∈ z(I, E)k (T ). On the other hand z(I, E) is also faithful to A by Proposition 5.35. The inclusion Z k (T ) ⊆ z(I, E)k (T ) then implies equality. ˆ k (T ) Example 5.40. Let A be a finite element system. Equip each X k with a continuous bilinear form a which is non-degenerate on A0 (T ) (e.g., ˆ k (T ) is continuously embedded in a Hilbert space). One can define a X mirror system by Z k (T ) = {a(·, v) : v ∈ Ak0 (T )}. When the FE system admits extensions, the associated interpolator can be interpreted as a recursive a-projection, starting from cells of minimal dimension, and continuing by incrementing dimension at each step, projecting with respect to a with given boundary conditions. Commuting interpolators. It is of interest to construct interpolators that commute with the exterior derivative. When T ⊆ T , restriction maps ˆ k (T ) → X ˆ k (T ), so that a k-mirror Z k (T ) on T can also be considered X as a k-mirror on T . If the mirror system is faithful it must be in direct sum with Z k (T ). Proposition 5.41. An interpolator commutes with the exterior derivative if and only if its mirror system satisfies ∀T ∈ T ∀l ∈ Z k (T )

l ◦ d ∈ Z k−1 (T˜).

(5.12)

ˆ k−1 (T ). Suppose first that Φk−1 u = 0. Then, Proof. (If ) Pick u ∈ X k ˜ for all l ∈ Z (T ), l(du) = 0, hence Φk (du) = 0. In the general case, since Φk−1 (u − Φk−1 u) = 0, we deduce Φk (du − dΦk−1 u) = 0, so that Φk du = dΦk−1 u. ˆ k−1 (T ) such (Only if ) Suppose l ∈ Z k (T ) and l ◦ d ∈ Z k−1 (T˜). Pick u ∈ X k−1 that l(du) = 0, but for all l ∈ Z (T˜), l (u) = 0. Then I k du = 0 but I k−1 u = 0 (so dI k−1 u = 0).


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Example 5.42. The canonical mirror system of trimmed polynomials of order p yields a commuting interpolator. Suppose that Z is a mirror system on T such that (5.12) holds. Then we ˆ : l → l ◦ d from Z k (T ) to Z k−1 (T ). Denote by δ have a well-defined map d its adjoint, which maps from Z k−1 (T ) to Z k (T ) . Remark 5.43. The following diagram commutes: ˆ k−1 (T ) X

Φk−1

Z k−1 (T ) Proof.

d

δ

/X ˆ k (T )

Φk

/ Z k (T )

ˆ k−1 (T ). We have Pick l ∈ Z k (T ) and u ∈ X ˆ = (Φu)(l ◦ d) = l(du) = (Φdu)(l). (δΦu)(l) = (Φu)(dl)

This concludes the proof. ˆ k (T ) with a continuous scalar product Proposition 5.44. Equip each X a. For any compatible finite element system A, the following is a faithful mirror system yielding a commuting interpolator. For k = dim T ,

Z k (T ) = {a(·, v) : v ∈ dAk−1 0 (T )} + {R ·}. For k < dim T , k Z k (T ) = {a(·, v) : v ∈ dAk−1 0 (T )} + {a(d·, v) : v ∈ dA0 (T )}.

This is the natural generalization, to the adopted setting, of projectionbased interpolation, as defined in Demkowicz and Babuˇska (2003) and Demkowicz and Buffa (2005). When the scalar products a are all the L2 -product on forms, we call it harmonic interpolation. Suppose U is a cellular complex on M and V is a cellular complex on N giving rise to the product complex U × V on M × N . If Z is a mirror system on U and Y is a mirror system on V, then Z ⊗ Y is a mirror system on U × V defined by (Z ⊗ Y)k (U × V ) = Z l (U ) ⊗ Y(V )k−l . l

Proposition 5.45. faithful.

The tensor product of two faithful mirror systems is

Proof. We have three FE systems, A, B and A ⊗ B, all admitting extensions. Moreover, by Proposition 5.19, Al0 (U ) ⊗ B0k−l (V ). (A ⊗ B)k0 (U × V ) = l

88


Now apply Proposition 5.35. Extension–projection interpolators. It will also be of interest to construct extension operators which commute with the exterior derivative. More precisely, for a cell of dimension m a commuting extension operator is a family of operators E k : Ak (∂T ) → Ak (T ) for 0 ≤ k ≤ m − 1, together with a map E m : R → Am (T ) such that the following diagram commutes: 0

0

/R

/ A0 (∂T )

/R

/ ···

/ Am−1 (∂T )

/ ···

E0

/ A0 (T )

E m−1

/ Am−1 (T )

/R

/0

Em

/ Am (T )

/0

(5.13) When has an element with non-zero integral an such that the above diagram commutes, is uniquely determined by E m−1 and exists if and only if E m−1 sends elements with zero integral to closed forms. Am−1 (∂T )

Em

Proposition 5.46. Let A be a finite element system where each Ak (T ) is equipped with a scalar product a. Suppose T is a cell such that (5.4) is exact. Put m = dim T . If T admits extensions, the harmonic extension operators defined by Proposition 5.25 commute in the sense of diagram (5.13). We suppose that for each cell T , extension operators E : A• (∂T ) → A (T ) and projections P : X • (T ) → A• (T ) have been defined. Then E and P uniquely determine an interpolator J as follows. One constructs J inductively. The initialization on cells of dimension 0 (points) is trivial. Now let T be a cell and suppose that J has been constructed for all cells on ˆ k (T ) with k < dim T , its boundary. On T we define, for u ∈ X •

Ju = EJ∂u + (id −E∂)P u = P u + E(J∂u − ∂P u), and for k = dim T we simply put Ju = P u. The only thing to check is that J commutes with restriction from T to cells on the boundary, which is trivial. We call J the associated extension– projection (EP) interpolator. Let T be cell of dimension m. endomorphism F of X m (T )

We say that an m preserves integrals if F u = u for all u ∈ X (T ). An interpolator is said to preserve integrals if it preserves integrals on all cells. For an interpolator I, this is equivalent to commutation of the following diagram, involving

89


de Rham maps: / A• (T ) vv vv v vv ρ zv v

I

ˆ • (T ) X

II II II ρ III $

C • (T )

Furthermore, we would like the interpolator to commute with the exterior derivative. Proposition 5.47. Suppose that the projectors P : X • (T ) → A• (T ) commute with the exterior derivative, that the extensions E commute in the sense of diagram (5.13) and that P preserves integrals. Then the associated EP interpolator commutes with the exterior derivative and preserves integrals. Proof. We use induction on the dimension of cells. Let T be a cell of dimension m. ˆ k (T ) with k ≤ m − 2, the commutation dJu = Jdu follows If u ∈ X immediately from the commutation of P , E and ∂, as well as J on ∂T (which follows from the induction hypothesis). ˆ m−1 (T ) we have For u ∈ X dJu = dP u + dE(J∂u − ∂P u). But we have

(J∂u − ∂P u) =

∂T

∂T

∂u −

∂T

∂u −

dP u = T

∂u −

=

∂T

P du, T

du = 0. T

The first thing we used is that J preserves integrals on the boundary (which is an induction hypothesis). Hence, by (5.13), dE(J∂u − ∂P u) = 0. Thus dJu = dP u = P du = Jdu. That J preserves integrals simply from the fact that, for

on T follows

u ∈ X m (T ), we have Ju = P u = u. Compared with the use of mirror systems, the advantage of defining an interpolator from extensions and projections is that approximation properties of the interpolator follow directly from estimates on the extensions and projections. In the following we denote the Lq (U )-norm simply by · U .

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Proposition 5.48. Let (Tn ) be a sequence of cellular complexes, each ˜ k (T ) equipped with a compatible FE system. Suppose that for each T , X is equipped with a densely defined seminorm |·|T . We also suppose that we have functions λn : Tn → R∗+ and τn : Tn → R∗+ and extensions and projections satisfying u − Pn uT # τn (T )|u|T , u − Pn u∂T # τn (T )λn (T )−1 |u|T , En uT # λn (T )u∂T . We suppose that for T, T ∈ Tn , if T ⊆ T then λn (T ) λn (T ) and τn (T ) τn (T ). Then the associated EP interpolator satisfies u − In uT # τn (T ) λn (T )dim T −dim T |u|T , T ⊆T

where we sum over subcells T of T in Tn . We shall express this bound as order optimality. Proof. We use induction. Suppose T is a cell and that In is order-optimal ˆ k (T ) with k < dim T , we then have on its boundary. For u ∈ X u − In uT # u − Pn uT + λn (T )In ∂u − ∂Pn u∂T # u − Pn uT + λn (T )u − Pn u∂T + λn (T )u − In u∂T # τn (T )|u| + λn (T )u − In u∂T . This completes the proof. This proposition was designed with the p-version of the finite element method in mind. One can think of τn (T ) = p−1 and λn (T ) = p−1/q . The seminorms involved would correspond to Sobolev spaces, possibly weighted. Thus one would require extension operators whose Lq (∂T ) → Lq (T )-norm is of order p−1/q . The construction used to prove Proposition 3.3 in the preprint of Christiansen (2007) might be useful here. 5.3. Quasi-interpolators Let S be a domain in Rd . Let (Tn ) be a sequence of cellular complexes on S, each equipped with a compatible FE system A[n]. We define Xnk = A[n]k (Tn ). Definition 5.49.

Consider a sequence of maps Qkn : Lq (S) → Xnk .

• We say that they are stable if, for u ∈ Lq (S), Qkn uLq (S) # uLq (S) .


91

• We say that they are order-optimal if, for u ∈ W,q (S), u − Qkn uLq (S) # τn uW,q (S) , where τn is the order of best approximation on Xnk in the W,q (S) → Lq (S)-norm. • We say that they are quasi-projections if, for some α < 1, we have for all u ∈ Xnk u − Qkn uLq (S) ≤ αuLq (S) . • We say that they commute if they commute with the exterior derivative. We shall construct stable quasi-interpolators of the form Q = IRE, where I is an interpolator, R is a regularization (smoother) approximating the identity and E is an extension operator (usually R requires values outside S). The following technique will be referred to as scaling. Lemma 5.50. For a cell T of diameter hT and barycentre bT , consider the scaling map σT : x → hT x + bT and let Tˆ be the pre-image of T by σT , called the reference cell. Let m be the dimension of T (0 ≤ m ≤ d). Let u be a k-multilinear form on T and let u ˆ = σT u be the pullback of u to Tˆ. Then we have −k+m/q

uLq (T ) = hT

ˆ uLq (Tˆ) .

We adopt the h-setting, in that the differential elements A[n]k (T ), when pulled back to reference domains Tˆ, belong to compact families: see Arnold et al. (2006b, Remark p. 64). This hypothesis excludes, for instance, methods where the polynomial degree is unbounded as n → ∞, and normally requires the cells to be shape-regular. Proposition 5.51. Suppose the finite elements A[n]k (T ) contain polynomials of degree − 1. A sequence of commuting interpolators In can be constructed to satisfy +(dim T −dim T )/q u − In uLq (T ) # hT ∇ uLq (T ) , (5.14) T ⊆T

where we sum over subcells T of T in Tn . We shall express this bound as order optimality.

92


Proof. Choose commuting interpolators In , which when pulled back to the reference cell satisfy u − Iˆn uLq (Tˆ) # ∇ uLq (Tˆ ) . T ⊆T

Estimate (5.14) follows by scaling. Such interpolators can be constructed from mirror systems (e.g., harmonic interpolation), or from extensions and projections as in Proposition 5.48. Regularizer. We consider regularizing operators constructed as follows. We require a function ψ supported in the unit ball Bd and a function Φ:

Bd × Rd → Rd , (y, x) → Φy (x).

For any given y we denote by Φy the pullback by Φy : Rd → Rd . That is, for any k-multilinear form u, (Φy u)[x](ξ1 , . . . , ξk ) = u[Φy (x)](Dx Φy (x)ξ1 , . . . , Dx Φy (x)ξk ). We define

Ru = Bd

ψ(y)Φy u dy.

(5.15)

For the function ψ we choose one which is smooth, rotationally invariant, non-negative, with support in the unit ball and with integral 1. Later on, we will require convolution by ψ to preserve polynomials up to a certain degree (see (5.24)), but until then this hypothesis will be irrelevant. In what follows, when we integrate with respect to the variable y it will always be on the unit ball Bd , so we omit this from expressions such as (5.15). Given such a ψ we are interested in how properties of Φ reflect upon R. For this purpose we call R defined by (5.15) the regularizer associated with Φ. To emphasize its dependence on Φ we sometimes denote the regularizer by R[Φ]. Formally, R[Φ] will commute with the exterior derivative, since pullbacks do. The minimal regularity we assume of Φ is to have continuous second derivatives. We also suppose that, for given x, y → Φy (x) is a diffeomorphism from Bd to its range. This is enough for the commutation to hold. In what follows we shall be a bit more careful about the regularizing effects of R[Φ] for given Φ. Let B(x, δ) denote the ball with centre x and radius δ. For any subset U of Rd , its δ-neighbourhood is defined by V δ (U ) = ∪{B(x, δ) : x ∈ U }. As a first result we state the following.


93

Proposition 5.52. For any δ > 0 and C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, and Dy Φy (x)−1 , Dx Φy (x) ≤ C, the associated regularizer satisfies (R[Φ]u)(x) ≤ C uL1 (V δ (x)) . Proof.

(5.16)

With x fixed, the Jacobian of the map Bd % y → Φy (x) ∈ V δ (x)

has an inverse bounded by C. We are interested in estimates on derivatives of R[Φ]u when u is a k-form. We have (∇R[Φ]u)(x) = ψ(y)Dx (Φy u)[x] dy, (5.17) where we can substitute the expression Dx Φy u[x](ξ0 , . . . , ξk )

(5.18)

= Du[Φy (x)](Dx Φy (x)ξ0 , . . . , Dx Φy (x)ξk ) + ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2xx Φy (x)(ξ0 , ξi ), . . . , Dx Φy (x)ξk ). The purpose of the following lemma is to get an integral expression for ∇R[Φ]u not involving any derivatives of u. Essentially, in (5.17), derivatives acting on u are transferred to other terms under the integral sign, using integration by parts. Without the integral sign, this corresponds to identifying the derivatives of u as a ‘total’ divergence, up to expressions involving no derivatives of u. Lemma 5.53.

We have

Dx Φy u[x](ξ0 , . . . , ξk ) = ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2xx Φy (x)(ξ0 , ξi ), . . . , Dx Φy (x)ξk ) − ki=1 u[Φy (x)](Dx Φy (x)ξ1 , . . . , D2yx Φy (x)(Ξy (x)ξ0 , ξi ), . . . , Dx Φy (x)ξk ) + Dy Φy u[x](Ξy (x)ξ0 , ξ1 , . . . , ξk ), with Ξy (x) defined by Ξy (x)ξ = Dy Φy (x)−1 Dx Φy (x)ξ.

94


We also have ψ(y)Dy Φy u[x](Ξy (x)ξ0 , ξ1 , . . . , ξk ) = divy ψ(y)Ξy (x)ξ0 Φy u[x](ξ1 , . . . , ξk ) − divy ψ(y)Ξy (x) ξ0 Φy u[x](ξ1 , . . . , ξk ). The following proposition shows that the regularizer maps forms of L1 regularity to continuously differentiable forms, while being careful about the operator norm. Proposition 5.54. For any δ > 0 and C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, and

Dy Φy (x)−1 , Dx Φy (x), D2xx Φy (x)

!

D2yx Φy (x), Dy (Dy Φy (x)−1 )

≤ C,

the associated regularizer satisfies (∇R[Φ]u)(x) ≤ C uL1 (V δ (x)) .

(5.19)

Proof. The preceding lemma gives an expression for (∇R[Φ]u)(x), from which the claim follows. In the estimate (5.19) one has one order more of differentiation on the left-hand side. In the following proposition we consider an equal amount of differentiation on both sides. Proposition 5.55. Pick an integer ≥ 0, and δ > 0. For any C > 0, there exists C > 0 such that, for all Φ satisfying, at a point x, for all y ∈ Bd , |Φy (x) − x| ≤ δ, −1

Dy Φy (x)

≤ C,

(5.20) (5.21)

and Dx Φy (x), D2xx Φy (x), . . . , D+1 x...x Φy (x) ≤ C,

(5.22)

the associated regularizer satisfies (∇ R[Φ]u)(x) ≤ C uW,1 (V δ (x)) . Proof. For = 0 one uses the change of variable formula in the definition (5.15), the Jacobian being taken care of by estimate (5.21). The case = 1 follows from expression (5.18). Differentiating this expression several times gives the claimed result for general (an expression for the differential of order of a pullback will be given later).


95

From now on we suppose Φ has the form Φy (x) = x + φ(x)y,

(5.23)

for some function φ : S → R. Most of the discussion would work with matrix-valued maps φ : S → Rd×d , which could be important for anisotropic meshes. But for simplicity we do not consider anisotropic meshes here, and thus scalar φ are sufficient. We also assume that ψ satisfies ψ(y)f (y) dy = f (0), (5.24) for all polynomials f of degree at most p + d, for some integer p ≥ 0. The purpose of these hypotheses is to make the regularizer preserve polynomials of degree up to p. To see this, first note that property (5.24) guarantees that convolution by ψ preserves polynomials of degree p + d. We state the following. Proposition 5.56. Suppose |φ(x)| ≤ δ for some δ > 0. If u is also a polynomial of degree at most p on V δ (x), then ∇ Ru(x) = ∇ u(x) for all . Proof.

We have Dx Φy (x)ξ = ξ + (Dφ(x)ξ)y.

Suppose u is a k-form which is a polynomial of degree at most p. Recall that (Φy u)[x](ξ1 , . . . , ξk ) = u[Φy (x)](Dx Φy (x)ξ1 , . . . , Dx Φy (x)ξk ). As a function of y this is a polynomial of degree at most p + k ≤ p + d. Its value at y = 0 is u[x](ξ1 , . . . , ξk ). This gives the case = 0 of the proposition. For = 1 one uses expression (5.18). Greater are obtained by further differentiation of this expression. To see to what extent Propositions 5.52, 5.54 and 5.55 can be applied, we remark that Φy (x) − x = φ(x)y, Dy Φy (x)ξ = φ(x)ξ, Dy (Dy Φy (x)−1 ) = 0, Dx Φy (x)ξ = (Dφ(x)ξ)y, D2yx Φy (x)(ξ, ξ ) = (Dφ(x)ξ)ξ , Dx...x Φy (x)(ξ1 , . . . , ξ ) = D φ(x)(ξ1 , . . . , ξ ) y.

96


Proposition 5.57. Pick ≤ p + 1. For any δ > 0 and any C > 0, there exists C > 0 such that, for all φ satisfying, at some point x, |φ(x)| ≤ δ, |φ(x)−1 | ≤ C, and Dx φ(x), D2xx φ(x), . . . , D+1 x...x φ(x) ≤ C, the associated regularizer satisfies (∇ Ru)(x) ≤ C ∇ uL1 (V δ (x)) . Proof. By the Deny–Lions lemma there exists C > 0 such that, for all u ∈ W,1 (V δ (x)), inf u − f W,1 (V δ (x)) # ∇ uL1 (V δ (x)) .

f ∈P−1

The regularizer R preserves the space P−1 of polynomials of degree up to − 1. Combining this with Proposition 5.55 gives the claimed result. The regularizer is adapted to the mesh Tn as follows. We choose φn such that, for all x ∈ T ∈ Tn , φn (x) hT , D

1+r

φn (x) #

h−r T ,

(5.25) for 0 ≤ r ≤ .

(5.26)

We introduce a parameter > 0 and consider the regularizations Rn = R[Φ] associated with the maps Φy (x) = x + φn (x)y. We define Vn (T ) = ∪{B(x, φn (x)) : x ∈ T }. We choose fixed but small. Proposition 5.58.

Fix > 0. For any T ∈ Tn we have an estimate

1+(d−dim T )/q ∇Rn uLq (T ) hT (d−dim T )/q ∇ Rn uLq (T ) hT

# uLq (Vn (T )) ,

(5.27)

# ∇ uLq (Vn (T )) .

(5.28)

For T of maximal dimension we also have u − Rn uLq (T ) # hT ∇ uLq (Vn (T )) .

(5.29)

Proof. Pick T ∈ Tn . Its diameter is hT and its barycentre bT . Consider the scaling map σT : x → hT x + bT and let Tˆ be the pre-image of T by σT , called the reference cell.


97

Note that, quite generally, the regularization R transforms as follows under pullback by a diffeomorphism σ: R[Φ] = (σ )−1 R[σ −1 ◦ Φ• ◦ σ]σ , where σ −1 ◦ Φ• ◦ σ : (y, x) → σ −1 (Φy (σ(x))). In our case the operator Rn [σT−1 ◦ Φ• ◦ σT ] is regularizing on the reference ˆ . We have cell Tˆ and we denote it by R n

(σT−1

◦ Φy ◦ σT )(x) = x + h−1 T φn (hT x + bT )y.

The conditions (5.25), (5.26) put us in a position to conclude from Propositions 5.54 and 5.55 that ˆ n u ∞ ˆ # u 1 −1 ∇R L (σ V (T )) , L (T ) n

T

ˆ n u ∞ ˆ # ∇ u 1 −1 ∇ R L (σ V (T )) . L (T ) n

T

From this we deduce ˆ u q ˆ # u q −1 ∇R n L (σ V (T )) , L (T ) T

n

ˆ n u q ˆ # ∇ u q −1 ∇ R L (σ V (T )) . L (T )

T

n

Then the estimates (5.27) and (5.28) follow from scaling. Let T ∈ Tn have dimension d. From Proposition 5.52, we get ˆ Ru Lq (Tˆ) # uLq (σ −1 V (T )) . T

n

Preservation of polynomials and the Deny–Lions lemma then gives ˆ u − Ru Lq (Tˆ) # ∇ uLq (σ −1 V (T )) . T

n

Scaling then gives (5.29). For a cell T ∈ T we denote by Mn (T ) the macro-element surrounding T in Tn , that is, the union of the cells T ∈ Tn touching T : Mn (T ) = ∪{T ∈ Tn : T ∩ T = ∅}. Choose so small that, for all n and all T ∈ Tn , Vn (T ) ∩ S ⊆ Mn (T ). Proposition 5.59.

For u defined on Vn (S) we have estimates In Rn uLq (S) # uLq (Vn (S)

and, for ≤ p + 1, u − In Rn uLq (S) # hn uLq (Vn (S) .

98


Extension. We shall define extension operators which extend differential forms on S to some neighbourhood of S, preserve polynomials up to a certain degree p, commute with the exterior derivative and are continuous in W,q -norms for ≤ p + 1. For this purpose we will use maps Φs depending on a parameter s, defined outside S with values in S, and pull back by these maps. By taking judiciously chosen linear combinations of such pullbacks we meet the requirements of continuity and polynomial preservation. First we derive some formulas for the derivative of order of the pullback of a differential form. Antisymmetry in the variables will not be important for these considerations, so we consider multilinear rather than differential forms. Consider then a smooth map Φ : Rd → Rd and u, a k-multilinear form on Rd , with k ≥ 1. We want to give an expression for (∇ Φ u)[x](ξ1 , . . . , ξk+ ), as a linear combination of terms of the form (∇r u)[Φ(x)] Dm1 Φ(x)(ζ1 , . . . , ζm1 ), Dm2 Φ(x)(ζm1 +1 , . . . , ζm1 +m2 ), . . . , (5.30) where m1 + · · · + mk+r = k + and (ζ1 , . . . , ζk+ ) = (ξσ(1) , . . . , ξσ(k+) ), for any permutation σ of the indexes {1, . . . , k + }. To handle the combinatorics behind this problem, we let D be the set of pairs (σ, m) where m = (m1 , . . . , mv(m) ) is a multi-index of valence v(m) ≥ 1 and weight w(m) = m1 +· · ·+mv(p) , such that mi ≥ 1 for each 1 ≤ i ≤ v(m), and σ is a permutation of the indexes (1, . . . , w(m)). We define the partial weights |m|0 = 0 and |m|i = m1 + · · · + mi for 1 ≤ i ≤ v(m). We define (σ, m)◦ [Φ, x](ξ1 , . . . , ξw(m) ) to be the v(m)-multivector: v(m)

⊗i=1 Dmi Φ(x)(ζ|m|i−1 +1 , . . . , ζ|m|i ), with (ζ1 , . . . , ζw(m) ) = (ξσ(1) , . . . , ξσ(w(m)) ). Thus (σ, m)◦ [Φ, x] ∈ L(⊗w(m) Rd , ⊗v(m) Rd ).

(5.31)

Let D be the free group generated by D. We define some operations in D, which correspond to differentiating (5.31) with respect to x. For a given (σ, m) we define, for j = 1, . . . , v(m), ∂j (σ, m) to consist of

99


the multi-index (m1 , . . . , mj + 1, . . . , mv(m) ) and the permutation σ(i) if σ(i) ≤ |m|j , σ(i) + 1 if σ(i) > |m|j ,

1 ≤ i ≤ w(m) →

w(m) + 1 → |m|j + 1. Defining

v(m)

D(σ, m) =

∂j (σ, m),

j=1

we have

D((σ, m)◦ [Φ, x]) = (D(σ, m))◦ [Φ, x].

On the left, D is ordinary differentiation with respect to x, and on the right, D is an operation in D. If u is a k-multilinear form and ζ = ζ1 ⊗ . . . ⊗ ζk ∈ ⊗k Rd , we define the contraction u : ζ = u(ζ1 , . . . , ζk ). Contraction is bilinear. Given a k-multilinear form u, we want to differentiate, with respect to x, expressions of the form (∇r u)[Φ(x)] : (σ, m)◦ [Φ, x], where v(m) = k+r. This corresponds to (5.30). Since contraction is bilinear and we know how to differentiate the right-hand side, it remains to differentiate the left-hand side. For this purpose we introduce one more operation in D. Define e such that e(σ, m) consists of the multi-index (m1 , . . . , mv(m) , 1) and the permutation 1 ≤ i ≤ w(m) → σ(i), w(m) + 1 → w(m) + 1. Then we have ∇ (∇r u)[Φ(x)] : (σ, m)◦ [Φ, x] = (∇

r+1

(5.32) ◦

r

◦

u)[Φ(x)] : (e(σ, m)) [Φ, x] + (∇ u)[Φ(x)] : (D(σ, m)) [Φ, x].

In the free group D we now define Γr [k] for ≥ 0 and 0 ≤ r ≤ recursively. We initialize by Γ00 [k] = (id, (1, . . . , 1)), with the identity permutation and k terms in the multi-index. Then we define for ≥ 0 Γ+1 0 [k] = eΓ0 [k], Γ+1 i [k] = eΓi [k] + DΓi−1 [k], for 1 ≤ i ≤ , Γ+1 +1 [k] = DΓ [k].

100


We check that Γ0 [k] = Γ00 [k + ], Γ [k] = D Γ00 [k]. Proposition 5.60.

We have, for a given k-form u,

∇ (Φ u)[x] =

(∇−i u)[Φ(x)] : Γi [k]◦ [Φ, x].

i=0

The expression k + . Proof.

Γi [k]

∈ D is a sum of terms with valence k + − i and weight

We use induction on , using (5.32).

For x outside S, let δ(x) denote the distance from x to S. Given a polynomial degree p, we will construct an extension operator E as a linear combination of pullbacks: E= ψs Φs , with Φs : V (S) \ S → S, s∈I

subject to the following conditions. • The index set I is a finite subset of the interval [2, 3] and the coefficients (ψs )s∈I are chosen such that, for any polynomial f of degree at most p + d, ψs f (s) = f (0). s∈I

• There is a function φ : V (S) \ S → Rd such that, for all s and x, Φs (x) = x + sφ(x),

(5.33)

and, moreover, φ(x) δ(x), D1+r φ(x) # δ(x)−r ,

for 0 ≤ r ≤ p.

• Finally, for s ∈ [2, 3] we should have DΦs (x)−1 # 1, and the Φs should determine diffeomorphisms from V (S) \ S for some > 0, to an interior neighbourhood of ∂S. In order to show that the above list of conditions can be met, we need some results of a general nature that are given in the Appendix. Proposition 5.61.

The conditions listed above can be met.


101

Proof. Let I consist of p + d + 1 points in the interval [2, 3]. Numbers ψs are then determined by solving a linear system with a Vandermonde matrix. If ∂S had been smooth, the orthogonal projection ℘ onto ∂S would be well-defined and smooth on a neighbourhood. Then we could have taken φ(x) = ℘(x) − x. In the following we modify this construction to allow for Lipschitz boundaries. Choose a smooth vector field ν according to Proposition A.3, pointing outwards on ∂S, so that for some > 0 the following map is a Lipschitz isomorphism onto its open range: g:

∂S×] − , [ (z, t)

→ Rd , → z + tν(z).

Define f on V (S) \ S by f (g(z, t)) = z − g(z, t) = −tν(z). The problem with f , to serve as φ in (5.33), is its lack of regularity. From Theorem 2 in Stein (1970, p. 171) we get a regularized distance function δ˜ defined outside S, such that ˜ δ(x) δ(x), and for all r ≥ 1 ˜ Dr δ(x) # δ(x)1−r . We regularize f by a variant of (5.15), (5.23). We put ˜ φ(x) = ψ(y)f (x − δ(x)y) dy, ˜ where the parameter is chosen to satisfy δ(x) ≤ 1/2δ(x), so that f is evaluated far enough from S. For an illustration we refer to Figure 5.1. The conditions are then met. In the following we choose an integer ≤ p. Proposition 5.62.

We have

∇ Eu = ∇ u[Φ2 (x)]

+

ψs (∇ u[Φs (x)] − ∇ u[Φ2 (x)]) : ⊗k+ (id +sDφ(x))

s

+

(5.34a)

1

ψs 0

s i=1

∇ u[tΦs (x) + (1 − t)Φ2 (x)] :

⊗i (s − 2)φ(x) ⊗ Γi [k]◦ [Φs , x]

(1 − t)i−1 dt. (i − 1)!

(5.34b) (5.34c)

(5.34d)

102


n ν Φ2 (x) Φ3 (x)

x s=0

s=1

n

s=2 s=3 S

Figure 5.1. Definition of Φs (x), without smoothing.

Proof.

We have ∇

(Φs u)[x]

= (∇−i u)[Φs (x)] : Γi [k]◦ [Φs , x].

(5.35)

i=0

Concerning Γi [k]◦ [Φs , x], note that DΦs (x) = id +sDφ(x), Dr Φs (x) = sDr φ(x) for r ≥ 2. Therefore Γi [k]◦ [Φs , x] is a polynomial in s of degree at most k + − i. For i = 0 the value in s = 0 is Γ0 [k]◦ [Φs , x]|s=0 = ⊗k+ (id +sDφ(x))|s=0 = ⊗k+ id. For i ≥ 1 we have a sum of terms that are products where at least one derivative of order at least two appears, so the value at s = 0 is 0. In (5.35) the i = 0 term gives rise to ψs ∇ u[Φs (x)] : Γ0 [k]◦ [Φs , x] = ∇ u[Φ2 (x)] s

+

ψs (∇ u[Φs (x)] − ∇ u[Φ2 (x)]) : ⊗k+ (id +sDφ(x)).

s

This corresponds to (5.34a,b).

103


For i ≥ 1, Taylor’s formula with integral remainder gives i−1

(∇−i u)[Φs (x)] =

∇−i+j u[Φ2 (x)] : ⊗j (s − 2)φ(x)

j=0

1

+ 0

1 j!

∇ u[tΦs (x) + (1 − t)Φ2 (x)] : ⊗i (s − 2)φ(x)

(5.36)

(1 − t)i−1 dt. (i − 1)!

When this expression is contracted with Γi [k]◦ [Φs , x], the sum (5.36) consists of polynomials in s with degree j + k + − i ≤ + d, with value 0 at s = 0. Therefore ψs (∇−i u)[Φs (x)] : Γi [k]◦ [Φs , x] s

=

1

ψs 0

s

∇ u[tΦs (x) + (1 − t)Φ2 (x)] :

⊗i (s − 2)φ(x) ⊗ Γi [k]◦ [Φs , x]

(1 − t)i−1 dt. (i − 1)!

Summing over 1 ≤ i ≤ , this corresponds to (5.34c,d). From this formula several conclusions can be drawn. Proposition 5.63. • If u is of class C (S), Eu is of class C (V (S)). • If u is a polynomial of degree at most p, Eu is too. More precisely, if T is a cell touching S, and u is a polynomial on its macro-element, of degree at most p, then Eu is a polynomial of degree at most p on V (T ). • E is bounded W,q (S) → W,q (V (S)). For cells T ∈ Tn touching ∂S, ∇ EuLq (Vn (T )) # ∇ uLq (Mn (T )) . Proof.

For (σ, m) ∈ D, (σ, m)◦ [Φs , x] # δ(x)v(m)−w(m) ,

from which it follows that Γi [k]◦ [Φs , x] # δ(x)−i . This gives ⊗i φ(x) ⊗ Γi [k]◦ [Φs , x] # 1.

(5.37)

We now check continuity properties on ∂S. Suppose x0 ∈ ∂S and x → x0 . Then we have ∇ u[Φs (x)] − ∇ u[Φ2 (x)] → 0,

104


so that (5.34b) converges to 0. If the integrals (5.34c) had been evaluated at ∇ u[Φ2 (x)], their sum over s ∈ I would be 0. An argument similar to the above, combined with (5.37), then shows that (5.34c,d) also converges to 0. We are left with the term on (5.34a), which converges to ∇ u(x0 ). Boundedness properties of E follow from (5.37) and the assumption that the Φs determine diffeomorphisms with uniformly bounded Jacobian determinants. Quasi-interpolator. Putting together the pieces, we get the following. Theorem 5.64. For any > 0, the operators Q n = In Rn E satisfy local estimates, for T ∈ Tnd , Q n uLq (T ) # uLq (Mn (T )) , u − Q n uLq (T ) # hT ∇ uLq (Mn (T )) , as the corresponding global ones: Q n uLq (S) # uLq (S) , u − Q n uLq (S) # h ∇ uLq (S) . Moreover, for any , choosing small enough will yield, for u ∈ Xnk , u − Q n uLq (T ) ≤ uLq (Mn (T )) , u − Q n uLq (S) # uLq (S) . Finally, Q n commutes with the exterior derivative (when it is in Lq (S)). When is chosen so small that (id −Q n )|Xnk Lq (S)→Lq (S) ≤ 1/2, Q n |Xnk is invertible with norm less than 2. We define operators: Pn = (Q n |Xnk )−1 Q n . Proposition 5.65. The operators Pn are Lq (S)-stable projections onto Xnk which commute with the exterior derivative. The case q = 2 leads to eigenvalue convergence for the operator d d discretized by the Galerkin method on Xnk and therefore for the Hodge– Laplacian in mixed form. For a discussion of eigenvalue convergence we refer to Boffi (2010), Arnold et al. (2010) and Christiansen and Winther (2010). 5.4. Sobolev injection and translation estimate In this subsection we prove a Sobolev injection theorem generalizing the one we introduced in Christiansen and Scheid (2011). The proof technique is slightly different, and we generalize to differential forms in all dimensions. We also prove a translation estimate of the type introduced in Karlsen and


105

Karper (2010, Theorem A.1). Compared with that paper we get an optimal bound and a generalization to all known mixed finite elements in the hversion. Given a cellular complex T , we define the broken H1 -seminorm |·| by 2 |u|2 = ∇u2L2 (T ) + h−1 (5.38) T [u]T L2 (T ) . T ∈T d

T ∈T d−1

Given a simplex T ∈ T d−1 , [u]T denotes the jump of u across T . The scaling factor in front of the jump terms is chosen so that the two terms that are summed scale in the same way: see Lemma 5.50. We assume we have a sequence of cellular complexes (Tn ) and that each Tn is equipped with a compatible FE system A[n]. We also assume that they are of the type discussed in Section 5.3 so that those constructions apply. We define Xnk = A[n]k (Tn ). The broken seminorm (5.38) will be defined relative to one of the Tn and, since the particular Tn should be clear from the context, we omit dependence on Tn from our notation. In the L2 (S) case define X k = {u ∈ L2 (S) : du ∈ L2 (S)}, W k = {u ∈ X k : du = 0}, V k = {u ∈ X k : ∀w ∈ W k

u · w = 0}.

Define Wnk = {u ∈ Xnk : du = 0}, Vnk = {u ∈ Xnk : ∀w ∈ Wnk

u · w = 0}. k

Denote by H the L2 -orthogonal projection onto V , the completion of V k in L2 (S). This operator realizes a Hodge decomposition of u in the form u = (u − Hu) + Hu. The following well-known trick is also useful in the proof of eigenvalue convergence. Proposition 5.66.

We have, for u ∈ Vnk ,

u − HuL2 (S) ≤ Hu − Pn HuL2 (S) . Proof.

We have u − Pn Hu = Pn (u − Hu) ∈ Wnk .

Now write Hu − Pn Hu = (u − Pn Hu) − (u − Hu), and note that the two terms on the left-hand side are orthogonal.

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Proposition 5.67.

For u ∈ Xnk , |u| ∇Rn EuL2 (S) ,

u − Proof.

Rn EuL2 (S)

# h|u|.

(5.39) (5.40)

The parameter is chosen so small that, for all u ∈ Xnk , |u − Rn Eu| ≤ 1/2|u|.

Then we have |u| ≤ 2|Rn Eu|, |Rn Eu| ≤ 3/2|u|. Since Rn Eu is smooth, we have |Rn Eu| = ∇Rn EuL2 (S) . This gives (5.39). The other estimate is proved locally by scaling from a reference macroelement. Proposition 5.68.

For u ∈ H1 (S), |Pn u| # ∇uL2 (S) .

Proof.

By restriction to reference simplexes and scaling, |Qn u| # ∇uL2 (S) .

Choosing small enough, we also have, for u ∈ Xnk , |u − Qn u| ≤ 1/2|u|. Combining the two, we get the proposition. Proposition 5.69. Suppose S is convex and that the meshes are quasiuniform. Then, for all u ∈ Vnk , |u| # duL2 (S) . Proof.

We have, for u ∈ Vnk , |u| = |Pn u| ≤ |Pn (u − Hu)| + |Pn Hu| ≤ h−1 Pn (u − Hu)L2 + ∇HuL2 # h−1 u − HuL2 + ∇HuL2 # ∇HuL2 .

From this (5.41) follows.

(5.41)

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Proposition 5.70. Let q be the relevant Sobolev exponent, so that H1 (S) is contained in Lq (S). For all u ∈ Xnk , uLq (S) # |u| + uL2 (S) . Proof.

We have uLq (S) # Rn EuLq (S) # ∇Rn EuL2 (S) + Rn EuL2 (S) # |u| + uL2 (S) ,

as claimed. Denote by τy the translation by the vector y, so that when u is defined in x − y we have (τy u)(x) = u(x − y). Proposition 5.71.

For all u ∈ Xnk ,

u − τy EuL2 (S) # (|y| + h1/2 |y|1/2 )|u|. On a reference simplex Tˆ we can write for u ∈ Xnk pulled back: ˆ2L2 (Tˆ) # |ˆ y |2 ∇ˆ u2L2 (T ) + |ˆ y| [ˆ u]T 2L2 (T ) . ˆ u − τyû

Proof.

T ∈Mn (Tˆ)d

Scaling back to T of size h, we get, with y = hˆ y, ∇u2L2 (T ) + |y| u − τy u2L2 (T ) # |y|2 T ∈Mn (T )d

T ∈Mn (Tˆ)d−1

[u]T 2L2 (T ) ,

T ∈Mn (T )d−1

so that u − τy u2L2 (T ) # (|y|2 + h|y|)|u|2Mn (T ) . This estimate comes with a restriction of the type |y| ≤ h/C, ensuring that one does not translate T out of its associated macro-element. For |y| ≥ h/C we can write u − τy Eu = (u − Rn Eu) + (Rn Eu − τy ERn Eu) + τy E(Rn Eu − u). From this we deduce u − τy EuL2 # u − Rn EuL2 + Rn Eu − τy ERn EuL2 # h|u| + |y|∇Rn EuL2 # |y| |u|. This concludes the proof.

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Acknowledgements First author. On finite elements, I have benefited from the insights of, in particular, Annalisa Buffa, Jean-Claude Nédélec and Ragnar Winther. On algebraic topology I am grateful for the help of John Rognes and Bjørn Jahren, particularly with Propositions 5.16 and 5.21. I am also grateful to Martin Costabel for a helpful introduction to Stein’s construction of universal extension operators. Example 5.30 is from a discussion with Thomas Dubos (shallow water on the sphere) in June 2009. Trygve Karper’s remarks on upwinding were also helpful. Second author. I would, in particular, like to thank Brett Ryland for his important contribution to the development of the theory of generalized Chebyshev polynomials and also for doing a major part of the computer implementations. Furthermore, I thank Morten Nome for his efforts in realizing the spectral element implementations. I am also grateful to Daan Huybrechs for important contributions to the understanding of generalized symmetries and Chebyshev polynomials of ‘other kinds’. Finally, thanks to Krister ˚ Ahlander for his contributions to developing both the theory and the computer codes for numerical treatment of equivariance. Third author. Morten Dahlby and Takaharu Yaguchi have contributed substantially to the section on integral-preserving methods. Elena Celledoni has proofread parts of this paper and made numerous suggestions for improvement. This work, conducted as part of the award ‘Numerical Analysis and Simulations of Geometric Wave Equations’, made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program. The work is also supported by the Norwegian Research Council through the project ‘Structure Preserving Algorithms for Differential Equations: Applications, Computation and Education’ (SpadeACE).

Appendix Lemma A.1. In a Banach space E, let U be an open set and let f : U → E be a contraction mapping, i.e., for some δ < 1, f (x) − f (y) ≤ δx − y.

(A.1)

Then the map g : U → E, x → x+f (x) has an open range V and determines a Lipschitz bijection U → V , with a Lipschitz inverse. Proof. Suppose that g(x0 ) = y0 . For y − y0 ≤ find the solution x of g(x) = y as a fixed point of the map z → y − f (z). More precisely, construct


109

a sequence starting at x0 and defined by xn+1 = y − f (xn ). Then, as long as it is defined (in U ), we have xn+1 − xn ≤ δxn − xn−1 ≤ δ n−1 . If is chosen so small that the closed ball with centre x0 and radius (1−δ)−1 is included in U , the sequence is defined for all n, and converges to a limit x ∈ U solving g(x) = y. It follows that g is an open mapping. In particular, the range V is open. Moreover, g(x) − g(y) ≥ (1 − δ)x − y. This gives injectivity, so that g : U → V is bijective. The inverse is Lipschitz with constant no worse than (1 − δ)−1 . Lemma A.2. In some Euclidean space E, let S be a bounded domain whose boundary ∂S is locally the graph of a Lipschitz function. Let n be the outward-pointing normal on ∂S. Suppose m is a unit vector, that x0 ∈ ∂S, and that for x in a neighbourhood of x0 in ∂S we have n(x) · m ≥ , for some > 0. Then there is a neighbourhood of x0 in ∂S which is a Lipschitz graph above the plane orthogonal to m. Proof. We know that for a certain outward-pointing unit vector m0 , a neighbourhood U0 of x0 is a Lipschitz graph above an open ball B0 in m⊥ 0. Choose θ ∈ [0, π/2[ such that, for x ∈ U0 , n(x) · m0 ≥ cos θ, and moreover ≥ cos θ. Let f : B0 → R be the function such that U1 is the range of B0 y

→ E, → y + f (y)m0 .

Since, for y ∈ B0 , n(y + f (y)m0 ) = (m0 − grad f (y))/(1 + | grad f (y)|2 )1/2 , we get | grad f (x)| ≤ tan θ. Choose y, y ∈ B0 and put x = y + f (y)m0 and x = y + f (y )m0 . For s, s ∈ R we have |(x + sm0 ) − (x + s m0 )|2 = |y − y |2 + (f (y) − f (y ))2 + 2(f (y) − f (y ))(s − s ) + (s − s )2 .

110


Then note that, for M > 0, 2(f (y) − f (y ))(s − s ) ≤ (1 + M −2 )(f (y) − f (y ))2 + (1 + M −2 )−1 (s − s )2 ≤ (f (y) − f (y ))2 + M −2 tan2 θ|y − y |2 + (1 + M −2 )−1 (s − s )2 . In particular, with M = tan θ we get |(x + sm0 ) − (x + s m0 )|2 ≥ (1 − (1 + tan−2 θ)−1 )(s − s )2 , which simplifies to |(x + sm0 ) − (x + s m0 )| ≥ cos θ|s − s |. Define the function g0 :

U0 × R x

→ E, → y + sm0 .

(A.2)

Its range is B0 + Rm0 , and it is bijective onto it. Consider now a unit vector m1 such that n(x) · m1 ≥ cos θ for x ∈ U0 . Define g1 as in (A.2), replacing m0 by m1 . We shall show that g1 is open and bi-Lipschitz, when |m1 − m0 | < cos θ. Define f1 on B0 + Rm0 , by f1 (x) = g1 ◦ g0−1 − x. With the preceding notation, we have f1 (x + sm0 ) − f1 (x + s m0 ) = (s − s )(m1 − m0 ), and hence |f1 (x + sm0 ) − f1 (x + s m0 )| ≤ |m1 − m0 |/ cos θ|(x + sm0 ) − (x + s m0 )|. We then apply Lemma A.1 and deduce that g1 is open and bi-Lipschitz. It follows that there is a ball B1 in m⊥ 1 above which a neighbourhood U1 ⊆ U0 of x0 is a graph. We may repeat the above considerations to construct a sequence of such vectors m1 , m2 , . . . , mk reaching m in a finite number of steps. Proposition A.3. In some Euclidean space E, let S be a bounded domain whose boundary ∂S is locally the graph of a Lipschitz function. Then there exists a smooth vector field ν on E, of unit length and outward-pointing on ∂S, such that, for some > 0, the map ∂S×] − , [ (x, s)

→ E, → x + sν(x)

has open range and determines a Lipschitz isomorphism onto it.

(A.3)


111

Proof. Cover ∂S with a finite number of orthogonal cylinders Ci directed along a unit vector ni pointing out of S, and with a base Ui , such that above Ui , ∂S is the graph of a Lipschitz function. Let n denote the outwardpointing normal on ∂S. For some θ ∈ [0, π/2[, we have for all i and all x ∈ ∂S ∩ Ci ni · n(x) ≥ cos(θ). (A.4) Choose smooth functions αi on E whose restrictions to ∂S form a partition of unity. Define ν˜(x) = αi (x)ni , i

and normalize by putting ν(x) = ν˜(x)/|ν(x)|. We have ν(x) · n(x) ≥ cos(θ), for all x ∈ ∂S. Denote by g the function g:

∂S × R (x, s)

→ E, → x + sν(x).

Pick x0 ∈ ∂S and put m0 = ν(x0 ). By Lemma A.2, ∂S is locally a Lipschitz graph above the plane orthogonal to m0 . Let U (x0 ) be the corresponding neighbourhood of x0 in ∂S. Denote by g0 the function g0 :

∂S ∩ U (x0 ) × R (x, s)

→ E, → x + sm0 .

It is a Lipschitz isomorphism onto its range, which is open in E. Define f by f (x) = g ◦ g0−1 (x) − x. We have f (x + sm0 ) = s(ν(x) − m0 ), so that f (x + sm0 ) − f (y + tm0 ) = (s − t)(ν(x) − m0 ) + t(ν(x) − ν(y)). It follows that, for a small enough and possibly reducing U(x0 ), f is short on g0 (U(x0 )×] − , [). By Lemma A.1 it follows that g ◦ g0−1 restricted to g0 (U (x0 )×] − , [), for some small enough , called (x0 ), has open range and determines a Lipschitz isomorphism onto it. Hence g restricted to U (x0 )×] − (x0 ), (x0 )[ has open range and determines a Lipschitz isomorphism onto it. In particular, g is an open mapping. The open subsets U(x0 ) associated with each x0 ∈ ∂S cover ∂S. Choose a finite subset F of ∂S such that the sets U (x) for x ∈ F cover ∂S. Pick µ > 0 such that if |x − x | ≤ µ they belong to a common U (x) for x ∈ F. In (A.3), choose smaller than each (x) for x ∈ F, and also smaller than

112


µ/3. Now pick two points x, x ∈ ∂S, and s, s in ] − , [. If |x − x | ≤ µ, they belong to a common U (x), x ∈ F. If, on the other hand, |x − x | ≥ µ, we have |g(x, s) − g(x , s )| ≥ |x − x | − |s| − |s |, ≥ µ/3. Based on these two cases we may conclude that, for some global m, |g(x, s) − g(x , s )| ≥ m(|x − x |2 + |s − s |2 ). The lemma follows.

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Acta Numerica (2011), pp. 121–209 doi:10.1017/S0962492911000031


Mathematical and computational methods for semiclassical Schr¨ odinger equations∗ Shi Jin† Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: [email protected]

Peter Markowich‡ Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK E-mail: [email protected]

Christof Sparber§ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607, USA E-mail: [email protected] We consider time-dependent (linear and nonlinear) Schr¨ odinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schr¨ odinger equation in the semiclassical regime. ∗ †

‡ §

Colour online available at journals.cambridge.org/anu. Partially supported by NSF grant no. DMS-0608720, NSF FRG grant DMS-0757285, a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the University of Wisconsin–Madison. Supported by a Royal Society Wolfson Research Merit Award and by KAUST through a Investigator Award KUK-I1-007-43. Partially supported by the Royal Society through a University Research Fellowship.

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S. Jin and P. Markowich and C. Sparber

CONTENTS 1 Introduction 2 WKB analysis for semiclassical Schr¨ odinger equations 3 Wigner transforms and Wigner measures 4 Finite difference methods for semiclassical Schrödinger equations 5 Time-splitting spectral methods for semiclassical Schrödinger equations 6 Moment closure methods 7 Level set methods 8 Gaussian beam methods: Lagrangian approach 9 Gaussian beam methods: Eulerian approach 10 Asymptotic methods for discontinuous potentials 11 Schrödinger equations with matrix-valued potentials and surface hopping 12 Schr¨ odinger equations with periodic potentials 13 Numerical methods for Schrödinger equations with periodic potentials 14 Schrödinger equations with random potentials 15 Nonlinear Schrödinger equations in the semiclassical regime References

122 125 130 134 140 146 151 155 160 167 173 178 183 190 193 200

1. Introduction This goal of this article is to give an overview of the currently available numerical methods used in the study of highly oscillatory partial differential equations (PDEs) of Schr¨ odinger type. This type of equation forms a canonical class of (nonlinear) dispersive PDEs, i.e., equations in which waves of different frequency travel with different speed. The accurate and efficient numerical computation of such equations usually requires a lot of analytical insight, and this applies in particular to the regime of high frequencies. The following equation can be seen as a paradigm for the PDEs under consideration: iε∂t uε = −

ε2 ∆uε + V (x)uε , 2

uε (0, x) = uεin (x),

(1.1)

for (t, x) ∈ R × Rd , with d ∈ N denoting the space dimension. In addition, ε ∈ (0, 1] denotes the small semiclassical parameter (the scaled Planck’s constant), describing the microscopic/macroscopic scale ratio. Here, we have already rescaled all physical parameters, such that only one dimensionless

¨ dinger equations Methods for semiclassical Schro

123

parameter ε 1 remains. The unknown uε = uε (t, x) ∈ C is the quantum mechanical wave function whose dynamics is governed by a static potential function V = V (x) ∈ R (time-dependent potentials V (t, x) can usually also be taken into account without requiring too much extra work, but for the sake of simplicity we shall not do so here). In this article, several different classes of potentials, e.g., smooth, discontinuous, periodic, random, will be discussed, each of which requires a different numerical strategy. In addition, possible nonlinear effects can be taken into account (as we shall do in Section 15) by considering nonlinear potentials V = f (|uε |2 ). In the absence of V (x) a particular solution to the Schrödinger equation is given by a single plane wave, i t 2 ε , u (t, x) = exp ξ · x − |ξ| ε 2 for any given wave vector ξ ∈ Rd . We see that uε features oscillations with frequency 1/ε in space and time, which inhibit strong convergence of the wave function in the classical limit ε → 0+ . In addition, these oscillations pose a huge challenge for numerical computations of (2.1); in particular, they strain computational resources when run-of-the-mill numerical techniques are applied in order to numerically solve (1.1) in the semiclassical regime ε 1. For the linear Schrödinger equation, classical numerical analysis methods (such as the stability–consistency concept) are sufficient to derive meshing strategies for discretizations (say, of finite difference, finite element or even time-splitting spectral type) which guarantee (locally) strong convergence of the discrete wave functions when the semiclassical parameter ε is fixed: see Chan, Lee and Shen (1986), Chan and Shen (1987), Wu (1996) and Dörfler (1998); extensions to nonlinear Schrödinger equations can be found in, e.g., Delfour, Fortin and Payre (1981), Taha and Ablowitz (1984) and Pathria and Morris (1990). However, the classical numerical analysis strategies cannot be employed to investigate uniform in ε properties of discretization schemes in the semiclassical limit regime. As we shall detail in Section 4, even seemingly reasonable, i.e., stable and consistent, discretization schemes, which are heavily used in many practical application areas of Schrödinger-type equations, require huge computational resources in order to give accurate physical observables for ε 1. The situation gets even worse when an accurate resolution of uε itself is required. To this end, we remark that time-splitting spectral methods tend to behave better than finite difference/finite element methods, as we shall see in more detail in Section 5. In summary, there is clearly a big risk in using classical discretization techniques for Schr¨ odinger calculations in the semiclassical regime. Certain schemes produce completely incorrect observables under seemingly reasonable meshing strategies, i.e., an asymptotic resolution of the oscillation is

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not always enough. Even worse, in these cases there is no warning from the scheme (such as destabilization) that something has gone wrong in the computation (since local error control is computationally not feasible in the semiclassical regime). The only safety anchor here lies in asymptotic mathematical analysis, such as WKB analysis, and/or a physical insight into the problem. They typically yield a (rigorous) asymptotic description of uε for small ε 1, which can consequently be implemented on a numerical level, providing an asymptotic numerical scheme for the problem at hand. In this work, we shall discuss several asymptotic schemes, depending on the particular type of potentials V considered. While one can not expect to be able to pass to the classical limit directly in the solution uε of (1.1), one should note that densities of physical observables, which are the quantities most interesting in practical applications, are typically better behaved as ε → 0, since they are quadratic in the wave function (see Section 2.1 below). However, weak convergence of uε as ε → 0 is not sufficient for passing to the limit in the observable densities (since weak convergence does not commute with nonlinear operations). This makes the analysis of the semiclassical limit a mathematically highly complex issue. Recently, much progress has been made in this area, particularly by using tools from micro-local analysis, such as H-measures (Tartar 1990) and Wigner measures (Lions and Paul 1993, Markowich and Mauser 1993, Gérard, Markowich, Mauser and Poupaud 1997). These techniques go far beyond classical WKB methods, since the latter suffers from the appearance of caustics: see, e.g., Sparber, Markowich and Mauser (2003) for a recent comparison of the two methods. In contrast, Wigner measure techniques reveal a kinetic equation in phase space, whose solution, the so-called Wigner measure associated to uε , does not exhibit caustics (see Section 3 for more details). A word of caution is in order. First, a reconstruction of the asymptotic description for uε itself (for ε 1) is in general not straightforward, since, typically, some phase information is lost when passing to the Wigner picture. Second, phase space techniques have proved to be very powerful in the linear case and in certain weakly nonlinear regimes, but they have not yet shown much strength when applied to nonlinear Schrödinger equations in the regime of supercritical geometric optics (see Section 15.2). There, classical WKB analysis (and in some special cases techniques for fully integrable systems) still prevails. The main mathematical reason for this is that the initial value problem for the linear Schrödinger equation propagates only one ε-scale of oscillations, provided the initial datum in itself is ε-oscillatory (as is always assumed in WKB analysis). New (spatial) frequencies ξ may be generated during the time evolution (typically at caustics) but no new scales of oscillations will arise in the linear case. For nonlinear Schrödinger problems this is different, as new oscillation scales may be generated through the

125


nonlinear interaction of the solution with itself. Further, one should note that this important analytical distinction, i.e., no generation of new scales but possible generation of new frequencies (in the linear case), may not be relevant on the numerical level, since, say, 100ε is analytically just a new frequency, but numerically a new scale. Aside from semiclassical situations, modern research in the numerical solution of Schrödinger-type equations goes in a variety of directions, of which the most important are as follows. (i) Stationary problems stemming from, e.g., material sciences. We mention band diagram computations (to be touched upon below in Sections 12 and 13) and density functional theory for approximating the full microscopic Hamiltonian (not to be discussed in this paper). The main difference between stationary and time-dependent semiclassical problems is given by the fact that in the former situation the spatial frequency is fixed, whereas in the latter (as mentioned earlier) new frequencies may arise over the course of time. (ii) Large space dimensions d 1, arising, for example, when the number of particles N 1, since the quantum mechanical Hilbert space for N indistinguishable particles (without spin) is given by L2 (R3N ). This is extremely important in quantum chemistry simulations of atomistic/molecular applications. Totally different analytical and numerical techniques need to be used and we shall not elaborate on these issues in this paper. We only remark that if some of the particles are very heavy and can thus be treated classically (invoking the so-called Born–Oppenheimer approximation: see Section 11), a combination of numerical methods for both d 1 and ε 1 has to be used.

2. WKB analysis for semiclassical Schr¨ odinger equations 2.1. Basic existence results and physical observables We recall the basic existence theory for linear Schr¨ odinger equations of the form ε2 iε∂t uε = − ∆uε + V (x)uε , uε (0, x) = uεin (x). (2.1) 2 For the sake of simplicity we assume the (real-valued) potential V = V (x) to be continuous and bounded, i.e., V ∈ C(Rd ; R) :

|V (x)| ≤ K.

The Kato–Rellich theorem then ensures that the Hamiltonian operator H ε := −

ε2 ∆ + V (x) 2

(2.2)

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is essentially self-adjoint on D(−∆) = C0∞ ⊂ L2 (Rd ; C) and bounded from below by −K: see, e.g., Reed and Simon (1975). Its unique self-adjoint extension (to be denoted by the same symbol) therefore generates a strongly ε continuous semi-group U ε (t) = e− itH /ε on L2 (Rd ), which ensures the global existence of a unique (mild) solution uε (t) = U ε (t)uin of the Schrödinger equation (2.1). Moreover, since U ε (t) is unitary, we have uε (t, ·)2L2 = uεin 2L2 ,

∀ t ∈ R.

In quantum mechanics this is interpreted as conservation of mass. In addition, we also have conservation of the total energy ε2 ε ε 2 |∇u (t, x)| dx + V (x)|uε (t, x)|2 dx, (2.3) E[u (t)] = 2 Rd d R which is the sum of the kinetic and the potential energies. In general, expectation values of physical observables are computed via quadratic functionals of uε . To this end, denote by aW (x, εDx ) the operator corresponding to a classical (phase space) observable a ∈ Cb∞ (Rd × Rd ), obtained via Weyl quantization, x+y 1 W , εξ f (y) e i(x−y)·ξ dξ dy, (2.4) a a (x, εDx )f (x) := (2π)m 2 Rd ×Rd where εDx := −iε∂x . Then the expectation value of a in the state uε at time t ∈ R is given by a[uε (t)] = uε (t), aW (x, εDx )uε (t) L2 ,

(2.5)

where ·, · L2 denotes the usual scalar product on L2 (Rd ; C). Remark 2.1. The convenience of the Weyl calculus lies in the fact that an (essentially) self-adjoint Weyl operator aW (x, εDx ) has a real-valued symbol a(x, ξ): see Hörmander (1985). The quantum mechanical wave function uε can therefore be considered only an auxiliary quantity, whereas (real-valued) quadratic quantities of uε yield probability densities for the respective physical observables. The most basic quadratic quantities are the particle density ρε (t, x) := |uε (t, x)|2 ,

(2.6)

and the current density

j ε (t, x) := ε Im uε (t, x)∇uε (t, x) .

(2.7)

It is easily seen that if uε solves (2.1), then the following conservation law holds: ∂t ρε + div j ε = 0.

(2.8)


127

In view of (2.3) we can also define the energy density 1 eε (t, x) := |ε∇uε (t, x)|2 + V (x)ρε (t, x). (2.9) 2 As will be seen (see Section 5), computing these observable densities numerically is usually less cumbersome than computing the actual wave function uε accurately. From the analytical point of view, however, we face the problem that the classical limit ε → 0 can only be regarded as a weak limit (in a suitable topology), due to the oscillatory nature of uε . Quadratic operations defining densities of physical observables do not, in general, commute with weak limits, and hence it remains a challenging task to identify the (weak) limits of certain physical observables, or densities, respectively. 2.2. Asymptotic description of high frequencies In order to gain a better understanding of the oscillatory structure of uε we invoke the following WKB approximation (see Carles (2008) and the references given therein): ε→0

uε (t, x) ∼ aε (t, x) e iS(t,x)/ε ,

(2.10)

with real-valued phase S and (possibly) complex-valued amplitude aε , satisfying the asymptotic expansion ε→0

aε ∼ a + εa1 + ε2 a2 + · · · .

(2.11)

Plugging the ansatz (2.10) into (2.1), one can determine an approximate solution to (2.1) by subsequently solving the equations obtained in each order of ε. To leading order, i.e., terms of order O(1), one obtains a Hamilton–Jacobi equation for the phase function S: 1 ∂t S + |∇S|2 + V (x) = 0, S(0, x) = Sin (x). (2.12) 2 This equation can be solved by the method of characteristics, provided V (x) is sufficiently smooth, say V ∈ C 2 (Rd ). The characteristic flow is given by the following Hamiltonian system of ordinary differential equations: x(t, ˙ y) = ξ(t, y), ˙ y) = −∇x V (x(t, y)), ξ(t,

x(0, y) = y, ξ(0, y) = ∇Sin (y).

(2.13)

Remark 2.2. The characteristic trajectories y → x(t, y) obtained via (2.13) are usually interpreted as the rays of geometric optics. The WKB approximation considered here is therefore also regarded as the geometric optics limit of the wave field uε .

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Figure 2.1. Caustics generated from initial data ∂x Sin (x) = − sin(πx)| sin(πx)|p−1 : (a) p = 1, and the solution becomes triple-valued; (b) p = 2, and we exhibit single-, triple- and quintuple-valued solutions.

By the Cauchy–Lipschitz theorem, this system of ordinary differential equations can be solved at least locally in time, and consequently yields the phase function t 1 |∇S(τ, y(t, x))|2 − V (τ, y(t, x)) dτ, S(t, x) = S(0, x) + 0 2 where y(τ, x) denotes the inversion of the characteristic flow Xt : y → x(t, y). This yields a smooth phase function S ∈ C ∞ ([−T, T ] × Rd ) up to some time T > 0 but possibly very small. The latter is due to the fact that, in general, characteristics will cross at some finite time |T | < ∞, in which case the flow map Xt : Rd → Rd is no longer one-to-one. The set of points at which Xt ceases to be a diffeomorphism is usually called a caustic set. See Figure 2.1 (taken from Gosse, Jin and Li (2003)) for examples of caustic formulation. Ignoring the problem of caustics for a moment, one can proceed with our asymptotic expansion and obtain at order O(ε) the following transport equation for the leading-order amplitude: a ∂t a + ∇S · ∇a + ∆S = 0, a(0, x) = ain (x). (2.14) 2 In terms of the leading-order particle density ρ := |a|2 , this reads ∂t ρ + div(ρ∇S) = 0,

(2.15)

which is reminiscent of the conservation law (2.8). The transport equation (2.14) is again solved by the methods of characteristics (as long as S is smooth, i.e., before caustics) and yields ain (y(t, x)) a(t, x) = , Jt (y(t, x))

|t| ≤ T,

(2.16)


129

where Jt (y) := det ∇y x(t, u) denotes the Jacobi determinant of the Hamiltonian flow. All higher-order amplitudes an are then found to be solutions of inhomogeneous transport equations of the form an ∂t an + ∇S · ∇an + ∆S = ∆an−1 . (2.17) 2 These equations are consequently solved by the method of characteristics. At least locally in time (before caustics), this yields an approximate solution of WKB type, uεapp (t, x) = e iS(t,x)/ε a(t, x) + εa1 (t, x) + ε2 a2 (t, x) + · · · , including amplitudes (an )N n=1 up to some order N ∈ N. It is then straightforward to prove the following stability result. Theorem 2.3. form,

Assume that the initial data of (2.1) are given in WKB uεin (x) = ain (x) e iSin (x)/ε ,

C ∞ (Rd ),

(2.18)

S(Rd ),

and let ain ∈ i.e., smooth and rapidly decaying. with Sin ∈ Then, for any closed time interval I ⊂ T , before caustic onset, there exists a C > 0, independent of ε ∈ (0, 1], such that sup uε (t) − uεapp (t)L2 ∩L∞ ≤ CεN . t∈I

The first rigorous result of this type goes back to Lax (1957). Its main drawback is the fact that the WKB solution breaks down at caustics, where S develops singularities. In addition, the leading-order amplitude a blows up in L∞ (Rd ), in view of (2.16) and the fact that limt→T Jt (y) = 0. Of course, these problems are not present in the exact solution uε but are merely an artifact of the WKB ansatz (2.10). Caustics therefore indicate the appearance of new ε-oscillatory scales within uε , which are not captured by the simple oscillatory ansatz (2.10). 2.3. Beyond caustics At least locally away from caustics, though, the solution can always be described by a superposition of WKB waves. This can be seen rather easily in the case of free dynamics where V (x) = 0. The corresponding solution of the Schrödinger equation (2.1) with WKB initial data is then explicitly given by 1 uε (t, x) = ain (y) e iϕ(x,y,ξ,t)/ε dy dξ, (2.19) (2πε)d d d R ×R with phase function t ϕ(x, y, ξ, t) := (x − y) · ξ + |ξ|2 + Sin (y). 2

(2.20)

130


The representation formula (2.19) comprises an oscillatory integral, whose main contributions stem from stationary phase points at which ∂y,ξ ϕ(x, t) = 0. In view of (2.20) this yields ξ = ∇S,

y = x − tξ.

The corresponding map y → x(t, y) is the characteristic flow of the free Hamilton–Jacobi equation 1 ∂t S + |∇S|2 = 0, S(0, x) = Sin (x). 2 Inverting the relation y → x(t, y) yields the required stationary phase points {yj (t, x)}j∈N ∈ Rd for the integral (2.19). Assuming for simplicity that there are only finitely many such points, then 1 ε ain (y) eiϕ(x,y,ξ,t)/ε dy dξ u (t, x) = (2πε)d d d R ×R (2.21) J ain (yj (t, x)) iS(yj (t,x))/ε+ iπmj /4 ε→0 e ∼ , Jt (yj (t, x)) j=1 with constant phase shifts mj = sgn D2 S(yj (t, x))) ∈ N (usually referred to as the Keller–Maslov index). The right-hand side of this expression is usually referred to as multiphase WKB approximation. The latter can be interpreted as an asymptotic description of interfering wave trains in uε . Remark 2.4. The case of non-vanishing V (x), although similar in spirit, is much more involved in general. In order to determine asymptotic description of uε beyond caustics, one needs to invoke the theory of Fourier integral operators: see, e.g., Duistermaat (1996). In particular, it is in general very hard to determine the precise form and number of caustics appearing throughout the time evolution of S(t, x). Numerical schemes for capturing caustics have been developed in, e.g., Benamou and Solliec (2000), or Benamou, Lafitte, Sentis and Solliec (2003) and the references therein.

3. Wigner transforms and Wigner measures 3.1. The Wigner-transformed picture of quantum mechanics Whereas WKB-type methods aim for approximate solutions of uε , the goal of this section is to directly identify the weak limits of physical observable densities as ε → 0. To this end, one defines the so-called Wigner transform of uε , as given in Wigner (1932): 1 ε ε ε ε ε ε u x + η u x − η eiξ·η dη. (3.1) w [u ](x, ξ) := 2 2 (2π)d Rd

131


Plancherel’s theorem together with a simple change of variables yields wε L2 (R2d ) = ε−d (2π)−d/2 uε 2L2 (Rd ) . The real-valued Wigner transform wε ∈ L2 (Rdx × Rdξ ) can be interpreted as a phase space description of the quantum state uε . In contrast to classical phase space distributions, wε in general also takes negative values (except for Gaussian wave functions). Applying this transformation to the Schrödinger equation (2.1), the timedependent Wigner function wε (t, x, ξ) ≡ wε [uε (t)](x, ξ) is easily seen to satisfy ∂t wε + ξ · ∇x wε − Θε [V ]wε = 0,

ε wε (0, x, ξ) = win (x, ξ),

(3.2)

where Θε [V ] is a pseudo-differential operator, taking into account the influence of V (x). Explicitly, it is given by i ε δV ε (x, y)f (x, ξ ) eiη(ξ−ξ ) dη dξ , (3.3) Θ [V ]f (x, ξ) := d (2π) Rd ×Rd where the symbol δV ε reads 1 ε ε δV ε := V x− y −V x+ y . ε 2 2 Note that in the free case where V (x) = 0, the Wigner equation becomes the free transport equation of classical kinetic theory. Moreover, if V ∈ C 1 (Rd ) we obviously have that ε→0

δV ε −→ y · ∇x V, in which case the ε → 0 limit of (3.2) formally becomes the classical Liouville equation in phase space: see (3.7) below. The most important feature of the Wigner transform is that it allows for a simple computation of quantum mechanical expectation values of physical observables. Namely, ε W ε u (t), a (x, εD)u (t) L2 = a(x, ξ)wε (t, x, ξ) dx dξ, (3.4) Rd ×Rd

where a(x, ξ) is the classical symbol of the operator aW (x, εDx ). In addition, at least formally (since wε ∈ L1 (Rd × Rd ) in general), the particle density (2.6) can be computed via wε (t, x, ξ) dξ, ρε (t, x) = Rd

and the current density (2.7) is given by ε ξwε (t, x, ξ) dξ. j (t, x) = Rd

132


Similarly, the energy density (2.9) is ε H(x, ξ)wε (t, x, ξ) dξ, e (t, x) = Rd

where the classical (phase space) Hamiltonian function is denoted by 1 H(x, ξ) = |ξ|2 + V (x). (3.5) 2 Remark 3.1. It can be proved that the Fourier transform of wε with respect to ξ satisfies w ε ∈ C0 (Rdy ; L1 (Rdx )) and likewise for the Fourier transformation of wε with respect to x ∈ Rd . This allows us to define the integrals of wε via a limiting process after convolving wε with Gaussians; see Lions and Paul (1993) for more details. 3.2. Classical limit of Wigner transforms The main point in the formulae given above is that the right-hand side of (3.4) involves only linear operations of wε , which is compatible with weak limits. To this end, we recall the main result proved in Lions and Paul (1993) and Gérard et al. (1997). Theorem 3.2. ε, that is,

Let uε (t) be uniformly bounded in L2 (Rd ) with respect to sup uε (t)L2 < +∞,

0 0, choosing h such that = hε → ∞ corresponds to asymptotically resolving the oscillations of wavelength O(ε) in the solution uε (t, x) to the Schr¨ odinger-type equation (4.1). In the case = const., i.e., putting a fixed number of grid points per oscillation, the symbol Hh,ε (x, ξ) ≡ H (x, ξ) is independent of h and ε, i.e., H (x, ξ) = |k| aγ,k (−i)|k| Hk (x) e iγ·ξ/ . (4.6) |k|≤K

γ∈Γk

ε,h→0

In the case −→ 0, which corresponds to a scheme ignoring the εoscillations, we find γ·ξ h,ε→0 H0 (x), Hh,ε ∼ aγ,0 cos γ∈Γ0


137

and hence Hh,ε (x, εDx )W does not approximate H(x, εDx )W . We thus cannot expect reasonable numerical results in this case (which will not be investigated further). 4.3. Temporal discretization and violation of gauge invariance For the temporal discretization one can employ the Crank–Nicolson scheme with time step ∆t > 0. This is a widely used time discretization scheme for the Schrödinger equation, featuring some desirable properties (see below). We shall comment on the temporal discretizations below. The scheme reads uσn+1 − uσn 1 σ W 1 σ ε + u = 0, n = 0, 1, 2, . . . , (4.7) + iHh,ε (x, εDx ) u ∆t 2 n+1 2 n subject to initial data uσin = uεin (x), where from now on, we shall denote the vector of small parameters by σ = (ε, h, ∆t). Note that the self-adjointness of Hh,ε (x, εDx )W implies that the operator 1 + isHh,ε (x, εDx )W is invertible on L2 (Rd ) for all s ∈ R. Therefore the scheme (4.7) gives welldefined approximations uσn for n = 1, 2, . . . if uεin ∈ L2 (Rd ). Moreover, we remark that it is sufficient to evaluate (4.7) at x ∈ hΓ in order to obtain discrete equations for uσn (hγ), γ ∈ Γ. Remark 4.2. For practical computations, one needs to impose artificial ‘far-field’ boundary conditions. Their impact, however, will not be taken into account in the subsequent analysis. By taking the L2 scalar product of (4.7) with 12 uσn+1 + 12 uσn , one can directly infer the following stability result. Lemma 4.3.

The solution of (4.7) satisfies uσn L2 = uεin L2 ,

n = 0, 1, 2, . . . .

In other words, the physically important property of mass-conservation also holds on the discrete level. On the other hand, the scheme can be seen to violate the gauge invariance of (4.1). More precisely, one should note that expectation values of physical observables, as defined in (2.5), are invariant under the substitution (gauge transformation) v ε (t, x) = uε (t, x) e iωt/ε ,

ω ∈ R.

In other words, the average value of the observable in the state uε is equal to its average value in the state v ε .

138


Remark 4.4. Note that in view of (3.1), the Wigner function is seen to also be invariant under this substitution, i.e., ∀ω ∈ R : wε [uε (t)] = wε [uε (t) e iωt/ε ] ≡ wε [v ε (t)]. On the other hand, using this gauge transformation the Schrödinger equation (4.1) transforms to (4.8) iε∂t v ε = H(x, εDx )W + ω v ε , v ε (0, x) = uεin (x), which implies that the zeroth-order term H0 (x) in (4.2) is replaced by H0 (x) + ω, while the other coefficients Hk (x), k = 0, remain unchanged. In physical terms, H0 (x) corresponds to a scalar (static) potential V (x). The corresponding force field obtained via F (x) = ∇H0 (x) = ∇(H0 (x) + ω) is unchanged by the gauge transformation and thus (4.8) can be considered (physically) equivalent to (4.1). The described situation, however, is completely different for the difference scheme outlined above. Indeed, a simple calculation shows that the discrete gauge transformation vnσ = uσn e iωtn /ε does not commute with the discretization (4.7), up to adding a real constant to the potential. Thus, the discrete approximations of average values of observables depend on the gauging of the potential. In other words, the discretization method is not time-transverse-invariant. 4.4. Stability–consistency analysis for FD in the semiclassical limit The consistency–stability concept of classical numerical analysis provides a framework for the convergence analysis of finite difference discretizations of linear partial differential equations. Thus, for ε > 0 fixed it is easy to prove that the scheme (4.7) is convergent of order N in space and order 2 in time if the exact solution uε (t, x) is sufficiently smooth. Therefore, again for fixed ε > 0, we conclude convergence of the same order for average values of physical observables provided a(x, ξ) is smooth. However, due to the oscillatory nature solutions to (4.1) the local discretization error of the finite difference schemes and, consequently, also the global discretization error, in general tend to infinity as ε → 0. Thus, the classical consistency–stability theory does not provide uniform results in the classical limit. Indeed, under the reasonable assumption that, for all multi-indices j1 and j2 ∈ Nd , ∂ |j1 |+|j2 | ε ε→0 u (t, x) ∼ ε−|j1 |−j2 ∂tj1 ∂xj2

in L2 (Rd ),

locally uniformly in t ∈ R, the classical stability–consistency analysis gives


139

the following bound for the global L2 -discretization error: N (∆t)2 h O + O . ε3 εN +1 The situation is further complicated by the fact that for any fixed t ∈ R, the solution uε (t, ·) of (4.1) and its discrete counterpart uσn in general converge only weakly in L2 (Rd ) as ε → 0, respectively, σ → 0. Thus, the limit processes ε → 0, σ → 0 do not commute with the quadratically nonlinear operation (2.5), needed to compute the expectation value of physical observables a[uε (t)]. In practice, one is therefore interested in finding conditions on the mesh size h and the time step ∆t, depending on ε in such a way that the expectation values of physical observables in discrete form approximate a[uε (t)] uniformly as ε → 0. To this end, let tn = n∆t, n ∈ N, and denote aσ (tn ) := a(·, εDx )W uσn , uσn . The function aσ (t), t ∈ R, is consequently defined by piecewise linear interpolation of the values aσ (tn ). We seek conditions on h, k such that, for all m a ∈ S(Rm x × Rξ ), lim (aσ (t) − a[uε (t)]) = 0 uniformly in ε ∈ (0, ε0 ],

h,∆t→0

(4.9)

and locally uniformly in t ∈ R. A rigorous study of this problem will be given by using the theory of Wigner measures applied in a discrete setting. Denoting the Wigner transformation (on the scale ε) of the finite difference solution uσn by wσ (tn ) := wε [uσn ], and defining, as before, wσ (t), for any t ∈ R, by the piecewise linear interpolation of wσ (tn ), we conclude that (4.9) is equivalent to proving, locally uniformly in t, lim (wσ (t) − wε (t)) = 0 in S (Rdx × Rdξ ), uniformly in ε ∈ (0, ε0 ], (4.10)

h,∆t→0

where wε (t) is the Wigner transform of the solution uε (t) of (4.1). We shall now compute the accumulation points of the sequence {wσ (t)}σ as σ → 0. We shall see that for any given subsequence {σn }n∈N , the set of Wigner measures of the difference schemes µ(t) := lim wσn (t) n→∞

depends decisively on the relative sizes of ε, h and ∆t. Clearly, in those cases in which µ = w, where w denotes the Wigner measure of the exact solution uε (t), the desired property (4.10) follows. On the other hand (4.10) does not hold if the measures µ and w are different. Such a Wigner-measure-based

140


study of finite difference schemes has been conducted in Markowich et al. (1999) and Markowich and Poupaud (1999). The main result given there is as follows. Theorem 4.5. Fix a scale ε > 0 and denote by µ the Wigner measure of the discretization (4.7) as σ → 0. Then we have the following results. Case 1. Let h/ε → 0 (or, equivalently, → ∞). (i) If ∆t/ε → 0, then µ satisfies ∂t µ + {H, µ} = 0,

µ(0, x, ξ) = win (x, ξ).

R+ ,

(ii) If ∆t/ε → ω ∈ then µ solves ∂ 2 ω µ+ arctan H , µ = 0, ∂t ω 2

µ(0, x, ξ) = win (x, ξ).

(iii) If ∆t/ε → ∞ and if in addition there exists C > 0 such that |H(x, ξ)| ≥ C, ∀x, ξ ∈ Rd , then µ is constant in time, i.e., µ(t, x, ξ) ≡ µin (x, ξ),

∀ t ∈ R.

Case 2. If h/ε → 1/ ∈ R+ , then the assertions (i)–(iii) hold true, with H replaced by H defined in (4.6). The proof of this result proceeds similarly to the derivation of the phase space Liouville equation (3.7), in the continuous setting. Note that Theorem 4.5 implies that, as ε → 0, expectation values for physical observables in the state uε (t), computed via the Crank–Nicolson finite difference scheme, are asymptotically correct only if both spatial and temporal oscillations of wavelength ε are accurately resolved. Remark 4.6. Time-irreversible finite difference schemes, such as the explicit (or implicit) Euler scheme, behave even less well, as they require ∆t = o(ε2 ) in order to guarantee asymptotically correct numerically computed observables; see Markowich et al. (1999).

5. Time-splitting spectral methods for semiclassical Schr¨ odinger equations 5.1. Basic setting, first- and second-order splittings As has been discussed before, finite difference methods do not perform well in computing the solution to semiclassical Schrödinger equations. An alternative is given by time-splitting trigonometric spectral methods, which will be discussed in this subsection; see also McLachlan and Quispel (2002) for a broad introduction to splitting methods. For the sake of notation, we shall introduce the method only in the case of one space dimension, d = 1.


141

Generalizations to d > 1 are straightforward for tensor product grids and the results remain valid without modifications. In the following, we shall therefore study the one-dimensional version of equation (2.1), i.e., iε∂t uε = −

ε2 ∂xx uε + V (x)uε , 2

uε (0, x) = uεin (x),

(5.1)

for x ∈ [a, b], 0 < a < b < +∞, equipped with periodic boundary conditions uε (t, a) = uε (t, b),

∂x uε (t, a) = ∂x uε (t, b),

∀ t ∈ R.

We choose the spatial mesh size ∆x = h > 0 with h = (b − a)/M for some M ∈ 2N, and an ε-independent time step ∆t ≡ k > 0. The spatio-temporal grid points are then given by xj := a + jh, j = 1, . . . , M,

tn := nk, n ∈ N.

In the following, let uε,n be the numerical approximation of uε (xj , tn ), for j j = 1, . . . , M .

First-order time-splitting spectral method (SP1 ) The Schr¨ odinger equation (5.1) is solved by a splitting method, based on the following two steps. Step 1. From time t = tn to time t = tn+1 , first solve the free Schrödinger equation iε∂t uε +

ε2 ∂xx uε = 0. 2

(5.2)

Step 2. On the same time interval, i.e., t ∈ [tn , tn+1 ], solve the ordinary differential equation (ODE) iε∂t uε − V (x)uε = 0,

(5.3)

with the solution obtained from Step 1 as initial data for Step 2. Equation (5.3) can be solved exactly since |u(t, x)| is left invariant under (5.3), u(t, x) = |u(0, x)| e iV (x)t . In Step 1, the linear equation (5.2) will be discretized in space by a (pseudo-) spectral method (see, e.g., Fornberg (1996) for a general introduction) and consequently integrated in time exactly. More precisely, one obtains at time t = tn+1 u(tn+1 , x) ≈ uε,n+1 = e iV (xj )k/ε uε,∗ j j ,

j = 0, 1, 2, . . . , M,

142


ε with initial value uε,0 j = uin (xj ), and

1 M

uε,∗ j = where γ =

2πl b−a

and

u ε,n =

u ε,n M

M/2−1 2

iγ (xj −a) e iεkγ /2 u ε,n , e

=−M/2

denote the Fourier coefficients of uε,n , i.e., −iγ (xj −a) uε,n , j e

=−

j=1

M M ,..., − 1. 2 2

Note that the only time discretization error of this method is the splitting error, which is first-order in k = ∆t, for any fixed ε > 0. Strang splitting (SP2 ) In order to obtain a scheme which is second-order in time (for fixed ε > 0), one can use the Strang splitting method , i.e., on the time interval [tn , tn+1 ] we compute = e iV (xj )k/2ε uε,∗∗ uε,n+1 j j ,

j = 0, 1, 2, . . . , M − 1,

where uε,∗∗ = j with

u ε,∗

1 M

M/2−1 2

iγ (xj −a) e iεkγ /2 u ε,∗ , e

=−M/2

denoting the Fourier coefficients of uε,∗ given by iV (xj )k/2ε ε,n uj . uε,∗ j =e

Again, the overall time discretization error comes solely from the splitting, which is now (formally) second-order in ∆t = k for fixed ε > 0. Remark 5.1. Extensions to higher-order (in time) splitting schemes can be found in the literature: see, e.g., Bao and Shen (2005). For rigorous investigations of the long time error estimates of such splitting schemes we refer to Dujardin and Faou (2007a, 2007b) and the references given therein. In comparison to finite difference methods, the main advantage of such splitting schemes is that they are gauge-invariant: see the discussion in Section 4 above. Concerning the stability of the time-splitting spectral approximations with variable potential V = V (x), one can prove the following lemma (see Bao, Jin and Markowich (2002)), in which we denote U = (u1 , . . . , uM ) and · l2 the usual discrete l2 -norm on the interval [a, b], i.e., 1/2 M b−a 2 U l2 = |uj | . M j=1


143

Lemma 5.2. The time-splitting spectral schemes SP1 and SP2 are unconditionally stable, i.e., for any mesh size h and any time step k, we have ε U ε,n l2 = U ε,0 l2 ≡ Uin l2 ,

n ∈ N,

and consequently ε,0 uε,n int L2 (a,b) = uint L2 (a,b) ,

n ∈ N,

where uε,n int denotes the trigonometric polynomial interpolating ε,n ε,n {(x1 , uε,n 1 ), (x1 , u1 ), . . . , (xM , uM )}.

In other words, time-splitting spectral methods satisfy mass-conservation on a fully discrete level. 5.2. Error estimate of SP1 in the semiclassical limit To get a better understanding of the stability of spectral methods in the classical limit ε → 0, we shall establish the error estimates for SP1. Assume that the potential V (x) is (b − a)-periodic, smooth, and satisfies m d ≤ Cm , (A) dxm V ∞ L [a,b] for some constant Cm > 0. Under these assumptions it can be shown that the solution uε = uε (t, x) of (5.1) is (b−a)-periodic and smooth. In addition, we assume m +m ∂ 1 2 ε Cm +m ≤ m11+m22 , (B) ∂tm1 ∂xm2 u ε 2 C([0,T ];L [a,b])

for all m, m1 , m2 ∈ N ∪ {0}. Thus, we assume that the solution oscillates in space and time with wavelength ε, but no smaller. Remark 5.3. The latter is known to be satisfied if the initial data uεin only invoke oscillations of wavelength ε (but no smaller). Theorem 5.4. Let V (x) satisfy assumption (A) and let uε (t, x) be a solution of (5.1) satisfying (B). Denote by uε,n int the interpolation of the discrete approximation obtained via SP1. Then, if ∆t ∆x = O(1), = O(1), ε ε as ε → 0, we have that, for all m ∈ N and tn ∈ [0, T ], m ε T ∆x CT ∆t u (tn ) − uε,n 2 , + int L (a,b) ≤ Gm ∆t ε(b − a) ε

(5.4)

where C > 0 is independent of ε and m and Gm > 0 is independent of ε, ∆x, ∆t.

144


1.2

1

1 0.5

0.8 0.6

0

0.4 −0.5

0.2 0

−1

0

0.2

0.4

0.6

0.8

1

0

0.2

(a)

0.4

0.6

0.8

1

(b)

Figure 5.1. Numerical solution of (a) ρε and (b) j ε at t = 0.54, as given in Example 5.5. In the figure, the solution computed by using SP2 for 1 , is superimposed, with the limiting ρ and j obtained by ε = 0.0008, h = 512 taking moments of the Wigner measure solution of (3.8).

The proof of this theorem is given in Bao et al. (2002), where a similar result is also shown for SP2. Now let δ > 0 be a desired error bound such that uε (tn ) − uε,n int L2 [a,b] ≤ δ holds uniformly in ε. Then Theorem 5.4 suggests the following meshing strategy on O(1) time and space intervals: ∆t ∆x = O(δ), = O δ 1/m (∆t)1/m , (5.5) ε ε where m ≥ 1 is an arbitrary integer, assuming that Gm does not increase too fast as m → ∞. This meshing is already more efficient than what is needed for finite differences. In addition, as will be seen below, the conditions (5.5) can be strongly relaxed if, instead of resolving the solution uε (t, x), one is only interested in the accurate numerical computation of quadratic observable densities (and thus asymptotically correct expectation values). Example 5.5. This is an example from Bao et al. (2002). The Schr¨ odinger equation (2.1) is solved with V (x) = 10 and the initial data ρin (x) = exp(−50(x − 0.5)2 ), 1 Sin (x) = − ln(exp(5(x − 0.5)) + exp(−5(x − 0.5))), 5

x ∈ R.


145

The computational domain is restricted to [0, 1] equipped with periodic boundary conditions. Figure 5.1 shows the solution of the limiting position density ρ and current density j obtained by taking moments of w, satisfying the Liouville equation (3.8). This has to be compared with the oscillatory ρε and j ε , obtained by solving the Schrödinger equation (2.1) using SP2. As one can see, these oscillations are averaged out in the weak limits ρ, j. 5.3. Accurate computation of quadratic observable densities using time-splitting We shall again invoke the theory of Wigner functions and Wigner measures. To this end, let uε (t, x) be the solution of (5.1) and let wε (t, x, ξ) be the corresponding Wigner transform. Keeping in mind the results of Section 3, we see that the first-order splitting scheme SP1 corresponds to the following time-splitting scheme for the Wigner equation (3.2). Step 1. For t ∈ [tn , tn+1 ], first solve the linear transport equation ∂t wε + ξ ∂x wε = 0.

(5.6)

Step 2. On the same time interval, solve the non-local (in space) ordinary differential equation ∂t wε − Θε [V ]wε = 0,

(5.7)

with initial data obtained from Step 1 above. In (5.6), the only possible ε-dependence stems from the initial data. In addition, in (5.7) the limit ε → 0 can be easily carried out (assuming sufficient regularity of the potential V (x)) with k = ∆t fixed. In doing so, one consequently obtains a time-splitting scheme of the limiting Liouville equation (3.8) as follows. Step 1. For t ∈ [tn , tn+1 ], solve ∂t w + ξ ∂x w0 = 0. Step 2. Using the outcome of Step 1 as initial data, solve, on the same time interval, ∂t w − ∂x V ∂ξ w0 = 0. Note that in this scheme no error is introduced other than the splitting error, since the time integrations are performed exactly. These considerations, which can easily be made rigorous (for smooth potentials), show that a uniform time-stepping (i.e., an ε-independent k = ∆t) of the form ∆t = O(δ)

146


combined with the spectral mesh size control given in (5.5) yields the following error: ε,n L2 (a,b) ≤ δ, wε (tn ) − wint

uniformly in ∆t as ε → 0. Essentially this implies that a fixed number of grid points in every spatial oscillation of wavelength ε combined with ε-independent time-stepping is sufficient to guarantee the accurate computation of (expectation values of) physical observables in the classical limit. This strategy is therefore clearly superior to finite difference schemes, which require k/ε → 0 and h/ε → 0, even if one only is interested in computing physical observables. Remark 5.6. Time-splitting methods have been proved particularly successful in nonlinear situations: see the references given in Section 15.4 below.

6. Moment closure methods We have seen before that a direct numerical calculation of uε is numerically very expensive, particularly in higher dimensions, due to the mesh and time step constraint (5.5). In order to circumvent this problem, the asymptotic analysis presented in Sections 2 and 3 can be invoked in order to design asymptotic numerical methods which allow for an efficient numerical simulation in the limit ε → 0. The initial value problem (3.8)–(3.9) is the starting point of the numerical methods to be described below. Most recent computational methods are derived from, or closely related to, this equation. The main advantage is that (3.8)–(3.9) correctly describes the limit of quadratic densities of uε (which in itself exhibits oscillations of wavelength O(ε)), and thus allows a numerical mesh size independent of ε. However, we face the following major difficulties in the numerical approximation. (1) High-dimensionality. The Liouville equation (3.8) is defined in phase space, and thus the memory requirement exceeds the current computational capability in d ≥ 3 space dimensions. (2) Measure-valued initial data. The initial data (3.9) form a delta measure and the solution at later time remains one (for a single-valued solution), or a summation of several delta functions (for a multivalued solution (3.10)). In the past few years, several new numerical methods have been introduced to overcome these difficulties. In the following, we shall briefly describe the basic ideas of these methods.


147

6.1. The concept of multivalued solutions In order to overcome the problem of high-dimensionality one aims to approximate w(t, x, p) by using averaged quantities depending only on t, x. This is a well-known technique in classical kinetic theory, usually referred to as moment closure. A basic example is provided by the result of Theorem 3.4, which tells us that, as long as the WKB analysis of Section 2.2 is valid (i.e., before the appearance of the first caustic), the Wigner measure is given by a mono-kinetic distribution in phase space, i.e., w(t, x, ξ) = ρ(t, x)δ(ξ − v(t, x)), where one identifies ρ = |a|2 and v = ∇S. The latter solve the pressure-less Euler system ∂t ρ + div(ρv) = 0, ∂t v + (v · ∇)v + ∇V = 0,

ρ(0, x) = |ain |2 (x), v(0, x) = ∇Sin (x),

(6.1)

which, for smooth solutions, is equivalent to the system of transport equation (2.14) coupled with the Hamilton–Jacobi equation (2.12), obtained through the WKB approximation. Thus, instead of solving the Liouville equation in phase space, one can as well solve the system (6.1), which is posed on physical space Rt × Rdx . Of course, this can only be done until the appearance of the first caustic, or, equivalently, the emergence of shocks in (6.1). In order to go beyond that, one might be tempted to use numerical methods based on the unique viscosity solution (see Crandall and Lions (1983)) for (6.1). However, the latter does not provide the correct asymptotic description – the multivalued solution – of the wave function uε (t, x) beyond caustics. Instead, one has to pass to so-called multivalued solutions, based on higher-order moment closure methods. This fact is illustrated in Figure 6.1, which shows the difference between viscosity solutions and multivalued solutions. Figures 6.1(a) and 6.1(b) are the two different solutions for the following eikonal equation (in fact, the zero-level set of S): ∂t S + |∇x S| = 0,

x ∈ R2 .

(6.2)

This equation, corresponding to H(ξ) = |ξ|, arises in the geometric optics limit of the wave equation and models two circular fronts moving outward in the normal direction with speed 1; see Osher and Sethian (1988). As one can see, the main difference occurs when the two fronts merge. Similarly, Figures 6.1(c) and 6.1(d) show the difference between the viscosity and the multivalued solutions to the Burgers equation ∂t v +

1 ∂x v 2 = 0, 2

x ∈ R.

(6.3)

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Eikonal equation (a)

(b)

Burgers equation (c)

(d)

Figure 6.1. Multivalued solution (left) versus viscosity solution (right). (a, b) Zero-level set curves (at different times) of solutions to the eikonal equation (6.2). (c, d) Two solutions to the Burgers equation (6.3) before and after the formation of a shock.

This is simply the second equation in the system (6.1) for V (x) = 0, written in divergence form. The solution begins as a sinusoidal function and then forms a shock. Clearly, the solutions are different after the shock formation. 6.2. Moment closure The moment closure idea was first introduced by Brenier and Corrias (1998) in order to define multivalued solutions to the Burgers equation, and seems to be the natural choice in view of the multiphase WKB expansion given in (2.21). The method was then used numerically in Engquist and Runborg (1996) (see also Engquist and Runborg (2003) for a broad review) and Gosse (2002) to study multivalued solutions in the geometrical optics regime of hyperbolic wave equations. A closely related method is given in Benamou (1999), where a direct computation of multivalued solutions to Hamilton– Jacobi equations is performed. For the semiclassical limit of the Schrödinger equation, this was done in Jin and Li (2003) and then Gosse et al. (2003).


149

In order to describe the basic idea, let d = 1 and define m (t, x) = ξ w(t, x, ξ) dξ, = 1, 2, . . . , L ∈ N,

(6.4)

R

i.e., the th moment (in velocity) of the Wigner measure. By multiplying the Liouville equation (3.8) by ξ and integrating over Rξ , one obtains the following moment system: ∂t m0 + ∂x m1 = 0, ∂t m1 + ∂x m2 = −m0 ∂x V, .. . ∂t mL−1 + ∂x mL = −(L − 1)mL−2 ∂x V. Note that this system is not closed , since the equation determining the th moment involves the ( + 1)th moment. The δ-closure As already mentioned in (3.10), locally away from caustics the Wigner measure of uε as ε → 0 can be written as w(t, x, ξ) =

J

ρj (t, x)δ(ξ − vj (t, x)),

(6.5)

j=1

where the number of velocity branches J can in principle be determined a priori from ∇Sin (x). For example, in d = 1, it is the total number of inflection points of v(0, x): see Gosse et al. (2003). Using this particular form (6.5) of w with L = 2J provides a closure condition for the moment system above. More precisely, one can express the last moment m2J as a function of all of the lower-order moments (Jin and Li 2003), i.e., m2J = g(m0 , m1 , . . . , m2J−1 ).

(6.6)

This consequently yields a system of 2J × 2J equations (posed in physical space), which effectively provides a solution of the Liouville equation, before the generation of a new phase, yielding a new velocity vj , j > J. It was shown in Jin and Li (2003) that this system is only weakly hyperbolic, in the sense that the Jacobian matrix of the flux is a Jordan block, with only J distinct eigenvalues v1 , v2 , . . . , vJ . This system is equivalent to J pressureless gas equations (6.1) for (ρj , vj ) respectively. In Jin and Li (2003) the explicit flux function g in (6.6) was given for J ≤ 5. For larger J a numerical procedure was proposed for evaluating g. Since the moment system is only weakly hyperbolic, with phase jumps which are under-compressive shocks (Gosse et al. 2003), standard shock-capturing schemes such as the Lax–Friedrichs scheme and the Godunov scheme

150


face severe numerical difficulties as in the computation of the pressure-less gas dynamics: see Bouchut, Jin and Li (2003), Engquist and Runborg (1996) or Jiang and Tadmor (1998). Following the ideas of Bouchut et al. (2003) for the pressure-less gas system, a kinetic scheme derived from the Liouville equation (3.8) with the closure condition (6.6) was used in Jin and Li (2003) for this moment system. The Heaviside closure Another type of closure was introduced by Brenier and Corrias (1998) using the following ansatz, called H-closure, to obtain the J-branch velocities vj , with j = 1, . . . , J: w(t, x, ξ) =

J

(−1)j−1 H(vj (t, x) − ξ).

(6.7)

j=1

This type of closure condition for (3.8) arises from an entropy-maximization principle: see Levermore (1996). Using (6.7), one arrives at (6.6) with L = J. The explicit form of the corresponding function g(m0 , . . . , m2J−1 ) for J < 5 is available analytically in Runborg (2000). Note that this method decouples the computation of velocities vj from the densities ρj . In fact, to obtain the latter, Gosse (2002) has proposed solving the following linear conservation law (see also Gosse and James (2002) and Gosse et al. (2003)): ∂t ρj + ∂x (ρj vj ) = 0,

for j = 1, . . . , N.

The numerical approximation to this linear transport with variable or even discontinuous flux is not straightforward. Gosse et al. (2003) used a semiLagrangian method that uses the method of characteristics, requiring the time step to be sufficiently small for the case of non-zero potentials. The corresponding method is usually referred to as H-closure. Note that in d = 1 the H-closure system is a non-strictly rich hyperbolic system, whereas the δ-closure system described before is only weakly hyperbolic. Thus one expects a better numerical resolution from the H-closure approach, which, however, is much harder to implement in the higher dimension. In d = 1, the mathematical equivalence of the two moment systems was proved in Gosse et al. (2003). Remark 6.1. Multivalued solutions also arise in the high-frequency approximation of nonlinear waves, for example, in the modelling of electron transport in vacuum electronic devices: see, e.g., Granastein, Parker and Armstrong (1999). There the underlying equations are the Euler–Poisson equations, a nonlinearly coupled hyperbolic–elliptic system. The multivalued solution of the Euler–Poisson system also arises for electron sheet initial data, and can be characterized by a weak solution of the Vlasov–Poisson equation: see Majda, Majda and Zheng (1994). Similarly, the work of Li,


151

Wöhlbier, Jin and Booske (2004) uses the moment closure ansatz (6.6) for the Vlasov–Poisson system; see also Wöhlbier, Jin and Sengele (2005). For multivalued (or multiphase) solution of the semiclassical limit of nonlinear dispersive waves using the closely related method of Whitham’s modulation theory, we refer to Whitham (1974) and Flaschka, Forest and McLaughlin (1980). Finally, we mention that multivalued solutions also arise in supply chain modelling: see, e.g., Armbruster, Marthaler and Ringhofer (2003). In summary, the moment closure approach yields an Eulerian method defined in the physical space which offers a greater efficiency compared to the computation in phase space. However, when the number of phases J ∈ N becomes very large and/or in dimensions d > 1, the moment systems become very complex and thus difficult to solve. In addition, in high space dimensions, it is very difficult to estimate a priori the total number of phases needed to construct the moment system. Thus it remains an interesting and challenging open problem to develop more efficient and general physicalspace-based numerical methods for the multivalued solutions.

7. Level set methods 7.1. Eulerian approach Level set methods have been recently introduced for computing multivalued solutions in the context of geometric optics and semiclassical analysis. These methods are rather general, and applicable to any (scalar) multi-dimensional quasilinear hyperbolic system or Hamilton–Jacobi equation (see below). We shall now review the basic ideas, following the lines of Jin and Osher (2003). The original mathematical formulation is classical; see for example Courant and Hilbert (1962). Computation of the multivalued phase Consider a general d-dimensional Hamilton–Jacobi equation of the form ∂t S + H(x, ∇S) = 0,

S(0, x) = Sin (x).

(7.1)

For example, in present context of semiclassical analysis for Schr¨ odinger equations, 1 H(x, ξ) = |ξ|2 + V (x), 2 while for applications in geometrical optics (i.e., the high-frequency limit of the wave equation), H(x, ξ) = c(x)|ξ|, with c(x) denoting the local sound (or wave) speed. Introducing, as before, a velocity v = ∇S and taking the gradient of (7.1), one gets an equivalent

152


equation (at least for smooth solutions) in the form (Jin and Xin 1998): ∂t v + (∇ξ H(x, v) · ∇)v + ∇x H(x, v) = 0,

v(0, x) = ∇x Sin (x).

(7.2)

Then, in d ≥ 1 space dimensions, define level set functions φj , for j = 1, . . . , d, via ∀(t, x) ∈ R × Rd :

φj (t, x, ξ) = 0 at ξ = vj (t, x).

In other words, the (intersection of the) zero-level sets of all {φj }dj=1 yield the graph of the multivalued solution vj (t, x) of (7.2). Using (7.2) it is easy to see that φj solves the following initial value problem: ∂t φj + {H(x, ξ), φj }φj = 0,

φj (x, ξ, 0) = ξj − vj (0, x),

(7.3)

which is simply the phase space Liouville equation. Note that in contrast to (7.2), this equation is linear and thus can be solved globally in time. In doing so, one obtains, for all t ∈ R, the multivalued solution to (7.2) needed in the asymptotic description of physical observables. See also Cheng, Liu and Osher (2003). Computation of the particle density It remains to compute the classical limit of the particle density ρ(t, x). To do so, a simple idea was introduced in Jin, Liu, Osher and Tsai (2005a). This method is equivalent to a decomposition of the measure-valued initial data (3.9) for the Liouville equation. More precisely, a simple argument based on the method of characteristics (see Jin, Liu, Osher and Tsai (2005b)) shows that the solution to (3.8)–(3.9) can be written as w(t, x, ξ) = ψ(t, x, ξ)

d

δ(φj (t, x, ξ)),

j=1

where φj (t, x, ξ) ∈ Rn , j = 1, . . . , d, solves (7.3) and the auxiliary function ψ(t, x, ξ) again satisfies the Liouville equation (3.8), subject to initial data: ψ(0, x, ξ) = ρin (x). The first two moments of w with respect to ξ (corresponding to the particle ρ and current density J = ρu) can then be recovered through d ρ(t, x) = ψ(t, x, ξ) δ(φj (t, x, ξ)) dξ, Rd

u(t, x) =

1 ρ(t, x)

j=1

ξψ(t, x, ξ) Rd

d

δ(φj (t, x, ξ)) dξ.

j=1

Thus the only time one has to deal with the delta measure is at the numerical output, while during the time evolution one simply solves for φj and ψ, both


153

of which are smooth L∞ -functions. This avoids the singularity problem mentioned earlier, and gives numerical methods with much better resolution than solving (3.8)–(3.9) directly, e.g., by approximating the initial delta function numerically. An additional advantage of this level set approach is that one only needs to care about the zero-level sets of φj . Thus the technique of local level set methods developed in Adalsteinsson and Sethian (1995) and Peng, Merriman, Osher, Zhao and Kang (1999) can be used. One thereby restricts the computational domain to a narrow band around the zero-level set, in order to reduce the computational cost to O(N ln N ), for N computational points in the physical space. This is a nice alternative for dimension reduction of the Liouville equation. When solutions for many initial data need to be computed, fast algorithms can be used: see Fomel and Sethian (2002) or Ying and Candès (2006). Example 7.1. Consider (2.1) in d = 1 with periodic potential V (x) = cos(2x + 0.4)), and WKB initial data corresponding to Sin (x) = sin(x + 0.15), 1 π 2 π 2 + exp − x − . ρin (x) = √ exp − x + 2 2 2 π Figure 7.1 shows the time evolution of the velocity and the corresponding density computed by the level set method described above. The velocity eventually develops some small oscillations with higher frequency, which require a finer grid to resolve. Remark 7.2. The outlined ideas have been extended to general linear symmetric hyperbolic systems in Jin et al. (2005a). So far, however, level set methods have not been formulated for nonlinear equations, except for the one-dimensional Euler–Poisson equations (Liu and Wang 2007), where a three-dimensional Liouville equation has to be used in order to calculate the corresponding one-dimensional multivalued solutions. 7.2. The Lagrangian phase flow method While the Eulerian level set method is based on solving the Liouville equation (3.8) on a fixed mesh, the Lagrangian (or particle) method , is based on solving the Hamiltonian system (3.6), which is simply the characteristic flow of the Liouville equation (3.7). In geometric optics this idea is referred to as ray tracing (Cerven´ y 2001), and the curves x(t, y, ζ), ξ(t, y, ζ) ∈ Rd , obtained by solving (3.6), are usually called bi-characteristics. Remark 7.3. Note that finding an efficient way to numerically solve Hamiltonian ODEs, such as (3.6), is a problem of great (numerical) interest in its own right: see, e.g., Leimkuhler and Reich (2004).

154


(a)

(b)

Figure 7.1. Example 7.1: (a) multivalued velocity v at time T = 0.0, 6.0, and 12.0, (b) corresponding density ρ.


155

Here we shall briefly describe a fast algorithm, called the phase flow method in Ying and Candès (2006), which is very efficient if multiple initial data, as is often the case in practical applications, are to be propagated by the Hamiltonian flow (3.6). Let Ft : R2d → R2d be the phase flow defined by Ft (y, ζ) = (x(t, y, ζ), ξ(t, y, ζ)), t ∈ R. A manifold M ⊂ Rdx × Rdξ is said to be invariant if Ft (M) ⊂ M. For the autonomous ODEs, such as (3.6), a key property of the phase map is the one-parameter group structure, Ft ◦ Fs = Ft+s . Instead of integrating (3.6) for each individual initial condition (y, ζ), up to, say, time T , the phase flow method constructs the complete phase map FT . To this end, one first constructs the Ft for small times using standard ODE integrators and then builds up the phase map for larger times via a local interpolation scheme together with the group property of the phase flow. Specifically, fix a small time τ > 0 and suppose that T = 2n τ . Step 1. Begin with a uniform or quasi-uniform grid on M. Step 2. Compute an approximation of the phase map Fτ at time τ . The value of Fτ at each grid point is computed by applying a standard ODE or Hamiltonian integrator with a single time step of length τ . The value of Fτ at any other point is defined via a local interpolation. Step 3. For k = 1, . . . , n, construct F2k τ using the group relation F2k τ = F2k−1 τ ◦ F2k−1 τ . Thus, for each grid point (y, ζ), F2k τ (y, ζ) = F2k−1 τ (F2k−1 τ (y, ζ)), while F2k τ is defined via a local interpolation at any other point. When the algorithm terminates, one obtains an approximation of the whole phase map at time T = 2n τ . This method is clearly much faster than solving each for initial condition independently.

8. Gaussian beam methods: Lagrangian approach A common numerical problem with all numerical approaches based on the Liouville equation with mono-kinetic initial data (3.8)–(3.9) is that the particle density ρ(t, x) blows up at caustics. Another problem is the loss of phase information when passing through a caustic point, i.e., the loss of the Keller–Maslov index (Maslov 1981). To this end, we recall that the Wigner measure only sees the gradient of the phase: see (3.10). The latter can be fixed by incorporating this index into a level set method as was done in Jin and Yang (2008)). Nevertheless, one still faces the problem that any numerical method based on the Liouville equation is unable to handle wave

156


interference effects. The Gaussian beam method (or Gaussian wave packet approach, as it is called in quantum chemistry: see Heller (2006)), is an efficient approximate method that allows an accurate computation of the wave amplitude around caustics, and in addition captures the desired phase information. This now classical method was developed in Popov (1982), Ralston (1982) and Hill (1990), and has seen increasing activity in recent years. In the following, we shall describe the basic ideas, starting with its classical Lagrangian formulation.

8.1. Lagrangian dynamics of Gaussian beams Similar to the WKB method, the approximate Gaussian beam solution is given in the form ϕε (t, x, y) = A(t, y) e iT (t,x,y)/ε ,

(8.1)

where the variable y = y(t, y0 ) will be determined below and the phase T (t, x, y) is given by 1 T (t, x, y) = S(t, y) + p(t, y) · (x − y) + (x − y) M (t, y)(x − y) + O(x − y|3 ). 2 This is reminiscent of the Taylor expansion of the phase S around the point y, upon identifying p = ∇S ∈ Rd , M = ∇2 S, the Hessian matrix. The idea is now to allow the phase T to be complex-valued (in contrast to WKB analysis), and choose the imaginary part of M ∈ Cn×n positive definite so that (8.1) indeed has a Gaussian profile. Plugging the ansatz (8.1) into the Schr¨ odinger equation (2.1), and ignoring the higher-order terms in both ε and (y − x), one obtains the following system of ODEs: dy = p, dt dM = −M 2 − ∇2y V, dt dS 1 = |p|2 − V, dt 2

dp = −∇y V, dt

(8.2) (8.3)

dA 1 = − Tr(M ) A, dt 2

(8.4)

where p, V, M, S and A have to be understood as functions of (t, y(t, y0 )). The latter defines the centre of a Gaussian beam. Equations (8.2)–(8.4) can be considered as the Lagrangian formulation of the Gaussian beam method, with (8.2) furnishing a classical ray-tracing algorithm. We further note that (8.3) is a Riccati equation for M . We state the main properties of (8.3), (8.4) in the following theorem, the proof of which can be found in Ralston (1982) (see also Jin, Wu and Yang (2008b)).


157

Theorem 8.1. Let P (t, y(t, y0 )) and R(t, y(t, y0 )) be the (global) solutions of the equations dP dR (8.5) = R, = −(∇2y V )P, dt dt with initial conditions P (0, y0 ) = Id,

R(0, y0 ) = M (0, y0 ),

(8.6)

where Id is the identity matrix and Im(M (0, y0 )) is positive definite. Assume that M (0, y0 ) is symmetric. Then, for each initial position y0 , we have the following properties. (1) P (t, y(t, y0 )) is invertible for all t > 0. (2) The solution to equation (8.3) is given by M (t, y(t, y0 )) = R(t, y(t, y0 ))P −1 (t, y(t, y0 )).

(8.7)

(3) M (t, y(t, y0 )) is symmetric and Im(M (t, y(t, y0 ))) is positive definite for all t > 0. (4) The Hamiltonian H = 12 |p|2 + V is conserved along the y-trajectory, as is (A2 det P ), i.e., A(t, y(t, y0 )) can be computed via 1/2 A(t, y(t, y0 )) = (det P (t, y(t, y0 )))−1 A2 (0, y0 ) , (8.8) where the square root is taken as the principal value. In particular, since (A2 det P ) is a conserved quantity, we infer that A does not blow up along the time evolution (provided it is initially bounded). 8.2. Lagrangian Gaussian beam summation It should be noted that a single Gaussian beam given by (8.1) is not an asymptotic solution of (2.1), since its L2 (R2d )-norm goes to zero, in the classical limit ε → 0. Rather, one needs to sum over several Gaussian beams, the number of which is O(ε−1/2 ). This is referred to as the Gaussian beam summation; see for example Hill (1990). In other words, one first needs to approximate given initial data through Gaussian beam profiles. For WKB initial data (2.18), a possible way to do so is given by the next theorem, proved by Tanushev (2008). Theorem 8.2.

Let the initial data be given by uεin (x) = ain (x) e iSin (x)/ε ,

with ain ∈ C 1 (Rd ) ∩ L2 (Rd ) and Sin ∈ C 3 (Rd ), and define ϕε (x, y0 ) = ain (y0 ) e iT (x,y0 )/ε ,

158


where 1 T (x, y0 ) = Tα (y0 ) + Tβ · (x − y0 ) + (x − y0 ) Tγ (x − y0 ), 2 Tα (y0 ) = Sin (y0 ), Tβ (y0 ) = ∇x Sin (y0 ), Tγ (y0 ) = ∇2x Sin (y0 ) + i Id. Then

ε uin − (2πε)−d/2

rθ (· − y0 )ϕ (·, y0 ) dy0 d

1

ε

R

L2

≤ Cε 2 ,

where rθ ∈ C0∞ (Rd ), rθ ≥ 0 is a truncation function with rθ ≡ 1 in a ball of radius θ > 0 around the origin, and C is a constant related to θ. In view of Theorem 8.2, one can specify the initial data for (8.2)–(8.4) as y(0, y0 ) = y0 ,

p(0, y0 ) = ∇x Sin (y0 ),

∇2x Sin (y0 )

M (0, y0 ) = S(0, y0 ) = Sin (y0 ),

+ i Id, A(0, y0 ) = ain (y0 ).

(8.9) (8.10) (8.11)

Then, the Gaussian beam solution approximating the exact solution of (2.1) is given by rθ (x − y(t, y0 ))ϕε (t, x, y(t, y0 )) dy0 . uεG (t, x) = (2πε)−d/2 Rd

In discretized form this reads uεG (t, x)

−d/2

≈ (2πε)

Ny0

rθ (x − y(t, y0j ))ϕε (t, x, y0j )∆y0 ,

j=1

where the y0j are equidistant mesh points, and Ny0 is the number of the beams initially centred at y0j . Remark 8.3. Note that the cut-off error introduced via rθ becomes large when the truncation parameter θ is taken too small. On the other hand, a big θ for wide beams makes the error in the Taylor expansion of T large. As far as we know, it is still an open mathematical problem to determine an optimal size of θ when beams spread. However, for narrow beams one can take a fairly large θ, which makes the cut-off error almost zero. For example, a one-dimensional constant solution can be approximated by ∆y0 −(x − y0 )2 −(x − y0j )2 1 √ √ 1= exp exp dy0 ≈ , 2ε 2ε 2πε 2πε R j in which rθ ≡ 1.


159

8.3. Higher-order Gaussian beams The above Gaussian beam method can be extended to higher order in ε: see Tanushev (2008), Jin et al. (2008b) and Liu and Ralston (2010). For notational convenience we shall only consider the case d = 1. Consider the Schrödinger equation (2.1) with initial data uεin (x) = e iSin (x)/ε

N

εj aj (x),

x ∈ R.

j=0

Let a ray y(t, y0 ) start at a point y0 ∈ R. Expand Sin (x) in a Taylor series around y0 : Sin (x) =

k+1

Sβ (y0 )(x − y0 )β + O(|x − y0 |β+1 ).

β=0

Then, a single kth-order Gaussian beam takes the form

[k/2]−1

ϕεk (t, x, y)

=

εj Aj (t, y) e iT (t,x,y)/ε ,

j=1

where the phase is given by 1 1 Tβ (t, y)(x−y)β , T (t, x, y) = T0 (t, y)+p(t, y)(x−y)+ M (t, y)(x−y)2 + 2 β! k+1

β=3

and the amplitude reads Aj =

k−2j−1 β=0

1 Aj,β (t, y)(x − y)β . β!

Here the bi-characteristic curves (y(t, y0 ), p(t, y(t, y0 ))) satisfy the Hamiltonian system (8.2) with initial data y(0, y0 ) = y0 ,

p(0, y0 ) = ∂y S0 (y0 ).

In addition, the equations for the phase coefficients along the bi-characteristic curves are given by dT0 p2 = − V, dt 2 dM = −M 2 − ∂y2 V, dt β dTβ (β − 1)! =− Tγ Tβ−γ+2 − ∂yβ V, dt (γ − 1)!(β − γ)! γ=2

160


for β = 3, . . . , k + 1. These equations are equipped with initial data T0 (0, y0 ) = S0 (y0 ),

M (0, y0 ) = ∂y2 S0 (y0 ) + i Id,

Tβ (0, y0 ) = Sβ (y0 ).

Finally, the amplitude coefficients are obtained recursively by solving the transport equations for Aj,β with β ≤ k − 2j − 1, starting from dA0,0 1 = − Tr(M (t, z))A0,0 , dt 2

(8.12)

with initial data Aj,β (0, y0 ) = aβ (y0 ). In d = 1, the kth-order Gaussian beam superposition is thus formed by ε −1/2 rθ (x − y(t, y0 ))ϕεk (t, x, y(t, y0 )) dy0 , (8.13) uG,k (t, x) = (2πε) R ∞ C0 (R; R)

is some cut-off function. For this type where, as before, rθ ∈ of approximation, the following theorem was proved in Liu, Runborg and Tanushev (2011). Theorem 8.4. If uε (t, x) denotes the exact solution to the Schrödinger equation (2.1) and uεG,k is the kth-order Gaussian beam superposition, then sup uε (t, ·) − uεG,k (t, ·) ≤ C(T )εk/2 ,

(8.14)

|t| 0.

9. Gaussian beam methods: Eulerian approach 9.1. Eulerian dynamics of Gaussian beams The Gaussian beam method can be reformulated in an Eulerian framework. To this end, let us first define the linear Liouville operator as L = ∂ t + ξ · ∇ y − ∇ y V · ∇ξ . In addition, we shall denote Φ := (φ1 , . . . , φd ), where φj is the level set function defined in (7.1) and (7.3). Using this, Jin et al. (2005b) and Jin and Osher (2003) showed that one can obtain from the original Lagrangian formulation (8.2)–(8.4) the following (inhomogeneous) Liouville equations for velocity, phase and amplitude, respectively: LΦ = 0, 1 LS = |ξ|2 − V, 2 1 LA = Tr (∇ξ Φ)−1 ∇y Φ A. 2

(9.1) (9.2) (9.3)


161

In addition, if one introduces the quantity (Jin et al. 2005b) f (t, y, ξ) = A2 (t, y, ξ) det(∇ξ Φ), then f (t, y, ξ) again satisfies the Liouville equation, i.e., Lf = 0.

(9.4)

Two more inhomogeneous Liouville equations, which are the Eulerian version of (8.5) for P and R, were introduced in Leung, Qian and Burridge (2007) to construct the Hessian matrix. More precisely, one finds LR = −(∇2y V )P,

(9.5)

LP = R.

(9.6)

Note that the equations (9.1)–(9.4) are real , while (9.5) and (9.6) are complex and consist of 2n2 equations. Gaussian beam dynamics using complex level set functions In Jin et al. (2008b) the following observation was made. Taking the gradient of the equation (9.1) with respect to y and ξ separately, we have L(∇y Φ) = ∇2y V ∇ξ Φ,

(9.7)

L(∇ξ Φ) = −∇y Φ.

(9.8)

Comparing (9.5)–(9.6) with (9.7)–(9.8), one observes that −∇y Φ and ∇ξ Φ satisfy the same equations as R and P , respectively. Since the Liouville operator is linear, one can allow Φ to be complex-valued and impose for −∇y Φ, ∇ξ Φ the same initial conditions as for R and P , respectively. By doing so, R = −∇x Φ, P = ∇ξ Φ hold true for any time t ∈ R. In view of (8.6) and (8.10), this suggests the following initial condition for Φ: Φ0 (y, ξ) = −iy + (ξ − ∇y S0 ).

(9.9)

With this observation, one can now solve (9.1) for complex Φ, subject to initial data (9.9). Then the matrix M can be constructed by M = −∇y Φ(∇ξ Φ)−1 ,

(9.10)

where the velocity v = ∇y S is given by the intersection of the zero-level contours of the real part of Φ, i.e., for each component φj , Re(φj (t, y, ξ)) = 0,

at ξ = v(t, y) = ∇y S.

(9.11)

Note that in order to compute v, S and M , one only needs to solve d complex-valued homogeneous Liouville equations (9.1). The Eulerian level

162


set method of Jin et al. (2008b) (in complex phase space) can then be summarized as follows. Step 1. Solve (9.1) for Φ complex , with initial condition (9.9), and obtain the velocity v by the intersection of the zero-level sets of Re φj , j = 1, . . . , n. Step 2. Use −∇y Φ and ∇ξ Φ to construct M by (9.10) (note that these quantities are already available from the first step after discretizing the Liouville equation for Φ). Step 3. Integrate the velocity v along the zero-level sets (Gosse 2002, Jin and Yang 2008) to get the phase S. To do so, one performs a numerical integration following each branch of the velocity. The integration constants are obtained by the boundary condition and the fact that the multivalued phase is continuous when passing from one branch to the other. For example, if one considers a bounded domain [a, b] in space dimension d = 1, the phase function is given by x 1 t 2 S(t, x) = −V (a)t − v (τ, a) dτ + v(t, y) dy + S(0, a). (9.12) 2 0 a For more details on this and its extension to higher dimensions, see Jin and Yang (2008). Step 4. Solve (9.4) with the initial condition f0 (y, ξ) = A20 (y, ξ). Then amplitude A is given by A = (det(∇ξ Φ)−1 f )1/2 ,

(9.13)

where the square root has to be understood as the principal value. (We also refer to Jin, Wu and Yang (2011) for a more elaborate computation of A.) Note that all functions appearing in Steps 2–4 only need to be solved locally around the zero-level sets of Re φj , j = 1, . . . , n. Thus, the entire algorithm can be implemented using the local level set methods of Osher, Cheng, Kang, Shim and Tsai (2002) and Peng et al. (1999). For a given mesh size ∆y, the computational cost is therefore O((∆y)−d ln(∆y)−1 ), about the same as the local level set methods for geometrical optics computation: see Jin et al. (2005b). Remark 9.1. If one is only interested in computing the classical limit of (the expectation values of) physical observables, one observes that the only in (9.12) which affects a quadratic observable density for fixed time t is term x a v(t, y) dy. Thus, as long as one is only interested in physical observables, one can simply take x S(t, x) = v(t, y) dy (9.14) a


163

in the numerical simulations. That M and A are indeed well-defined via (9.10) and (9.13) is justified by the following theorem (which can be seen as the Eulerian version of Theorem 8.1), proved in Jin et al. (2008b). Theorem 9.2. Let Φ = Φ(t, y, ξ) ∈ C be the solution of (9.1) with initial data (9.9). Then, the following properties hold: (1) ∇ξ Φ is non-degenerate for all t ∈ R, (2) Im −∇y Φ(∇ξ Φ)−1 is positive definite for all t ∈ R, y, ξ ∈ Rd . Although det Re(∇ξ Φ) = 0 at caustics, the complexified Φ makes ∇ξ Φ non-degenerate, and the amplitude A, defined in (9.13), does not blow up at caustics. 9.2. Eulerian Gaussian beam summation As before, we face the problem of Gaussian beam summation, i.e., in order to reconstruct the full solution uε a superposition of single Gaussian beams has to be considered. To this end, we define a single Gaussian beam, obtained through the Eulerian approach, by ϕ˜ε (t, x, y, ξ) = A(t, y, ξ) eiT (t,x,y,ξ)/ε ,

(9.15)

where A is solved via (9.13), and 1 T (t, x, y, ξ) = S(t, y, ξ) + ξ · (x − y) + (x − y) M (t, y, ξ)(x − y). 2 Then, the wave function is constructed via the following Eulerian Gaussian beam summation formula (Leung et al. 2007): u ˜εG (t, x)

−d/2

= (2πε)

Rd ×Rd

rθ (x − y)ϕ˜ε (t, x, y, ξ)

d

δ(Re(φj )) dξ dy,

j=1

which is consistent with the Lagrangian summation formula (8.2). Indeed, the above given double integral for uεG can be evaluated as a single integral in y as follows. We again denote by vj , j = 1, . . . , J the jth branch of the multivalued velocity and write J ϕ˜ε (t, x, y, vk ) ε −d/2 dy. (9.16) u ˜G (t, x) = (2πε) rθ (x − y) | det(Re(∇ξ Φ)ξ=vj )| Rd j=1

However, since det Re(∇ξ Φ) = 0 at caustics, a direct numerical integration of (9.16) loses accuracy around caustics. To get better accuracy, one can

164


split (9.16) into two parts, J (2πε)−d/2 rθ (x − y) I1 = j=1

I2 =

J j=1

where

Ω1

Ω2

(2πε)−d/2 rθ (x − y)

Ω1 := y

Ω2 := y

ϕ˜ε (t, x, y, vk ) dy, det(Re(∇ξ Φ)ξ=vj )

(9.17)

ϕ˜ε (t, x, y, vk ) dy, det(Re(∇ξ Φ)ξ=vj )

(9.18)

: det(Re(∇p φ)(t, y, pj )) ≥ τ , : det(Re(∇p φ)(t, y, pj )) < τ ,

with τ being a small parameter. The latter is chosen sufficiently small to minimize the cost of computing (9.18), yet large enough to make I1 a regular integral. The regular integral I1 can then be approximated by a standard quadrature rule, such as the trapezoid quadrature rule, while the singular integral I2 is evaluated by the semi-Lagrangian method introduced in Leung et al. (2007). Remark 9.3. When the velocity contours are complicated due to large numbers of caustics, implementation of the local semi-Lagrangian method is hard. In such situations one can use a discretized δ-function method for numerically computing (9.18), as was done in Wen (2010). In this method one needs to numerically solve (9.2) in order to obtain the phase function, since all values of φj near the support of δ(Re(φj )) are needed to evaluate (9.18). Example 9.4. This is an example from Jin et al. (2008b). It considers the free motion of particles in d = 1 with V (x) = 0. The initial conditions for the Schr¨ odinger equation (2.1) are induced by ρin (x) = exp(−50x2 ),

vin (x) = ∂x S0 (x) = − tanh(5x).

Figure 9.1 shows the l∞ -errors between the square modulus of uε , the exact solution of the Schrödinger equation (2.1), and the approximate solution constructed by: (i) the level set method described in Section 7, (ii) the level set method with Keller–Maslov index built in (Jin and Yang 2008), and (iii) the Eulerian Gaussian beam method described above. As one can see, method (ii) improves the geometric optics solution (i) away from caustics, while the Gaussian beam method offers a uniformly small error even near the caustics. Compared to the Lagrangian formulation based on solving the ODE system (8.2)–(8.4), the Eulerian Gaussian beam method has the advantage of maintaining good numerical accuracy since it is based on solving PDEs on


(a)

(b)

(c)

Figure 9.1. Example 9.4: numerical errors between the solution of the Schr¨ odinger equation and (a) the geometrical optics solution, (b) the geometrical optics with phase shift built in, and (c) the Gaussian beam method. Caustics are around x = ±0.18.

165

166


fixed grids. Moreover, higher-order (in ε) Eulerian Gaussian beam methods have been constructed: see Liu and Ralston (2010) and Jin et al. (2011). Remark 9.5. Research on Gaussian beam methods is of great recent interest in the applied mathematics community: see, e.g., Leung et al. (2007), Jin et al. (2008b) and Leung and Qian (2009) for Eulerian formulations, Tanushev (2008), Motamed and Runborg (2010), Liu and Ralston (2010), Bougacha, Akian and Alexandre (2009) and Liu et al. (2011) for error estimates, and Tanushev, Engquist and Tsai (2009), Ariel, Engquist, Tanushev and Tsai (2011), Qian and Ying (2010) and Yin and Zheng (2011) for initial data decompositions. 9.3. Frozen Gaussian approximations The construction of the Gaussian beam approximation is based on the truncation of the Taylor expansion of the phase T (t, x, y) (8.1) around the beam centre y. Hence it loses accuracy when the width of the beam becomes large, i.e., when the imaginary part of M (t, y) in (8.1) becomes small so that the Gaussian function is no longer localized . This happens, for example, when the solution of the Schrödinger equation spreads (namely, the ray determined by (8.2) diverges), which can be seen as the time-reversed situation of caustic formation. The corresponding loss in the numerical computation can be overcome by re-initialization every once in a while: see Tanushev et al. (2009), Ariel et al. (2011), Qian and Ying (2010) and Yin and Zheng (2011). However, this approach increases the computational complexity in particular, when beams spread quickly. The frozen Gaussian approximation (as it is referred to in quantum chemistry), first proposed in Heller (1981), uses Gaussian functions with fixed widths to approximate the exact solution uε . More precisely, instead of using Gaussian beams only in the physical space, the frozen Gaussian approximation uses a superposition of Gaussian functions in phase space. That is why the method is also known by the name coherent state approximation. To this end, we first decompose the initial data into several Gaussian functions in phase space, 1 2 ε ψ (y0 , p0 ) = uεin (y) e−( ip0 ·(y−y0 )− 2 |y−y0 | )/ε dy, Rd

and then propagate the centre of each function (y(t), p(t)) along the Hamiltonian flow (8.2), subject to initial data at (y0 , p0 ). The frozen Gaussian beam solution takes the form uεFG (t, x) = −3d/2

1

(2πε)

Rd ×Rd

a(t, y0 , p0 )ψ(y0 , p0 ) e( ip(t)·(x−y(t))− 2 |x−y(t)|

2 )/ε

dp0 dy0 ,


167

where the complex-valued amplitude a(t, y0 , p0 ), is the so-called Herman– Kluk pre-factor : see Herman and Kluk (1984). Since the width of the Gaussians is fixed, one does not encounter the problem of beam spreading here. However, since this method is based in phase space, the computational cost is considerably higher than the standard Gaussian beam methods. For subsequent developments in this direction, see Herman and Kluk (1984), Kay (1994, 2006), Robert (2010), Swart and Rousse (2009) and Lu and Yang (2011).

10. Asymptotic methods for discontinuous potentials Whenever a medium is heterogeneous, the potential V can be discontinuous, creating a sharp potential barrier or interface where waves can be partially reflected and transmitted (as in the Snell–Descartes Law of Refraction). This gives rise to new mathematical and numerical challenges not present in the smooth potential case. Clearly, the semiclassical limit (3.8)–(3.9) does not hold at the barrier. Whenever V is discontinuous, the Liouville equation (3.8) contains characteristics which are discontinuous or even measure-valued. To this end, we recall that the characteristic curves (x(t), ξ(t)) are determined by the Hamiltonian system (3.6). The latter is a nonlinear system of ODEs whose right-hand side is not Lipschitz due to the singularity in ∇x H(x, ξ), and thus the classical well-posedness theory for the Cauchy problem of the ODEs fails. Even worse, the coefficients in the Liouville equation are in general not even BV (i.e., of bounded variation), for which almost everywhere solutions were introduced by DiPerna and Lions (1989) and Ambrosio (2004). Analytical studies of semiclassical limits in situations with interface were carried out by, e.g., Bal, Keller, Papanicolaou and Ryzhik (1999b), Miller (2000), Nier (1995, 1996) and Benedetto, Esposito and Pulvirenti (2004) using more elaborate Wigner transformation techniques, such as two-scale Wigner measures. 10.1. The interface condition In order to allow for discontinuous potentials, one first needs to introduce a notion of solutions of the underlying singular Hamiltonian system (3.6). This can be done by providing interface conditions for reflection and transmission, based on Snell’s law. The solution so constructed will give the correct transmission and reflection of waves through the barrier, obeying the laws of geometrical optics. Figure 10.1 (taken from Jin, Liao and Yang (2008a)) shows typical cases of wave transmissions and reflections through an interface. When the interface is rough, or in higher dimensions, the scattering can be diffusive, in which the transmitted or reflected waves can move in all directions: see

168


Reflection

A

B

Reflection

A

B

Transmission

Transmission

Incident

Incident

(a)

(b)

Figure 10.1. Wave transmissions and reflections through an interface.

Figure 10.1(b). However, in this section we will not discuss the diffusive interface, which was treated analytically in Bal et al. (1999b) and numerically in Jin et al. (2008a). An Eulerian interface condition Jin and Wen (2006a) provide an interface condition connecting the Liouville equations at both sides of a given (sharp) interface. Let us focus here only on the case of space dimension d = 1 and consider a particle moving with velocity ξ > 0 towards the barrier. The interface condition at a given fixed time t is given by w(t, x+ , ξ + ) = αT w(t, x− , ξ − ) + αR w(t, x+ , −ξ + ).

(10.1)

Here the superscripts ‘±’ represent the right and left limits of the quantities, αT ∈ [0, 1] and αR ∈ [0, 1] are the transmission and reflection coefficients respectively, satisfying αR + αT = 1. For a sharp interface x+ = x− , however, ξ + and ξ − are connected by the Hamiltonian preservation condition H(x+ , ξ + ) = H(x− , ξ − ).

(10.2)

The latter is motivated as follows. In classical mechanics, the Hamiltonian H = 12 ξ 2 + V (x) is conserved along the particle trajectory, even across the barrier. In this case, αT , αR = 0 or 1, i.e., a particle can be either completely transmitted or completely reflected. In geometric optics (corresponding to H(x, ξ) = c(x)|ξ|), condition (10.2) is equivalent to Snell’s Law of Refraction for a flat interface (Jin and Wen 2006b), i.e., waves can be partially transmitted or reflected.


169

Remark 10.1. In practical terms, the coefficients αT and αR are determined from the original Schrödinger equation (2.1) before the semiclassical limit is taken. Usually one invokes stationary scattering theory to do so. Thus (10.1) represents a multiscale coupling between the (macroscopic) Liouville equation and the (microscopic) Schrödinger (or wave) equation. Furthermore, by incorporating the diffraction coefficients, determined from the geometrical theory of diffraction developed in Keller and Lewis (1995), into the interface condition, one could even simulate diffraction phenomena near boundaries, interfaces or singular geometries (Jin and Yin 2008a, 2008b, 2011). The well-posedness of the initial value problem of the singular Liouville equation with the interface condition (10.1) was established in Jin and Wen (2006a), using the method of characteristics. To determine a solution at (x, p, t) one traces back along the characteristics determined by the Hamiltonian system (3.6) until hitting the interface. At the interface, the solution bifurcates according to the interface condition (10.1). One branch of the solution thereby corresponds to the transmission of waves and the other to the reflection. This process continues until one arrives at t = 0. The interface condition (10.1) thus provides a generalization of the method of characteristics. A Lagrangian Monte Carlo particle method for the interface A notion of the solution of the (discontinuous) Hamiltonian system (3.6) was introduced in Jin (2009) (see also Jin and Novak (2006)) using a probability interpretation. One thereby solves the system (3.6) using a standard ODE or Hamiltonian solver, but at the interface, the following Monte Carlo solution can be constructed (we shall only give a solution in the case of ξ − > 0; the other case is similar). (1) With probability αR , the particle (wave) is reflected with x → x,

ξ − → −ξ − .

(10.3)

(2) With probability αT , the particle (wave) is transmitted, with x → x,

ξ + obtained from ξ − using (10.2).

(10.4)

Although the original problem is deterministic, this probabilistic solution allows one to go beyond the interface with the new value of (x, ξ) defined in (10.3)–(10.4). This is the Lagrangian formulation of the solution determined by using the interface condition (10.1). This is the basis of a (Monte-Carlobased) particle method for thin quantum barriers introduced in Jin and Novak (2007).

170


10.2. Modification of the numerical flux at the interface A typical one-dimensional semi-discrete finite difference method for the Liouville equation (3.7) is ∂t wij + ξj

− + wi+ − wi− 1 1 ,j ,j 2

2

∆x

− DVi

wi,j+ 1 − wi,j− 1 2

2

∆ξ

= 0.

Here wij is the cell average or pointwise value of w(t, xi , ξj ) at fixed t. The numerical fluxes wi+ 1 ,j , wi,j+ 1 are typically defined by a (first-, or higher2 2 order) upwind scheme, and DVi is some numerical approximation of ∂x V at x = xi . When V is discontinuous, such schemes face difficulties when the Hamiltonian is discontinuous, since ignoring the discontinuity of V in the actual numerical computation will result in solutions which are inconsistent with the notion of the physically relevant solution, defined in the preceding subsection. Even with a smoothed Hamiltonian, it is usually impossible (at least in the case of partial transmission and reflection) to obtain transmission and reflection with the correct transmission and reflection coefficients. A smoothed V will also give a severe time step constraint like ∆t ∼ O(∆x∆ξ): see, e.g., Cheng, Kang, Osher, Shim and Tsai (2004). This is a parabolic-type CFL condition, despite the fact that we are solving a hyperbolic problem. A simple method to solve this problem was introduced by Jin and Wen (2005, 2006a). The basic idea is to build the interface condition (10.1) into the numerical flux , as follows. Assume V is discontinuous at xi+1/2 . First one should avoid discretizing V across the interface at xi+1/2 . One possible discretization is DVk =

− + − Vk−1/2 Vk+1/2

∆x

,

for k = i, i + 1,

where, for example, ± Vk+1/2 =

lim

x→xk+1/1 ±0

V (x).

The numerical flux in the ξ direction, Wi,j±1/2 , can be the usual numerical flux (for example, the upwind scheme or its higher-order extension). To ± define the numerical flux wi+1/2,j , without loss of generality, consider the − case ξj > 0. Using the upwind scheme, wi+ = wij . However, 1 ,j 2

+ wi+1/2,j

=

+ w(x+ i+1/2 , ξj )

= =

− + + αT w(x− i+1/2 , ξj ) + αR w(xi+1/2 , −ξj ) αT wi (ξj− ) + αR wi+1,−j ,

while ξ − is obtained from (10.2) with ξj+ = ξj . Since ξ − may not be a


171

grid point, one has to define it approximately. A simple approach is to locate the two cell centres that bound it, then use a linear interpolation to approximate the needed numerical flux at ξj− . The case of ξj < 0 is treated similarly. The detailed algorithm to generate the numerical flux is given in Jin and Wen (2005, 2006a). This numerical scheme overcomes the aforementioned analytic and numerical difficulties. In particular, it possesses the following properties. (1) It produces the correct physical solution crossing the interface (as defined in the previous subsection). In particular, in the case of geometric optics, this solution is consistent with the Snell–Descartes Law of Refraction at the interface. (2) It allows a hyperbolic CFL condition ∆t = O(∆x, ∆ξ). The idea outlined above has its origin in so-called well-balanced kinetic schemes for shallow water equations with bottom topography: see Perthame and Simeoni (2001). It has been applied to computation of the semiclassical limit of the linear Schr¨ odinger equation with potential barriers in Jin and Wen (2005), and the geometrical optics limit with complete transmission/reflection in Jin and Wen (2006b), for thick interfaces, and Jin and Wen (2006a), for sharp interfaces. Positivity and both l1 and l∞ stabilities were also established, under a hyperbolic CFL condition. For piecewise constant Hamiltonians, an l1 -error estimate of the first-order finite difference of this type was established in Wen (2009), following Wen and Jin (2009). Remark 10.2. Let us remark that this approach has also been extended to high-frequency elastic waves (Jin and Liao 2006), and high-frequency waves in random media with diffusive interfaces (Jin et al. 2008a). When the initial data are measure-valued, such as (3.9), the level set method introduced in Jin et al. (2005b) becomes difficult for interfaces where waves undergo partial transmissions and reflections, since one needs to increase the number of level set functions each time a wave hits the interface. A novel method to get around this problem has been introduced in Wei, Jin, Tsai and Yang (2010). It involves two main ingredients. (1) The solutions involving partial transmissions and partial reflections are decomposed into a finite sum of solutions, obtained by solving problems involving only complete transmissions or complete reflections. For the latter class of problems, the method of Jin et al. (2005b) applies. (2) A re-initialization technique is introduced such that waves coming from multiple transmissions and reflections can be combined seamlessly as new initial value problems. This is implemented by rewriting the sum of several delta functions as one delta measure with a suitable weight, which can easily be implemented numerically.

172


10.3. Semiclassical computation of quantum barriers Correct modelling of electron transport in nano-structures, such as resonant tunnelling diodes, superlattices or quantum dots, requires the treatment of quantum phenomena in highly localized regions within the devices (so-called quantum wells), while the rest of the device can be dealt with by classical mechanics. However, solving the Schr¨ odinger equation in the entire physical domain is usually too expensive, and thus it is attractive to use a multiscale approach as given in Ben Abdallah, Degond and Gamba (2002): that is, solve the quantum mechanics only in the quantum well, and couple the solution to classical mechanics outside the well. To this end, the following semiclassical approach for thin quantum barriers was proposed in Jin and Novak (2006). Step 1. Solve the time-independent Schr¨ odinger equation (either analytically or numerically) for the local barrier well to determine the scattering data, i.e., the transmission and reflection coefficients αT , αR . Step 2. Solve the classical Liouville equation elsewhere, using the scattering data generated in Step 1 and the interface condition (10.1) given above. The results for d = 1 and d = 2 given in Jin and Novak (2006, 2007) demonstrate the validity of this approach whenever the well is either very thin (i.e., of the order of only a few ε) or well separated. In higher dimensions, the interface condition (10.1) needs to be implemented in the direction normal to interface, and the interface condition may be non-local for diffusive transmissions or reflections (Jin and Novak 2007). This method correctly captures both the transmitted and the reflected quantum waves, and the results agree (in the sense of weak convergence) with the solution obtained by directly solving the Schrödinger equation with small ε. Since one obtains the quantum scattering information only in a preprocessing step (i.e., Step 1), the rest of the computation (Step 2) is classical, and thus the overall computational cost is the same as for computing classical mechanics. Nevertheless, purely quantum mechanical effects, such as tunnelling, can be captured. If the interference needs to be accounted for, then such Liouville-based approaches are not appropriate. One attempt was made in Jin and Novak (2010) for one-dimensional problems, where a complex Liouville equation is used together with an interface condition using (complex-valued ) quantum scattering data obtained by solving the stationary Schr¨ odinger equation. Its extension to multi-dimensional problems remains to be done. A more general approach could use Gaussian beam methods, which do capture the phase information. This is an interesting line of research yet to be pursued.


173

11. Schr¨ odinger equations with matrix-valued potentials and surface hopping Problems closely related to those mentioned in Section 10 arise in the study of semiclassical Schrödinger equations with matrix-valued potentials. This type of potential can be seen as a highly simplified model of the full manybody quantum dynamics of molecular dynamics. Using the celebrated Born–Oppenheimer approximation to decouple the dynamics of the electrons from the one for the much heavier nuclei (see, e.g., Spohn and Teufel (2001)), one finds that the electrons are subject to external forces which can be modelled by a system of Schrödinger equations for the nuclei along the electronic energy surfaces. The nucleonic Schr¨ odinger system has matrixvalued potentials, which will be treated in this section. To this end, we consider the following typical situation, namely, a timedependent Schrödinger equation with R2×2 -matrix-valued potential (see, e.g., Spohn and Teufel (2001), or Teufel (2003)): 2 ε ε iε∂t u = − ∆x + V (x) uε , uε (0, x) = uεin ∈ L2 (R2 , C2 ), (11.1) 2 for (t, x) ∈ R+ × R2 and ∆x = diag(∆x1 + ∆x2 , ∆x1 + ∆x2 ). The unknown uε (t, x) ∈ C2 and V is a symmetric matrix of the form 1 v (x) v2 (x) , (11.2) V (x) = tr V (x) + 1 v2 (x) −v1 (x) 2 with v1 (x), v2 (x) ∈ R. The matrix V then has two eigenvalues, λ(±) (x) = tr V (x) ± v1 (x)2 + v2 (x)2 . Remark 11.1. In the Born–Oppenheimer approximation, m the dimensionless semiclassical parameter ε > 0 is given by ε = M , where m and M are the masses of an electron and a nucleus respectively (Spohn and Teufel 2001). Then, all oscillations are roughly characterized by the frequency 1/ε, which typically ranges between one hundred and one thousand. 11.1. Wigner matrices and the classical limit for matrix-valued potentials In this section, we shall discuss the influence of matrix-valued potentials on the semiclassical limit of the Schrödinger equations (11.1). Introduce the Wigner matrix as defined in Gérard et al. (1997): ε ε ε ε −2 ε ε ¯ W [u ](x, ξ) = (2π) ψ x− η ⊗ψ x+ η e iη·ξ dy, (x, ξ) ∈ R2x ×R2ξ . 2 2 2 R We also let W denote the corresponding (weak) limit W ε [uε ] −→ W ∈ L∞ (R; M+ (R2x × R2ξ ; R2 )). ε→0

174


(a)

(b)

Figure 11.1. The conical crossings for potentials (a) Viso and (b) VJT defined in (11.3). The crossing point is (0, 0) for both potentials.

In order to describe the the dynamics of the limiting matrix-valued measure W (t, x, ξ), first note that the complex 2 × 2 matrix-valued symbol of (11.1) is given by i P (x, ξ) = |ξ|2 + iV (x). 2 The two eigenvalues of −iP (x, ξ) are |ξ|2 |ξ|2 + tr V (x) ± v1 (x)2 + v2 (x)2 = + λ± (x). 2 2 These eigenvalues λn , n = 1, 2 govern the time evolution of the limiting measure W (t), as proved in Gérard et al. (1997). They act as the correct classical Hamiltonian function in phase space, corresponding to the two energy levels, respectively. In the following, we shall say that two energy levels cross at a point x∗ ∈ R2 if λ+ (x∗ ) = λ− (x∗ ). Such a crossing is called conical if the vectors ∇x v1 (x∗ ) and ∇x v2 (x∗ ) are linearly independent. In Figure 11.1 we give two examples of conical crossings for potentials of the form x1 x2 x1 x2 Viso = , VJT = . (11.3) x2 −x1 x2 −x1 λ1,2 (x, ξ) =

If all the crossings are conical, the crossing set S = {x ∈ R2 : λ+ (x) = λ− (x)} is a sub-manifold of co-dimension two in R2 : see Hagedorn (1994). Assume that the Hamiltonian flows with Hamiltonians λn leave invariant the set Ω = (R2x × R2ξ )\S.


175

For (x, ξ) ∈ Ω, denote by χn (x, ξ) the column eigenvector corresponding to the eigenvalue λn (x, ξ), and the matrix Πn (x, ξ) = χn (x, ξ)(χτ (x, ξ)) is the orthogonal projection onto the eigenspace associated to λn (x, ξ). By Theorem 6.1 of Gérard et al. (1997), the matrix-valued Wigner measure W (t) commutes with the projectors Πn , outside the crossing set S, and can thus be decomposed as W (t, ·) = Π1 W (t, ·)Π1 + Π2 W (t, ·)Π2 . Since the eigenspaces are both one-dimensional, the decomposition is simplified to be W (t, ·) = W1 (t, ·)Π1 + W2 (t, ·)Π2 . The scalar functions Wn (t, x, ξ) given by Wn (t, x, ξ) = tr(Πn W (t, x, ξ)) are then found to satisfy the Liouville equations ∂t Wn + ∇ξ λn · ∇x Wn − ∇x λn · ∇ξ Wn = 0,

(t, x, ξ) ∈ R+ × Ω,

(11.4)

subject to initial data Wn (0) = tr(Πn W ),

(x, ξ) ∈ Ω.

(11.5)

The scalar functions Wn , n = 1, 2, are the phase space probability densities corresponding to the upper and lower energy levels, respectively. One can recover from them the particle densities ρn via Wn (t, x, ξ) dξ, n = 1, 2. (11.6) ρn (t, x) = R2ξ

In other words, the Liouville equations (11.4) yield the propagation of the Wigner measures W1 (t, ·) and W2 (t, ·) on any given time interval, provided that their supports do not intersect the eigenvalue crossing set S. However, analytical and computational challenges arise when their supports intersect the set S. In S the dynamics of W1 , W2 are coupled due to the non-adiabatic transitions between the two energy levels, and an additional hopping condition is needed (analogous to the interface condition considered in Section 10 above). The Landau–Zener formula Lasser, Swart and Teufel (2007) give a heuristic derivation of the nonadiabatic transition probability. The derivation is based on the Hamiltonian

176


system corresponding to the Liouville equations (11.4), i.e., x˙ n (t) = ∇ξ λn (t) = ξn (t), ξ˙n (t) = −∇x λn (t), n = 1, 2.

(11.7)

The basic idea is to insert the trajectories (x(t), ξ(t)) of the Hamiltonian systems (11.7) into the trace-free part of the potential matrix (11.2) to obtain a system of ordinary differential equations, given by d ε v1 (x(t)) v2 (x(t)) uε (t). iε u (t) = v2 (x(t)) −v1 (x(t)) dt The non-adiabatic transitions occur in the region where the spectral gap between the eigenvalues becomes minimal. The function h(x(t)) = |λ+ (x(t)) − λ− (x(t))| = 2|ϑ(x(t))| measures the gap between the eigenvalues in phase space along the classical trajectory (x(t), ξ(t)), where ϑ(x) = (v1 (x), v2 (x)) and | · | denotes the Euclidean norm. The necessary condition for a trajectory to attain the minimal gap is given by d |ϑ(x(t))|2 = ϑ(x(t)) · ∇x ϑ(x(t))ξ(t) = 0, dt ˙ where ∇x ϑ(x(t) is the Jacobian matrix of the vector ϑ(x(t)), and ξ(t) = x(t). Hence, a crossing manifold in phase space containing these points is given by

S ∗ = (x, ξ) ∈ R2x × R2ξ : ϑ(x(t)) · ∇x ϑ(x(t))ξ(t) = 0 . The transition probability when one particle hits S ∗ is assumed to be given by π (ϑ(x0 ) ∧ ∇x ϑ(x0 )ξ0 )2 ε T (x0 , ξ0 ) = exp − , (11.8) ε |∇x ϑ(x0 )ξ0 |3 which is the famous Landau–Zener formula (Landau 1932, Zener 1932). Note that T decays exponentially in x and ξ and 1 (x, ξ) ∈ S ∗ }, lim T ε = T0 = ε→0 0 (x, ξ) ∈ / S∗. In other words, as ε → 0, the transition between the energy bands only occurs on the set S ∗ , which is consistent with the result in the previous subsections. 11.2. Numerical approaches Lagrangian surface hopping One widely used numerical approach to simulation of the non-adiabatic dynamics at energy crossings is the surface hopping method, first proposed by


177

Tully and Preston (1971), and further developed in Tully (1990) and Sholla and Tully (1998). The basic idea is to combine the classical transports of the system on the individual potential energy surfaces λ± (x) that follow (11.7) with an instantaneous transitions at S ∗ from one energy surface to another. The rate of transition is determined by the Landau–Zener formula (11.8) whenever available, or computed by some quantum mechanical simulation locally around S ∗ . The hoppings were performed in a Monte Carlo procedure based on the transition rates. For a review of surface hopping methods see Drukker (1999). More recently, surface hopping methods have generated increasing interest among the mathematical community. For molecular dynamical simulations, Horenko, Salzmann, Schmidt and Sch¨ utte (2002) adopted the partial Wigner transform to reduce a full quantum dynamics into the quantum–classical Liouville equation, and then the surface hopping is realized by approximating the quantum Liouville equation using phase space Gaussian wave packets. From the analytical point of view, Lasser and Teufel (2005) and Fermanian Kammerer and Lasser (2003) analysed the propagation through conical surface crossings using matrix-valued Wigner measures and proposed a corresponding rigorous surface hopping method based on the semiclassical limit of the time-dependent Born–Oppenheimer approximation. In Lasser et al. (2007) and Kube, Lasser and Weber (2009) they used a particle method to solve the Liouville equation, in which each classical trajectory was subject to a deterministic branching (rather than Monte Carlo) process. Branching occurs whenever a trajectory attains one of its local minimal gaps between the eigenvalue surfaces. The new branches are consequently re-weighted according to the Landau–Zener formula for conical crossings These Lagrangian surface hopping methods are very simple to implement, and in particular, very efficient in high space dimensions. However, they require either many statistical samples in a Monte Carlo framework or the increase of particle numbers whenever hopping occurs. In addition, as is typical for Lagrangian methods, a complicated numerical re-interpolation procedure is needed whenever the particle trajectories diverge, in order to maintain uniform accuracy in time. Eulerian surface hopping The Eulerian framework introduced in Jin, Qi and Zhang (2011) consists of solving the two Liouville equations (11.4), with a hopping condition that numerically incorporates the Landau–Zener formula (11.8). Note that the Schrödinger equation (11.1) implies conservation of the total mass, which in the semiclassical limit ε → 0, locally away from S ∗ , yields d (W1 + W2 )(t, x, ξ) dξ dx = 0. (11.9) dt

178


For this condition to hold for all x, ξ, the total flux in the direction normal to S ∗ needs to be continuous across S ∗ . To ensure this, the Landau–Zener transition at S ∗ should be formulated as a continuity condition for the total flux in the normal direction en . Define the flux function jn (x, ξ) ∈ R2 for each eigenvalue surface via jn (x, ξ) = (∇ξ λn , −∇x λn ) Wn (x, ξ),

n = 1, 2.

Assume, before hopping, that the particle remains on one of the eigenvalue surfaces, i.e., − − − (1) j1 (x− 0 , ξ0 ) = 0 and j2 (x0 , ξ0 ) = 0, or − − − (2) j1 (x− 0 , ξ0 ) = 0 and j2 (x0 , ξ0 ) = 0.

Then the interface condition is given by + − j1 (x− j1 (x+ T (x0 , ξ0 ) 0 , ξ0 ) · en = 1 − T (x0 , ξ0 ) 0 , ξ0 ) · e n , + − T (x0 , ξ0 ) 1 − T (x0 , ξ0 ) j2 (x+ j2 (x− 0 , ξ0 ) · e n 0 , ξ0 ) · e n where (x± , ξ ± ) are the (pre- and post-hopping) limits to (x0 , ξ0 ) ∈ S ∗ along the direction en . Remark 11.2. There is a restriction of this approach based on the Liouville equation and the Landau–Zener transition probability. The interference effects generated when two particles from different energy levels arrive at S ∗ at the same time are thereby not accounted for, and thus important quantum phenomena, such as the Berry phase, are missing. One would expect that Gaussian beam methods could handle this problem, but this remains to be explored.

12. Schr¨ odinger equations with periodic potentials So far we have only considered ε-independent potential V (x). The situation changes drastically if one allows for potentials varying on the fast scale y = x/ε. An important example concerns the case of highly oscillatory potentials VΓ (x/ε), which are periodic with respect to a periodic lattice Γ ⊂ Rd . In the following we shall therefore consider Schrödinger equations of the form x ε ε2 ε ε u + V (x)uε , uε (0, x) = uεin (x). iε∂t u = − ∆u + VΓ (12.1) 2 ε Here V ∈ C ∞ denotes some smooth, slowly varying potential and VΓ is a rapidly oscillating potential (not necessarily smooth). For definiteness we shall assume that for some orthonormal basis {ej }dj=1 , VΓ satisfies VΓ (y + 2πej ) = VΓ (y) i.e., Γ =

(2πZ)d .

∀ y ∈ Rd ,

(12.2)


179

Remark 12.1. Equations of this form arise in solid state physics, where they are used to describe the motion of electrons under the action of an external field and a periodic potential generated by the ionic cores. This problem has been extensively studied from both physical and mathematical points of view: see, e.g., Ashcroft and Mermin (1976), Bensoussan, Lions and Papanicolaou (1978), Teufel (2003) and the references given therein. One of the most striking dynamical effects due to the inclusion of a periodic potential VΓ is the occurrence of so-called Bloch oscillations. These are oscillations exhibited by electrons moving in a crystal lattice under the influence of a constant electric field V (x) = F · x, F ∈ Rd (see Section 12.2 below). 12.1. Emergence of Bloch bands In order to better understand the influence of VΓ , we recall here the basic spectral theory for periodic Schrödinger operators of the form (Reed and Simon 1976): 1 Hper = − ∆y + VΓ (y). 2 With VΓ obeying (12.2), we have the following. (1) The fundamental domain of the lattice Γ = (2πZ)d is the interval Y = [0, 2π]d . (2) The dual lattice Γ∗ can then be defined as the set of all wave numbers k ∈ R for which plane waves of the form e ik·x have the same periodicity as the potential VΓ . This yields Γ∗ = Zd in our case. (3) The fundamental domain of the dual lattice, Y ∗ , i.e., the (first) Brillouin zone, is the set of all k ∈ R closer to zero than to any other d dual lattice point. In our case Y ∗ = − 12 , 12 , equipped with periodic boundary conditions, i.e., Y ∗ Td . By periodicity, it is sufficient to consider the operator Hper on the fundamental domain Y only, where we impose the following quasi-periodic boundary conditions: f (y + 2πej ) = e2 ikj π f (y) ∀ y ∈ R, k ∈ Y ∗ . (12.3) It is well known (Wilcox 1978) that under mild conditions on VΓ , the operator H admits a complete set of eigenfunction {ψm (y, k)}m∈N , parametrized by k ∈ Y ∗ . These eigenfunctions provide, for each fixed k ∈ Y ∗ , an orthonormal basis in L2 (Y ). Correspondingly there exists a countable family of real eigenvalues {E(k)}m∈N , which can be ordered as E1 (k) ≤ E2 (k) ≤ · · · ≤ Em (k) ≤ · · · , where the respective multiplicities are accounted for in the ordering. The set {Em (k) : k ∈ Y } ⊂ R is called the mth energy band of the operator Hper .

180

S. Jin and P. Markowich and C. Sparber 9 8 8 7 6 7 5 6

4 3

5

2

4

1

3 2

0

1

−1 −0.5

−0.4

−0.3

−0.2

−0.1

0

k

0.1

0.2

0.3

0.4

0.5

Figure 12.1. The eigenvalues Em (k), m = 1, . . . , 8 for Mathieu’s model: VΓ = cos y.

Concerning the dependence on k ∈ Y ∗ , it has been shown (Wilcox 1978) that for any m ∈ N there exists a closed subset X ⊂ Y ∗ such that Em (k) is analytic and ψm (·, k) can be chosen to be a real analytic function for all k ∈ Y ∗ \X. Moreover, Em−1 < Em (k) < Em+1 (k)

∀ k ∈ Y ∗ \X.

If this condition indeed holds for all k ∈ Y ∗ , then Em (k) is called an isolated Bloch band. Moreover, it is known that meas X = meas {k ∈ Y ∗ : En (k) = Em (k), n = m} = 0.

(12.4)

This set of Lebesgue measure zero consists of the so-called band crossings. See Figure 12.1 for an example of bands for Mathieu’s model with potential VΓ = cos y. Note that due to (12.3) one can rewrite ψm (y, k) as ψm (y, k) = e ik·y χm (y, k)

∀ m ∈ N,

(12.5)

for some 2π-periodic function χm (·, k), usually called Bloch functions. In terms of χm (y, k) the spectral problem for Hper becomes (Bloch 1928) H(k)χm (y, k) = Em (k)χm (y, k), χm (y + 2πej , k) = χm (y, k) ∀ k ∈ Y ∗ ,

(12.6)

where H(k) denotes the shifted Hamiltonian 1 H(k) := e− ik·y Hper e ik·y = (−i∇y + k)2 + VΓ (y). 2

(12.7)


181

Most importantly, the spectral data obtained from (12.6) allow us to decompose the original space L2 (Rd ) into a direct sum of so-called band Hilbert ∞ 2 d spaces: L (R ) = m=1 Hm . This is the well-known Bloch decomposition method , which implies that ∀ f ∈ L2 (Rd ) : f = fm , fm ∈ H m . (12.8) m∈N

The corresponding projection of f ∈ L2 (Rd ) onto the mth band space Hm is thereby given via (Reed and Simon 1976) f (ζ)ψ m (ζ, k) dζ ψm (y, k) dk. (12.9) fm (y) = Y∗

Rd

In the following, we shall also denote Cm (k) := f (ζ)ψ m (ζ, k) dζ,

(12.10)

Rd

the coefficient of the Bloch decomposition. 12.2. Two-scale WKB approximation Equipped with the basic theory of Bloch bands, we recall here an extension of the WKB method presented in Section 2.2 to the case of highly oscillatory periodic potentials. Indeed, it has been shown in Bensoussan et al. (1978) and Guillot, Ralston and Trubowitz (1988) that solutions to (12.1) can be approximated (at least locally in time) by x ε→0 ε (12.11) u (t, x) ∼ a(t, x)χm , ∇S e iS(t,x)/ε + O(ε), ε where χm is parametrized via k = ∇S(t, x). The phase function S thereby solves the semiclassical Hamilton–Jacobi equation ∂t S + Em (∇S) + V (x) = 0,

S(0, x) = Sin (x).

(12.12)

The corresponding semiclassical flow Xtsc : y → x(t, y) is given by x(t, ˙ y) = ∇k Em (k(t, y)), ˙ y) = −∇x V (x(t, y)), k(t,

x(0, x) = y,

(12.13a)

k(0, y) = ∇Sin (y).

(12.13b)

The wave vector k ∈ Y ∗ is usually called crystal momentum. In the case of a constant electric field V = F · x, equation (12.13b) yields k(t, y) = k − tF,

k = ∇Sin (y).

Note that since k ∈ Y ∗ Td , this yields a periodic motion in time of x(t, y), the so-called Bloch oscillations.

182


In addition, the leading-order amplitude in (12.11) is found to be the solution of the semiclassical transport equation (Carles, Markowich and Sparber 2004), ∂t a + ∇k Em (∇S) · ∇a +

1 divx (∇k Em (∇x φm ))am = (βm · ∇x V (x)) am , 2 (12.14)

where βm (t, x) := χm (y, ∇S), ∇k χm (y, ∇S) L2 (Y )

(12.15)

denotes the so-called Berry phase term (Carles et al. 2004, Panati, Spohn and Teufel 2006), which is found to be purely imaginary βm (t, x) ∈ (iR)d . It is, importantly, related to the quantum Hall effect (Sundaram and Niu 1999). The amplitude a is therefore necessarily complex-valued and exhibits a non-trivial phase modulation induced by the geometry of VΓ ; see also Shapere and Wilczek (1989) for more details. Note that (12.14) yields the following conservation law for ρ = |a|2 : ∂t ρ + div(ρ∇k Em (∇S)) = 0. The outlined two-scale WKB approximation again faces the problem of caustics. Furthermore, there is an additional problem of possible bandcrossings at which ∇k Em is no longer defined. The right-hand side of (12.11) can therefore only be regarded as a valid approximation for (possibly very) short times only. Nevertheless, it shows the influence of the periodic potential, which can be seen to introduce additional high-frequency oscillations (Γ-periodic) within uε . Remark 12.2. These techniques have also been successfully applied in weakly nonlinear situations (Carles et al. 2004). 12.3. Wigner measures in the periodic case The theory of Wigner measures discussed in Section 3 can be extended to the case of highly oscillatory potentials. The theory of the so-called Wigner band series was developed in Markowich, Mauser and Poupaud (1994) and Gérard et al. (1997). The basic idea is to use Bloch’s decomposition and replace the continuous momentum variable ξ ∈ Rd by the crystal momentum k ∈ Y ∗. A more general approach, based on space adiabatic perturbation theory (Teufel 2003), yields in the limit ε → 0 a semiclassical Liouville equation of the form sc ∂t w + {Hm , w} = 0,

w(0, x, k) = win (x, k),

(12.16)

where w(t, x, k) is the mth band Wigner measure in the Γ-periodic phase

183


space Rdx × Y ∗ , {·, ·} denotes the corresponding Poisson bracket, and sc = Em (k) + V (x) Hm

(12.17)

denotes the mth band semiclassical Hamiltonian.

13. Numerical methods for Schr¨ odinger equations with periodic potentials 13.1. Bloch-decomposition-based time-splitting method

The introduction of a highly oscillatory potential VΓ xε poses numerical challenges for the numerical computation of semiclassical Schr¨ odinger equations. It has been observed by Gosse (2006) and Gosse and Markowich (2004) that conventional split-step algorithms do not perform well. More precisely, to guarantee convergence of the scheme, time steps of order O(ε) are required. In order to overcome this problem, a new time-splitting algorithm based on Bloch’s decomposition method has been proposed in Huang, Jin, Markowich and Sparber (2007) and further developed in Huang, Jin, Markowich and Sparber (2008) and Huang, Jin, Markowich and Sparber (2009). The basic idea is as follows. Step 1. For t ∈ [tn , tn+1 ] one first solves iε∂t uε = −

x ε ε2 ∆uε + VΓ u . 2 ε

(13.1)

The main point is that by using the Bloch decomposition method, Step 1 can be solved exactly, i.e., only up to numerical errors. In fact, in each band space Hm , equation (13.1) is equivalent to iε∂t uεm = Em (−i∇)uεm ,

uεm ∈ Hm ,

(13.2)

where uεm ≡ Pεm uε is the (appropriately ε-scaled) projection of uε ∈ L2 (Rd ) onto Hm defined in (12.9) and Em (−i∇) is the Fourier multiplier corresponding to the symbol Em (k). Using standard Fourier transformation, equation (13.2) is easily solved by uεm (t, x) = F −1 e iEm (k)t/ε (Fum )(0, k) , (13.3) where F −1 is the inverse Fourier transform. In other words, one can solve (13.1) by decomposing uε into a sum of band-space functions uεm , each of which is propagated in time via (13.3). After resummation this yields uε (tn+1 , x). Once this is done, we proceed as usual to take into account V (x). Step 2. On the same time interval as before, we solve the ODE iε∂t uε = V (x)uε ,

(13.4)

184


where the solution obtained in Step 1 serves as initial condition for Step 2. In this algorithm, the dominant effects from the dispersion and the periodic lattice potential are computed in one step. It thereby maintains their strong interaction and treats the non-periodic potential as a perturbation. Because the split-step error between the periodic and non-periodic parts is relatively small, the time steps can be chosen considerably larger than a conventional time-splitting algorithm (see below). As in a conventional splitting method (see Section 5), the numerical scheme conserves the particle density ρε = |uε |2 on the fully discrete level. More importantly, if V (x) = 0, i.e., no external potential, the algorithm preserves the particle density (and hence the mass) in each individual band space Hm . Remark 13.1. Clearly, the algorithm given above is only first-order in time, but this can easily be improved by using the Strang splitting method (see Section 5). In this case, the method is unconditionally stable and comprises spectral convergence for the space discretization as well as secondorder convergence in time. Numerical calculation of Bloch bands In the numerical implementation of this algorithm a necessary prerequisite is the computation of Bloch bands Em (k) and Bloch eigenfunction χm (y, k). This requires us to numerically solve the eigenvalue problem (12.6). In one space dimension d = 1 we proceed as in Gosse and Markowich (2004), by expanding VΓ ∈ C 1 (R) in its Fourier series 2π 1 iλy VΓ (y) = V (λ) e , V (λ) = VΓ (y) e− iλy dy. 2π 0 λ∈Z

Clearly, if VΓ ∈ C ∞ (R) the corresponding Fourier coefficients V (λ) decay faster than any power, as λ → ±∞, in which case we only need to take into account a few coefficients to achieve sufficient accuracy. Likewise, expand the Bloch eigenfunction χm (·, k) in its respective Fourier series χ m (λ, k) e iλy . χm (y, k) = λ∈Z

For λ ∈ {−Λ, . . . , Λ − 1} ⊂ Z, one consequently approximates the spectral problem (12.6) by the algebraic eigenvalue problem     χ m (−Λ) χ m (−Λ) χ χ    m (1 − Λ)  m (1 − Λ) H(k)  (13.5)  = Em (k)  , .. ..     . . χ m (Λ − 1)

χ m (Λ − 1)


185

where the 2Λ × 2Λ matrix H(k) is given by H(k) =  V (0) + 12 (k − Λ)2 V (−1) ··· 1 2  V (1) V (0) + 2 (k − Λ + 1) · · ·   . .. .. ..  . . V (2Λ − 1) V (2Λ − 2) ···

V (1 − 2Λ) V (2 − 2Λ) .. .

   . 

V (0) + 12 (k + Λ − 1)2

The matrix H(k) has 2Λ eigenvalues. Clearly, this number has to be large enough to have sufficiently many eigenvalues Em (k) for the simulation, i.e., we require m ≤ 2Λ. Note, however, that the number Λ is independent of the spatial grid (in particular, independent of ε), thus the numerical costs of this eigenvalue problem are often negligible compared to those of the evolutionary algorithms (see below for more details). In higher dimensions d > 1, computing the eigenvalue problem (12.6) along these lines becomes numerically too expensive to be feasible. In many physical applications, however, the periodic potential splits into a sum of one-dimensional potentials, i.e., VΓ (y) =

d

VΓ (yj ),

Vj (yj + 2π) = Vj (yj ),

j=1

where y = (y1 , y2 , . . . , yd ) ∈ Rd . In this case, Bloch’s spectral problem can be treated separately (using a fractional step-splitting approach) for each coordinate yj ∈ R, as outlined before. Remark 13.2. In practical applications, the accurate numerical computation of Bloch bands is a highly non-trivial task. Today, though, there exists a huge amount of numerical data detailing the energy band structure of the most important materials used in the design of semiconductor devices, for example. In the context of photonic crystals the situation is similar. Thus, relying on such data one can in principle completely avoid the above eigenvalue computations and their generalizations to more dimensions. To this end, one should also note that, given the energy bands Em (k), we do not need any knowledge about VΓ to solve (12.1) numerically. Also, we remark that it was shown in Huang et al. (2009) that the Bloch-decomposition-based time-splitting method is remarkably stable with respect to perturbations of the spectral data. Implementation of the Bloch-decomposition-based time-splitting method In the numerical implementation we shall assume that VΓ admits the decomposition (13.1). In this case we can solve (12.1) by using a fractional step method, treating each spatial direction separately, i.e., one only needs

186


to study the one-dimensional equation

x ε ε2 ε u iε∂t u = − ∂xx u + Vj 2 ε ε

(13.6)

on the time interval [tn , tn+1 ]. This equation will be considered on a one-dimensional computational domain (a, b) ⊂ R, equipped with periodic boundary conditions (necessary to invoke fast Fourier transforms). We suppose that there are L ∈ N lattice cells within (a, b) and numerically compute uε at L × R grid points, for some R ∈ N. In other words we assume that there are R grid points in each lattice cell, which yields the discretization 1 −1 k = − + , where = {1, . . . , L} ⊂ N, 2 L 2π(r − 1) , where r = {1, . . . , R} ⊂ N, yr = R and thus un ≡ u(tn ) are evaluated at the grid points x,r = ε(2π( − 1) + yr ).

(13.7)

(13.8)

Note that in numerical computations one can use R L whenever ε 1, i.e., only a few grid points are needed within each cell. Keeping in mind the basic idea of using Bloch’s decomposition, one has the problem that the solution uε of (13.6) does not in general have the same periodicity properties as ϕm . A direct decomposition of uε in terms of this new basis of eigenfunctions is therefore not possible. This problem can be overcome by invoking the following unitary transformation for f ∈ L2 (R): f (ε(y + 2πγ)) e− i2πkγ , y ∈ Y, k ∈ Y ∗ , f (y) → f(y, k) := γ∈Z

with the properties f(y + 2π, k) = e2 iπk f(y, k),

f(y, k + 1) = f(y, k).

In other words f(y, k) admits the same periodicity properties with respect to k and y as the eigenfunction ψm (y, k). In addition, the following inversion formula holds: f (ε(y + 2πγ)) = (13.9) f(y, k) e i2πkγ dk. Y∗

Moreover, one easily sees that the Bloch coefficient, defined in (12.10), can be equivalently written as f(y, k)ψ m (y, k) dy, Cm (k) = (13.10) Y

which, in view of (12.5), resembles a Fourier integral. In fact, all of these formulae can be easily implemented by using the fast Fourier transform.


187

The numerical algorithm needed to perform Step 1 outlined above is then as follows. Step 1.1. First compute u ε at time tn by u ε (tn , x,r , k ) =

L

uε (tn , xj,r ) e− i2πk ·(j−1) ,

j=1

where x,r is as in (13.8). ε (t , k ) via (13.10): Step 1.2. Compute the coefficient Cm n ε Cm (tn , k ) ≈

R 2π ε u (tn , x,r , k )χm (yr , k ) e− ik yr . R r=1

ε (t , k) up to time tn+1 according to (13.2), Step 1.3. Evolve Cm n ε ε (tn+1 , k ) = Cm (tn , k ) e− iEm (k )∆t/ε . Cm

Step 1.4. u ε can be obtained at time tn+1 by summing over all band contributions, u (tn+1 , x,r , k ) = ε

M

ε Cm (tn+1 , k )χm (yr , k ) e ik yr .

m=1

Step 1.5. Perform the inverse transformation (13.9): 1 ε u (tn+1 , xj,r , kj ) e i2πkj (−1) . u (tn+1 , x,r , k ) ≈ L L

ε

j=1

This concludes the numerical procedure performed in Step 1. The Bloch-decomposition-based time-splitting method was found to be converging for ∆x = O(ε) and ∆t = O(1); see Huang et al. (2007) for more details. In other words, the time steps can be chosen independently of ε, a huge advantage in comparison to the standard time-splitting method used in Gosse (2006), for instance. Moreover, the numerical experiments done in Huang et al. (2007) show that, of only a few Bloch bands, Em (k) are sufficient to achieve very high accuracy, even in cases where V (x) is no longer smooth (typically m = 1, . . . , M , with M ≈ 8 is sufficient). Applications of this method are found in the simulation of lattice Bose–Einstein condensates (Huang et al. 2008) and of wave propagation in (disordered) crystals (Huang et al. 2009). Remark 13.3. For completeness, we recall the numerical complexities for the algorithm outlined above: see Huang et al. (2007). The complexities of Steps 1.1 and 1.5 are O(RL ln L), the complexities of Steps 1.2 and 1.4

188


are O(M LR ln R), and for Step 1.3 it is O(M L). The complexity of the eigenvalue problem (13.5) is O(Λ3 ). However, since Λ (or R) is independent of ε and since (13.5) needs to be solved only once (as a preparatory step), the computational costs for this step are negligible. In addition, since M and R are independent of ε, one can choose R L and M L, whenever ε 1. Finally, one should notice that the complexities in each time step are comparable to the usual time-splitting method. 13.2. Moment closure in Bloch bands It is straightforward to adapt the moment closure method presented in Section 6 to the case of periodic potentials. To this end, one considers the semiclassical Liouville equation (12.16), i.e., ∂t w + ∇k Em (k) · ∇x w − ∇x V (x) · ∇k w = 0, and close it with the following ansatz for the Wigner measure: w(t, x, k) =

J

|aj (t, x)|2 δ (k − vj (t, x)),

j=1

where we let δ denote the Γ∗ -periodic delta distribution, i.e., δ(· − γ ∗ ). δ = γ ∗ ∈Γ∗

By following this idea, Gosse and Markowich (2004) showed the applicability of the moment closure method in the case of periodic potentials in d = 1 (see also Gosse (2006)). In addition, self-consistent Schrödinger–Poisson systems were treated in Gosse and Mauser (2006). As mentioned earlier, extending this method to higher space dimensions d > 1 is numerically challenging. 13.3. Gaussian beams in Bloch bands The Gaussian beam approximation, discussed in Sections 8 and 9, can also be extended to the Schrödinger equation with periodic potentials. To this end, one adopts the Gaussian beam within each Bloch band of the Schrödinger equation (12.1). In the following, we shall restrict ourselves to the case d = 1, for simplicity. Lagrangian formulation As for the two-scale WKB ansatz (12.11), we define x i Tm (t,x,ym )/ε ε ∗ ϕm (t, x, ym ) = Am (t, ym )χm ∂x Tm , e , ε

(13.11)


189

where ym = ym (t, y0 ), and 1 Tm (t, x, ym ) = Sm (t, ym ) + pm (t, ym )(x − ym ) + Mm (t, ym )(x − ym )2 . 2 Here Sm ∈ R, pm ∈ R, am ∈ C, Mm ∈ C. In addition, we denote by χ∗m the function obtained by evaluating the usual Bloch function χm (y, k) (with real-valued k ∈ Y ∗ ) at the point y = x/ε and k = ∂x Tm ∈ C. To this end, we impose the following condition: χ∗m (y, z) = χm (y, z)

for z ∈ R.

One can derive a similar derivation of the Lagrangian formulation as in Section 8.1 that corresponds to the semiclassical Hamiltonian (12.17). For more details we refer to Jin, Wu, Yang and Huang (2010b). Here we only mention that, in order to define the initial values for the Gaussian beams, one first decomposes the initial condition, which is assumed to be given in two-scale WKB form, i.e., x iSin (x)/ε ε e uin (x) = bin x, , ε in terms of Bloch waves with the help of the stationary phase method (Bensoussan et al. 1978): ∞ x ε→0 ε in , ∂x Sin e iSin (x)/ε + O(ε), uin (x) ∼ am (x)χm ε m=1

where the coefficient ain m (x)

= Y

bin (x, y)χm (y, ∂x Sin ) dy.

When one computes the Lagrangian beam summation integral, the complexvalued ∂x Tm = pm + (x − ym )Mm can be approximated by the real-valued pm√with a Taylor truncation error of order O(|x − ym |). Since |x − ym | is of O( ε) (see Tanushev (2008) and Jin et al. (2008b)), this approximation does not destroy the total accuracy of the Gaussian beam method, yet it provides the benefit that the eigenfunction χ∗m (k, z) is only evaluated for real-valued k (and thus consistent with the Bloch decomposition method). However, because of this, the extension of this method to higher order becomes a challenging task. Eulerian formulation Based on the ideas presented in Section 9, an Eulerian Gaussian beam method for Schr¨ odinger equations with periodic potentials has been introduced in Jin et al. (2010b). It involves solving the (multivalued) velocity

190


um = ∂x Sm from the zero-level set of the function Φm , which satisfies the homogeneous semiclassical Liouville equation (12.16) in the form Lm Φm = 0, where the mth band Liouville operator Lm is defined as L m = ∂ t + ∂ k Em ∂ x − ∂ x V ∂ ξ ,

(13.12)

and Φm is the complex-valued d-dimensional level set function for the velocity corresponding to the mth Bloch band. They also solve the following inhomogeneous Liouville equations for the phase Sm and amplitude am : Lm Sm = k∂k Em − V, 1 Lm am = − ∂k2 Em Mm am + βm am ∂y V, 2 where Lm is defined by (13.12) and βm denotes the Berry phase term as given by (12.15). Here, the Hessian Mm ∈ C is obtained from Mm = −

∂y Φm . ∂ξ Φm

14. Schr¨ odinger equations with random potentials Finally, we shall consider (small) random perturbations of the potential V (x). It is well known that in one space dimension, linear waves in a random medium get localized even when the random perturbations are small: see, e.g., Fouque, Garnier, Papanicolaou and Sølna (2007). Thus the analysis here is restricted to three dimensions. (The two-dimensional case is difficult because of criticality, i.e., the mean-field approximation outlined below is most likely incorrect.) 14.1. Scaling and asymptotic limit Consider the Schr¨ odinger equation with a random potential VR : √ x ε ε2 ε ε ε iε∂t u = − ∆u + V (x)u + ε VR u , x ∈ R3 . 2 ε

(14.1)

Here VR (y) is a mean zero, stationary random function with correlation length of order one. Its correlation length is assumed to be of the same √ order as the wavelength. The ε-scaling given above is critical in the sense that the influence of the random potential is of the same order as the one given by V (x) (see also the remark below). We shall also assume that the fluctuations are statistically homogeneous and isotropic so that VR (x)VR (y) = R(|x − y|),

(14.2)


191

where · · denotes statistical average and R(|x|) is the covariance of random fluctuations. The power spectrum of the fluctuations is defined by −3 e iξ·x R(x) dx. (14.3) R(ξ) = (2π) R3

is a function of |ξ| When (14.2) holds, the fluctuations are isotropic and R only. Remark 14.1. Because of the statistical homogeneity, the Fourier transform of the random potential VR is a generalized random process with orthogonal increments + p). VR (ξ)VR (p) = R(ξ)δ(ξ

(14.4)

If the amplitude of these fluctuations is large, then purely random scattering will dominate and waves will be localized: see Fr¨ ohlich and Spencer (1983). On the other hand if the random fluctuations are too weak, they will not affect the transport of waves at all. Thus, in order to have scattering induced by the random potential and the influence of the slowly varying background V (x) affect the (energy transport of the) waves in comparable ways, the √ fluctuations in the random potential must be of order ε. Using the ε-scaled Wigner transformation, we can derive the analogue of (3.2) in the following form: ∂t wε + ξ · ∇x wε − Θε [V + VR ]wε = 0,

(14.5)

where the pseudo-differential operator Θε is given by (3.3). The behaviour of this operator as ε → 0 is very different from the case without VR , as can already be seen on the level of formal multiscale analysis: see Ryzhik, Papanicolaou and Keller (1996). Let y = x/ε be a fast spatial variable, and introduce an expansion of wε (t) in the following form: wε (t, x, ξ) = w(t, x, ξ) + ε1/2 w(1) (t, x, y, ξ) + εw(2) (t, x, y, ξ) + · · · . (14.6) Note that we hereby assume that the leading term w does not depend on the fast scale. We shall also assume that the initial Wigner distribution ε (x, ξ) tends to a smooth, non-negative function w (x, ξ), which decays win in sufficiently fast at infinity. Then, as ε → 0, one formally finds that wε (t) , i.e., the averaged solution to (14.5), is close to a limiting measure w(t), which satisfies the following linear Boltzmann-type transport equation: ∂t w + ξ · ∇x w − ∇x V · ∇ξ w = Qw. Here, the linear scattering operator Q is given by − p)δ(|ξ|2 − |p|2 )(w(x, p) − w(x, ξ)) dp, R(ξ Qw(x, ξ) = 4π R3

(14.7)

192


with differential scattering cross-section − p)δ(|ξ|2 − |p|2 ) σ(k, p) = 4π R(ξ and total scattering cross-section − p)δ(|ξ|2 − |p|2 ) dp. Σ(k) = 4π R(ξ R3

(14.8)

(14.9)

Note that the transport equation (14.7) has two important properties. First, the total mass (or energy, depending on the physical interpretation) is conserved, i.e., E(t) = w(t, x, ξ) dξ dx = E(0). (14.10) R3 ×R3

Second, the positivity of w(t, x, ξ) is preserved. Rigorous mathematical results concerning the passage from (14.1) to the transport equation (14.7) can be found in Spohn (1977), Dell’Antonio (1983) and Ho, Landau and Wilkins (1993) (which contains extensive references), and, more recently, in Erdös and Yau (2000). 14.2. Coupling with other media One can also study the problems when there are other media, including periodic media and interfaces (flat or random). In the case of periodic media coupled with random media, one can use the above multiscale analysis combined with the Bloch decomposition method to derive a system of radiative transport equations: see Bal, Fannjiang, Papanicolaou and Ryzhik (1999a). In the limit system ε → 0, the transport part is governed by the semiclassical Liouville equation (12.16) in each Bloch band, while the right-hand side is a non-local scattering operator (similar to the one above) coupling all bands. One can also consider random high-frequency waves propagating through a random interface. Away from the interface, one obtains the radiative transport equation (14.7). At the interface, due to the randomness of the interface, one needs to consider a diffusive transmission or reflection process, in which waves can be scattered in all directions (see Figure 10.1). To this end, a non-local interface condition has to be derived: see Bal et al. (1999b). So far there have been few numerical works on random Schrödinger equations of the form (14.1). Bal and Pinaud (2006) studied the accuracy of the radiative transport equation (14.7) as an approximation to (14.1). Jin et al. (2008a) discretized a non-local interface condition for diffusive scattering, similar in spirit to the treatment described in Section 10.2. We also mention that the temporal resolution issue for time-splitting approximations of the Liouville equations with random potentials was rigorously studied in Bal


193

and Ryzhik (2004). Finally, we refer to Fouque et al. (2007) and Bal, Komorowski and Ryzhik (2010) for a comprehensive reading on high-frequency waves in random media.

15. Nonlinear Schr¨ odinger equations in the semiclassical regime So far we have only considered linear Schrödinger equations. Nonlinear models, however, are almost as important, since they describe a large number of physical phenomena in nonlinear optics, quantum superfluids, plasma physics or water waves: see, e.g., Sulem and Sulem (1999) for a general overview. The inclusion of nonlinear effects poses new challenges for mathematical and numerical study. 15.1. Basic existence theory In the following we consider nonlinear Schr¨ odinger equations (NLS) in the form ε2 ∆uε + V (x)uε + εα f (x, |uε |2 )uε , uε (0, x) = uεin (x), (15.1) 2 with α ≥ 0 some scaling parameter, measuring (asymptotically) the strength of the nonlinearity. More precisely, we shall focus on two important classes of nonlinearities: iε∂t uε = −

(1) local gauge-invariant nonlinearities, where f = ±|uε |2σ , with σ > 0, (2) Hartree-type nonlinearities, where f = V0 ∗ |uε |, with V0 = V0 (x) ∈ R some given convolution kernel. The first case of nonlinearities has important applications in Bose–Einstein condensation and nonlinear optics, whereas the latter typically appears as a mean-field description for the dynamics of quantum particles, say electrons 1 (in which case one usually has V0 (x) = ± |x| in d = 3). Concerning the existence of solutions to (15.1), we shall in the following only sketch the basic ideas. For more details we refer to Cazenave (2003) and Carles (2008). As a first step we represent the solution to (15.1) using Duhamel’s formula: t ε ε α u (t) = U (t)uin − iε U ε (t − s) f ε (x, |uε |2 )(s)uε (s) ds, (15.2) 0

U ε (t)

where is the unitary semi-group generated by the linear Hamiltonian 1 H = − 2 ∆ + V (see the discussion in Section 2.1). The basic idea is to prove that the right-hand side of (15.2) defines a contraction mapping (in some suitable topology). To this end, one has to distinguish between the case of sub- and supercritical nonlinearities. The existence of a unique solution

194


uε ∈ C([0, ∞); L2 (Rd )) can be guaranteed, provided (sub-critical case) σ < 2 ∞ d q d d and/or V0 sufficiently smooth and decaying, say V0 ∈ L (R ) ∩ L (R ), with q > d4 . Remark 15.1. Unfortunately, this subcritical range of nonlinearities does not allow inclusion of several physically important examples, such as a cubic nonlinearity σ = 1 in three space dimensions (needed, for instance, in the description of Bose–Einstein condensates). In the case of supercritical nonlinearities σ > d2 , the space L2 (Rd ) is generally too large to guarantee existence of solutions. Typically, one restricts the initial data to uε ∈ H 1 (Rd ), i.e., initial data with finite kinetic energy, in order to control the nonlinear terms by means of Sobolev’s embedding. Assuming that uε ∈ H 1 , local-in-time existence can be guaranteed by using Strichartz estimates (see Cazenave (2003) for more details), and one obtains uε ∈ C([−T ε , T ε ]; H 1 (Rd )), for some (possibly small) existence time T ε > 0. 2 in order to use Remark 15.2. Strictly speaking, we require σ < d−2 Sobolev’s embedding to control the nonlinearity. Note, however, that the case of cubic NLS in d = 3 is allowed.

Even though local-in-time existence is relatively easy to achieve even in the supercritical case, the existence of a global-in-time solution to (15.1) generally does not hold. The reason is the possibility of finite-time blow-up, i.e., the existence of a time |T ∗ | < ∞ such that lim ∇uε = +∞.

t→T ∗

For T > T ∗ , a strong solution to (15.1) ceases to exist, and one can only hope for weak solutions, which, however, are usually not uniquely defined. In order to understand better which situations blow-up can occur in, consider, for simplicity, the case of a cubic NLS with V (x) = 0: ε2 (15.3) ∆uε ± |uε |2 uε , uε (0, x) = uεin (x). 2 Given a local-in-time solution to this equation, one can invoke the conservation of the (nonlinear) energy, i.e., ε2 1 ε ε 2 |∇u (t, x)| dx ± |uε (t, x)|4 dx = E[uεin ]. E[u (t)] = 2 Rd 2 Rd iε∂t uε = −

Since E[uεin ] < const., by assumption, we can see that in the case where the nonlinearity comes with ‘+’, the a priori bound on the energy rules out the possibility of finite-time blow-up. These nonlinearities are usually referred to as defocusing. On the other hand, in the case of a focusing nonlinearity, i.e., f = −|uε |2 , we can no longer guarantee the existence of global-in-time


195

solutions (in the supercritical regime). Rather, we have to expect finitetime blow-up to occur in general. Clearly, this phenomenon will also have a significant impact on numerical simulations, in particular for ε 1. 15.2. WKB analysis of nonlinear Schr¨ odinger equations In order to understand the influence of nonlinear terms in the limit ε → 0, one can again invoke a WKB approximation of the form (2.10): ε→0

uε (t, x) ∼ aε (t, x) e iS(t,x)/ε . Let us assume for simplicity that α ∈ N. Then, plugging the ansatz given above into (15.1), and ordering equal powers in ε, we see that S solves 0 if α > 0, 1 2 (15.4) ∂t S + |∇S| + V (x) = 2 2 f (x, |a| ) if α = 0. We see that in the case α = 0 we can no longer solve the Hamilton–Jacobi equation for the phase S independently of the amplitude a. In other words, the amplitude influences the geometry of the rays or characteristics. This is usually referred to as supercritical geometric optics (Carles 2008), not to be confused with the supercritical regime concerning the existence of solutions, as outlined in Section 15.1 above. Weakly nonlinear geometric optics In contrast, the situation for α > 0 (subcritical geometric optics) is similar to the linear situation, in the sense that the rays of geometric optics are still given by (2.13) and thus independent of the nonlinearity. In this case, the method of characteristics guarantees the existence of a smooth S ∈ C ∞ ([−T, T ] × Rd ) up to caustics, and one can proceed with the asymptotic expansion to obtain the following transport equation for the leading-order amplitude: 0 if α > 1, a (15.5) ∂t a + ∇S · ∇a + ∆S = 2 2 −if (x, |a| ) if α = 1. One sees that if α > 1, nonlinear effects are asymptotically negligible for ε 1. The solution is therefore expected to be roughly the same as in the linear situation (at least before caustics). For α = 1, however, nonlinear effects show up in the leading-order amplitude. Note, however, that by multiplying (15.5) with a ¯ and taking the real part, one again finds ∂t ρ + div(ρ∇S) = 0, which is the same conservation law for ρ = |a|2 as in the linear case. The nonlinear effects for α = 1 are therefore given solely by nonlinear phase

196


modulations of the leading-order amplitude. In the case of a simple cubic nonlinearity, one explicitly finds (Carles 2000) a0 (y(t, x)) iG(t,x) e , a(t, x) = Jt (y(t, x))

|t| ≤ T,

(15.6)

where the slowly varying phase G is given by t |a0 (y(τ, x))|2 dτ. G(t, x) = − Jτ (y(τ, x)) 0 This regime is therefore often called weakly nonlinear geometric optics, and indeed it is possible to prove a rigorous approximation result analogous to Theorem 2.3 in this case too. Supercritical geometric optics On the other hand, the situation for α = 0 is much more involved. Indeed, it can be easily seen that a naive WKB approximation breaks down since the system of amplitude equation is never closed, i.e., the equation for an , obtained at O(εn ), involves an+1 , and so on (a problem which is reminiscent of the moment closure problem discussed in Section 6 above). This difficulty was overcome by Grenier (1998) and Carles (2007a, 2007b), who noticed that there exists an exact representation of uε in the form uε (t, x) = aε (t, x) e iS

ε (t,x)/ε

,

(15.7)

with real-valued phase S ε (possibly ε-dependent) and complex-valued amplitude aε . (Essentially, the right-hand side of (15.7) introduces an additional degree of freedom by invoking complex amplitudes.) Plugging (15.7) into (15.1) with α = 0, one has the freedom to split the resulting equations for aε and S ε as follows. On plugging the ansatz given above into (15.1), and ordering equal powers in ε, one sees that S solves 1 ∂t S ε + |∇S ε |2 + V (x) + f (x, |aε |2 ) = 0, 2 a ε ∂t aε + ∇S ε · ∇aε + ∆S ε − i ∆aε = 0. 2 2 Formally, we expect the limit of this system as ε → 0 to give a semiclassical approximation of uε , at least locally in time. Indeed this can be done by first rewriting these equations into a new system for ρε = |aε |2 and v ε = ∇S ε . Under some assumptions on the nonlinearity f (satisfied, for instance, in the cubic case), the obtained equations form a strictly hyperbolic system in which one can rigorously pass to the limit as ε → 0 to obtain ∂t v + v · ∇v + V (x) + f (ρ) = 0, ∂t ρ + div(ρv) = 0.

(15.8)


197

This is a system of (irrotational) Euler equations for a classical fluid with pressure law p(ρ) = f (ρ)/ρ. Following this approach, one can prove that as long as (15.8) admits (local-in-time) smooth solutions ρ, v, the quantum mechanical densities (2.6)–(2.7) indeed converge strongly in C([0, T ]; L1 (Rd )), with ε→0 ε→0 ρε −→ ρ, J ε −→ ρ. Reconstructing from these limits an approximate solution of the nonlinear Schrödinger equation in WKB form, i.e., uεapp (t, x) = ρ(t, x) e iS(t,x)/ε , v(t, x) := ∇S(t, x), generally requires some care (see Carles (2007b) for more details). Essentially one needs to take into account a higher-order corrector to ensure that the formal approximation is indeed correct up to errors of order O(ε), independent of t ∈ [0, T ]. The semiclassical limit for α = 0 for |t| > T , i.e., beyond the formation of shocks in (15.8), is a very challenging mathematical problem. In the one-dimensional case the only available result is given by Jin, Levermore and McLaughlin (1999), using the inverse scattering technique. The semiclassical limit of focusing NLS is more delicate, but can also be obtained by inverse scattering: see Kamvissis, McLaughlin and Miller (2003). Remark 15.3. Let us close this subsection by noticing that the analysis for Hartree-type nonlinearities is found to require slightly less sophistication than for local ones (Alazard and Carles 2007) and that the intermediate case 0 < α < 1 can be understood as a perturbation of the situation for α = 0 (Carles 2007b). In addition, a threshold condition for global smooth solutions to (15.8) has been determined in Liu and Tadmor (2002). 15.3. Wigner measure techniques for nonlinear potentials One might hope to extend the results for Wigner measures given in Section 3 to nonlinear problems. This would have the considerable advantage of avoiding problems due to caustics. Unfortunately, this idea has not been very successful so far, the reason being that Wigner measures are obtained as weak limits only, which in general is not sufficient to pass to the limit in nonlinear terms. Indeed, one can even prove an ill-posedness result for Wigner measures in the nonlinear case (Carles 2001). A notable exception is the case of Hartree nonlinearities f = V0 ∗ |uε | with smooth interaction kernels V0 ∈ Cb1 (Rd ) (Lions and Paul 1993). In this case the Wigner measure associated to uε is found to be a solution of the self-consistent Vlasov equation: ∂t w + ξ · ∇ξ w − ∇x (V0 ∗ ρ) · ∇ξ w = 0, ρ(t, x) = w(t, x, dξ). Rd

198


The physically more interesting case of non-smooth interaction kernels, such 1 as V0 ∼ |x| , which describes the coupling to a Poisson equation, is not covered by this result and can only be established in the particular case of the fully mixed quantum state; see Lions and Paul (1993) and Markowich and Mauser (1993) for more details, and also Zhang (2002) for a similar result valid for short times only.

15.4. Numerical challenges Due to the introduction of nonlinear effects, the numerical difficulties discussed in Sections 4 and 5 are enhanced. The main numerical obstacles are the formation of singularities in focusing nonlinear Schrödinger equations and the creation of new scales at caustics for both focusing and defocusing nonlinearities. Basic numerical studies of semiclassical NLS were conducted in Bao et al. (2002) and Bao, Jin and Markowich (2003b), with the result that finite difference methods typically require prohibitively fine meshes to even approximate observables well in semiclassical defocusing and focusing NLS computations. In the case when these very restrictive meshing constraints are bypassed, the usual finite difference schemes for NLS can deliver incorrect approximations in the classical limit ε → 0, without any particular sign of instability (Carles and Gosse 2007). Time-splitting spectral schemes are therefore the preferred method. To this end, we refer to Jin, Markowich and Zheng (2004) for the application of the time-splitting spectral method to the Zakharov system, to Huang, Jin, Markowich, Sparber and Zheng (2005) for the numerical solution of the Dirac–Maxwell system, and to Bao et al. (2003b) for numerical studies of nonlinear Schrödinger equations. In addition, let us mention Bao, Jaksch and Markowich (2003a, 2004), where numerical simulations of the cubically nonlinear Gross–Pitaevskii equation (appearing in the description of Bose–Einstein condensates) are given using time-splitting trigonometric spectral methods. A numerical study of ground state solutions of the Gross–Pitaevskii equation can be found in Bao, Wang and Markowich (2005). Remark 15.4. Note, however, that in the nonlinear case, even for ε > 0 fixed, a rigorous convergence analysis of splitting methods is considerably more difficult than for linear Schr¨ odinger equations: see, e.g., Lubich (2008), Gauckler and Lubich (2010) and Faou and Grebert (2010). Due to the nonlinear creation of new highly oscillatory scales in the limit ε → 0, time-splitting methods suffer from more severe meshing restrictions for NLS than for linear Schrödinger equations, particularly after the appearance of the first caustics in the corresponding WKB approximation; see Bao et al. (2003b) and Carles and Gosse (2007) for more details. In the weakly


199

nonlinear regime the following meshing strategy is sufficient: ∆x = O(ε),

∆t = O(ε)

(to be compared with (5.5)). In the regime of supercritical geometric optics, however, one typically requires (even for quadratic observable densities) ∆x = O(ε),

∆t = o(ε),

i.e., a severe restriction on the time step. In addition, one may need to invoke the Krasny filter (Krasny 1986), i.e., high Fourier-mode cut-offs, to avoid artifacts (such as symmetry breaking) in focusing NLS computations (Bao et al. 2003b). The latter, however, violates the conservation of mass: a clear drawback from the physics point of view. In order to overcome this problem, higher-order methods (in time) have to be deployed, such as exponential time-differencing or the use of integrating factors, and we refer to Klein (2008) for a comparison of different fourth-order methods for cubic NLS in the semiclassical regime. Remark 15.5. In the closely related problem of the complex Ginzburg– Landau equation in the large space and time limit, the situation is known to be slightly better, due to the dissipative nature of the equation; see Degond, Jin and Tang (2008) for a numerical investigation. Finally, we note that the cubic NLS in d = 1 is known to be fully integrable by means of inverse scattering. This feature can be used in the design of numerical algorithms, as has been done in Zheng (2006), for instance. A generalization to higher dimensions or more general nonlinearities seems to be out of reach at present. The lack of a clear mathematical understanding of the asymptotic behaviour of solutions to the semiclassical NLS beyond the formation of caustics has so far hindered the design of reliable asymptotic schemes. One of the few exceptions is the case of the Schr¨ odinger–Poisson equation in d = 1, which can be analysed using Wigner measures, and which has recently been studied numerically in Jin, Wu and Yang (2010a) using a Gaussian beam method. In addition, moment closure methods have been employed for this type of nonlinearity, since it is known that the underlying classical problem, i.e., the Euler–Poisson system, allows for a construction of multivalued solutions. Numerical simulations for the classical system have been conducted in Wöhlbier et al. (2005). In addition, the case of the Schr¨ odinger–Poisson equation with periodic potential is treated in Gosse and Mauser (2006).

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Tsunami modelling with adaptively refined finite volume methods∗ Randall J. LeVeque Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA E-mail: [email protected]

David L. George US Geological Survey, Cascades Volcano Observatory, Vancouver, WA 98683, USA E-mail: [email protected]

Marsha J. Berger Courant Institute of Mathematical Sciences, New York University, NY 10012, USA E-mail: [email protected] Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.

∗

Colour online for monochrome figures available at journals.cambridge.org/anu.

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R. J. LeVeque, D. L. George and M. J. Berger

CONTENTS 1 Introduction 2 Tsunamis and tsunami modelling 3 The shallow water equations 4 Finite volume methods 5 The nonlinear Riemann problem 6 Algorithms in two space dimensions 7 Source terms for friction 8 Adaptive mesh refinement 9 Interpolation strategies for coarsening and refining 10 Verification, validation, and reproducibilty 11 The radial ocean 12 The 27 February 2010 Chile tsunami 13 Final remarks References

212 217 226 238 242 248 251 252 256 265 268 273 282 283

1. Introduction Many fluid flow or wave propagation problems in geophysics can be modelled with two-dimensional depth-averaged equations, of which the shallow water equations are the simplest example. In this paper we focus primarily on the problem of modelling tsunamis propagating across an ocean and inundating coastal regions, but a number of related applications have also been tackled with depth-averaged approaches, such as storm surges arising from hurricanes or typhoons; sediment transport and coastal morphology; river flows and flooding; failures of dams, levees or weirs; tidal motions and internal waves; glaciers and ice flows; pyroclastic or lava flows; landslides, debris flows, and avalanches. These problems often share the following features. • The governing equations are a nonlinear hyperbolic system of conservation laws, usually with source terms (sometimes called balance laws). • The flow takes place over complex topography or bathymetry (the term used for topography below sea level). • The flow is of bounded extent: the depth goes to zero at the margins or shoreline and the ‘wet–dry interface’ is a moving boundary that must be captured as part of the flow. • There exist non-trivial steady states (such as a body of water at rest) that should be maintained exactly. Often the wave propagation or flow to be modelled is a small perturbation of this steady state. • There are multiple scales in space and/or time, requiring adaptively refined grids in order to efficiently simulate the full problem, even when

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two-dimensional depth-averaged equations are used. For geophysical problems it may be necessary to refine each spatial dimension by five orders of magnitude or more in some regions compared to the grid used on the full domain. Transoceanic tsunami modelling provides an excellent case study to explore the computational difficulties inherent in these problems. The goal of this paper is to discuss these challenges and present a set of computational techniques to deal with them. Specifically, we will describe the methods that are implemented in GeoClaw, open source software for solving this class of problems that is distributed as part of Clawpack (www3). The main focus is not on this specific software, however, but on general algorithmic ideas that may also be useful in the context of other finite volume methods, and also to problems outside the domain of geophysical flows that exhibit similar computational difficulties. For the interested reader, the software itself is described in more detail in Berger, George, LeVeque and Mandli (2010) and in the GeoClaw documentation (www7). We will also survey some uses of tsunami modelling and a few of the challenges that remain in developing this field, and geophysical flow modelling more generally. This is a rich source of computational and modelling problems with applicability to better understanding a variety of hazards throughout the world. The two-dimensional shallow water equations generally provide a good model for tsunamis (as discussed further below), but even so it is essential to use adaptive mesh refinement (AMR) in order to efficiently compute accurate solutions. At specific locations along the coast it may be necessary to model small-scale features of the bathymetry as well as levees, sea walls, or buildings on the scale of metres. Modelling the entire ocean with this resolution is clearly both impossible and unnecessary for a tsunami that may have originated thousands of kilometres away. In fact, the wavelength of a tsunami in the ocean may be 100 km or more, so that even in the region around the wave a resolution on the scale of several km is appropriate. In undisturbed regions of the ocean even larger grid cells are optimal. In Section 12.1 we show an example where the coarsest cells are 2◦ of latitude and longitude on each side. Five levels of mesh refinement are used, with the finest grids used only near Hilo, Hawaii, where the total refinement factor of 214 = 16 384 in each spatial dimension, so that the finest grid has roughly 10 m resolution. With adaptive refinement we can simulate the propagation of a tsunami originating near Chile (see Figure 1.1 and Section 12) and the inundation of Hilo (see Section 12.1) in a few hours on a single processor. The shallow water equations are a nonlinear hyperbolic system of partial differential equations and solutions may contain shock waves (hydraulic jumps). In the open ocean a tsunami has an extremely small amplitude

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Figure 1.1. The 27 February 2010 tsunami as computed using GeoClaw. In this computation a uniform 216 × 300 grid with ∆x = ∆y = 1/6 degree (10 arcminutes) is used. Compare to Figure 12.1, where adaptive mesh refinement is used. The surface elevation and bathymetry along the indicated transect is shown in Figure 1.2. The colour scale for the surface is in metres relative to mean sea level. The location of DART buoy 32412 discussed in the text is also indicated.

(relative to the depth of the ocean) and long wavelength. Hence the propagation is essentially linear, with variable coefficients due to varying bathymetry. As a tsunami approaches shore, however, the amplitude typically increases while the depth of the water decreases and nonlinear effects become important. It is thus desirable to use a method that handles the nonlinearity well (e.g., a high-resolution shock-capturing method), while also being efficient in the linear regime. In general we would like the method to conserve mass to the extent possible (the momentum equations contain source terms due to the varying bathymetry and possibly Coriolis and frictional drag terms). In this paper we focus on shock-capturing finite volume designed for nonlinear problems that are extensions of Godunov’s method. These methods are based on solving Riemann problems at the interfaces between grid cells, which consist of

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the given equations together with piecewise constant initial data determined by the cell averages on either side. Second-order correction terms are defined using limiters to avoid non-physical oscillations that might otherwise appear in regions of steep gradients (e.g., breaking waves or turbulent bores that arise as a tsunami approaches and inundates the shore). The methods exactly conserve mass on a fixed grid, but as we will see in Section 9.2 mass conservation is not generally possible or desirable near the shore when AMR is used. Even away from the shore, conserving mass when the grid is refined or de-refined requires some care when the bathymetry varies, as discussed in Section 9.1. Studying the effect of a tsunami requires accurate modelling of the motion of the shoreline; a major tsunami can inundate several kilometres inland in low-lying regions. This is a free boundary problem and the location of the wet–dry interface must be computed as part of the numerical solution; in fact this is one of the most important aspects of the computed solution for practical purposes. Most tsunami codes do not attempt to explicitly track the moving boundary, which would be very difficult for most realistic problems since the shoreline topology is constantly changing as islands and isolated lakes appear and disappear. Some tsunami models use a fixed shoreline location with solid wall boundary conditions and measure the depth of the solution at this boundary, perhaps converting this via empirical expression to estimates of the inundation distances and run-up (the elevation above sea level at the point of maximum inundation). Most recent codes, however, use some ‘wetting and drying’ algorithm. The computational grid covers dry land as well as the ocean, and each grid cell is allowed to be wet (h > 0) or dry (h = 0) in the shallow water equations. The state of each cell can change dynamically in each time step as the wave advances or retreats. Of course accurate modelling of the inundation also requires detailed models of the local topography and bathymetry on a scale of tens of metres or less, while the water depth must be resolved to a fraction of a metre. Again this generally requires the use of mesh refinement to achieve a suitable resolution at the coast. In the context of a Godunov-type method, it is necessary to develop a robust Riemann solver that can deal with Riemann problems in which one cell is initially dry, as well as the case where a cell dries out as the water recedes. This must be done in a manner that does not result in undershoots that might lead to negative fluid depth. For tsunami modelling it is essential to accurately capture small perturbations to undisturbed water at rest; the ocean is 4 km deep on average while even a major tsunami has an amplitude less than 1 m in the open ocean. Moreover, the wavelength may be 100 km or more, so that over 1 km, for example, the ocean surface elevation in a tsunami wave varies by less than 1 cm while the bathymetry (and hence the water depth) may vary

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Figure 1.2. Cross-section of the Pacific Ocean on a transect at constant latitude 25◦ S, as shown in Figure 1.1. (b) Full depth of the ocean. (a) Zoom of the surface elevation from −20 cm to 20 cm showing the small amplitude and long wavelength of the tsunami, 2.5 hours after the earthquake. Note the difference in vertical scales and that in both figures the vertical scale is greatly exaggerated relative to the horizontal scale. The bathymetry and surface elevation are shown as piecewise constant functions over the finite volume cells used, in order to illustrate the large jump in bathymetry between neighbouring grid cells.

by hundreds of metres. This is illustrated in Figure 1.2, which shows a crosssection of the Pacific Ocean along the transect indicated in Figure 1.1, along with a zoomed view of the top surface exhibiting the long wavelength of the tsunami. This extreme difference in scales makes it particularly important that a numerical method be employed that can maintain the steady state of the ocean at rest, and that accurately captures small perturbations to this steady state. Such methods are often called ‘well-balanced’, because the balance between the flux gradient and the source terms must be maintained numerically. This must also be done in a way that remains well-balanced in the context of AMR, with no spurious waves generated at mesh refinement boundaries. We discuss this difficulty and our approach to well-balancing further in Section 3.1. Two-dimensional finite volume methods can be applied either on regular (logically rectangular) quadrilateral grids or on unstructured grids such as triangulations. Unstructured grids have the advantage of being able to fit complicated geometries more easily, and for complex coastlines this may seem the obvious approach. For a fixed coastline this might be true, but

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when inundation is modelled using a wetting and drying approach the advantage is no longer clear. Logically rectangular grids (indexed by (i, j)) in fact have several advantages: high accuracy is often easier to obtain (at least for smoothly varying grids), and refinement on rectangular patches is natural and relatively easy to perform. The GeoClaw software uses patchbased logically rectangular grids following the approach of Berger, Colella and Oliger (Berger and Colella 1989, Berger and Oliger 1984, Berger and LeVeque 1998). This approach to AMR has been extensively used over the past three decades in many applications and software packages, including Clawpack as well as Chombo (www2), AMROC (www1), SAMRAI (www11), and FLASH (www6). We review this approach in Section 8 and discuss several difficulties that arise in applications to tsunamis. For many geophysical flow problems it is natural to use either purely Cartesian coordinates (over relatively small domains) or latitude–longitude coordinates on the sphere. The latter is generally used for tsunami propagation problems, for which the region of interest is usually far from the poles. For problems on the full sphere, other grids may be more appropriate, as discussed briefly in Section 6.2. For problems such as flooding of a serpentine river, it may be most appropriate to use a coarse grid that broadly follows the river valley, together with AMR to focus computational cells in the region where the river actually lies. In Section 6 we discuss a class of two-dimensional wave propagation algorithms that maintain stability and accuracy on general quadrilateral grids. When developing methods to simulate complex geophysical flows it is very important to perform validation and verification studies, as discussed in Section 10. This requires both tests on synthetic problems where the accuracy of the solvers can be judged as well as comparison to observations from real events. Sections 11 and 12 present computational results of each type in order to illustrate the application of these methods.

2. Tsunamis and tsunami modelling The term tsunami (which means ‘harbour wave’ in Japanese) generally refers to any impulse-generated gravity wave. Tsunamis can arise from many different sources. Most large tsunamis are generated by vertical displacement of the ocean floor during megathrust earthquakes on subduction zones. At a subduction zone, one plate (typically an oceanic plate) descends beneath another (typically continental) plate. The rate of this plate motion is on the order of centimetres per year. In the shallow part of the subduction zone, at depths less than 40 km, the plates are usually stuck together and the leading edge of the upper plate is dragged downwards. Slip during an earthquake releases this part of the plate, generally causing both upward and downward deformation of the ocean floor, and hence

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the entire water column above it. The vertical displacement can be several metres, and it can extend across areas of tens of thousands of square kilometres. Displacing this quantity of water by several metres injects an enormous amount of potential energy into the ocean (as much as 1023 ergs for a large tsunami, equivalent to roughly 10 megatonnes of TNT). The potential energy is given by η(x,y) ρgz dz dx dy Potential energy = (2.1) 0 1 2 ρgη (x, y) dx dy, = 2 where ρ is the density of water, g is the gravitational constant, and η(x, y) is the displacement of the surface from sea level. Here x and y are horizontal Cartesian coordinates and z is the vertical direction. This energy is carried away by propagating waves that tend to wreak their greatest havoc nearby, but if the tsunami is large enough can also cause severe flooding and damage thousands of kilometres away. Long-range tsunamis are often termed teletsunamis or far-field tsunamis, to distinguish them from local tsunamis that affect only regions near the source. For example, the Aceh–Andaman earthquake on 26 December 2004 generated a tsunami along the zone where the Indian plate is subducting beneath the Burma platelet. The rupture extended for a length of roughly 1500 km and displaced water over a region approximately 150 km wide, with a vertical displacement of several metres. More recently, the Chilean earthquake of 27 February 2010 set off a tsunami along part of the South American subduction zone, where the Nazca plate descends beneath the South American plate. The fault-rupture length was shorter, perhaps 450 km, and fault displacement was also less, yielding a tsunami considerably smaller than the Indian Ocean tsunami of 2004. The fact that a megathrust earthquake displaces the entire water column over a large surface area is advantageous to modellers, since it means that use of the two-dimensional shallow water equations is well justified. These equations, introduced in Section 3, model gravity waves with long wavelength (relative to the depth of the fluid) in which the entire water column is moving. These conditions are well satisfied as the tsunami propagates across an ocean. A secondary source of tsunamis is submarine landslides, also called subaqueous landslides; see for example Bardet, Synolakis, Davies, Imamura and Okal (2003), Masson, Harbitz, Wynn, Pedersen and Løvholt (2006), Ostapenko (1999) and Watts, Grilli, Kirby, Fryer and Tappin (2003). These often occur on the continental slope, which can be several kilometres high and quite steep. The displacement of a large mass on the seafloor causes a corresponding perturbation of the water column above this region, which

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again results in gravity waves that can appear as tsunamis. The local displacements may be much larger than in a megathrust earthquake, but usually over much smaller areas and so the resulting tsunamis have far less energy and rapidly dissipate as they radiate outwards. However, they can still do severe damage to nearby coastal regions. For example, an earthquake in 1998 resulted in a tsunami that destroyed several villages and killed more than 2000 people along a 30 km stretch of the north shore of Papua New Guinea. In this case it is thought that the tsunami was caused by a coseismic submarine landslide rather than by the earthquake itself (Synolakis et al. 2002). In the case of large earthquakes, it is possible that in addition to the seismic event itself, thousands of coseismic landslides may also occur, leading to additional tsunamis. As an example, Plafker, Kachadoorian, Eckel and Mayo (1969) documented numerous tsunamis in Alaskan fjords in connection with the 1964 earthquake. There have also been submarine slumps of epic proportions that have caused large-scale destruction. An example is the Storegga slide roughly 8200 years ago on the Norwegian shelf, in which as much as 3000 km3 of mass was set in motion, creating a tsunami that inundated areas as far away as Scotland (Dawson, Long and Smith 1988, Haflidason, Sejrup, Nyg˚ ard, Mienert and Bryn 2004). Subaerial landslides occurring along the coast can also cause localized tsunamis when the landslide debris enters the water. For example, a largescale landslide on Lituya Bay in Alaska in 1958 caused a landslide within the bay that washed trees away to an elevation of 500 m on the far side of the bay, as documented by Miller (1960) and studied for example in Mader and Gittings (2002), Weiss, Fritz and W¨ unnemann (2009). Tsunamis and seiches can also arise in lakes as a result of earthquakes or landslides. As an example see McCulloch (1966). The example we use in this paper is the tsunami generated by the Chilean earthquake of 27 February 2010. The computational advantages of the AMR techniques discussed in this paper are particularly dramatic in modelling far-field effects of transoceanic tsunamis, but are also important in modelling localized tsunamis or the near-field region (which is hardest hit by any tsunami). Typically much higher resolution is needed along a small portion of the coast of primary interest than elsewhere, and over much of the computational domain there is dry land or quiescent water where a very coarse grid can be used. 2.1. Available data sets Modelling a tsunami requires not only a set of mathematical equations and computational techniques, it also requires data sets, often very large ones. We must specify the bathymetry of the ocean and coastal regions, the topo-

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graphy onshore in regions that may be inundated, and the motion of the seafloor that initiates the tsunami. For validation studies we also need observed data from past events, which might include DART buoy (Deepocean Assessment and Reporting of Tsunamis) or tide gauge data as well as post-tsunami field surveys of run-up and inundation. Fortunately there are now ample sources of real data available online that are relatively easy to work with. One of the goals of our own work has been to provide tools to facilitate this, and to provide templates that may be useful in setting up and solving a new tsunami problem. This is still work in progress, but some pointers and documentation are provided in the GeoClaw documentation (www7). Large-scale bathymetry at the resolution of 1 minute (1/60 degree) for the entire Earth is available from the National Geophysical Data Center (NGDC). The National Geophysical Data Center (NGDC) GEODAS Grid Translator (www9) allows one to specify a rectangular latitude–longitude domain and download bathymetry at a choice of resolutions. Note that one degree of latitude is about 111 km and one degree longitude varies from 111 km at the equator to half that at 60◦ North, for example. For modelling transoceanic propagation we have found that 10-minute data, with a resolution of roughly 18 km, are often sufficient. In coastal regions greater resolution is required. In particular, in order to model inundation of a target region it may be necessary to have data sets with a resolution of tens of metres or less. The availability of such data varies greatly. In some countries coastal bathymetry is virtually impossible to obtain. In other locations it is easily available online. In particular, many coastal regions of the US are covered by data sets available from NOAA DEMs (www10). In addition to bathymetry, it is necessary to have matching onshore topography for regions where inundation is to be studied. Unfortunately bathymetry and topography are generally measured by different techniques and sometimes the data sets do not match up properly at the coastline, which of course is exactly the region of primary interest in modelling inundation. Often a great deal of work has already gone into creating the public data sets in order to reconcile these differences, but an awareness of potential difficulties is valuable. When studying landslide-induced tsunamis, an additional difficulty is that detailed bathymetry of the region around the slide is typically obtained only after the slide has occurred. Without pre-slide bathymetry at the same resolution it can be difficult to determine the correct initial bathymetry or the mass of the slide, which of course is crucial to know in order to generate the correct tsunami numerically. For subduction zone events it is also necessary to know the seafloor displacement in order to generate the tsunami. In this case the modeller is aided by the fact that the mechanics of some earthquakes have been well

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studied. For large events there are generally ample seismic data available from around the world that can be used to attempt to reconstruct the focal mechanism of the quake: the direction of slip and orientation of the fault, along with the depth at which the rupture occurred, the length of the rupture, the magnitude of the displacement, etc. An event can sometimes be modelled by a simplified representation consisting of a few such parameters, for example the USGS model of the Chile 2010 earthquake (www12) that we use in some of our examples later in this paper. To convert these parameters into seafloor deformation in each grid cell would require solving three-dimensional elasticity equations with a dislocation within the earth, and would require detailed knowledge of the elastic parameters and the geological substructure of the earth in the region of the quake. Instead, a simplified model is generally used to quickly convert parameters into approximate seafloor deformation, such as the well-known model introduced by Mansinha and Smylie (1971) and later modified by Okada (1985, 1992). We use a Python implementation of the Okada model that we based on the models in the COMCOT (www4) (Liu, Woo and Cho 1998). Larger events are often subdivided into a finite collection of such parametrizations, by breaking the fault into pieces with different sets of parameters. For each piece, the focal mechanism parameters can then be converted into the resulting motion of the seafloor, and these can be summed to obtain the approximate seafloor deformation resulting from the earthquake. It may also be necessary to use time-dependent deformations for large events, such as the 2004 Aceh–Andaman event, which lasted more than 10 minutes as the rupture propagated northwards. Although large earthquakes are well studied, determining the correct mechanism is non-trivial and there are often several different mechanisms proposed that may be substantially different, particularly in regard to the tsunamis that they generate. One use of tsunami modelling is to aid in the study of earthquakes, providing additional constraints on the mechanism beyond the seismic evidence; see for example Hirata, Geist, Satake, Tanioka and Yamaki (2003). However, the existence of competing descriptions of the earthquake can also make it more difficult to validate a numerical method for the tsunami itself. In addition to seismic data, real-time data during a tsunami are also measured by tide gauges at many coastal locations, from which the amplitude and waveform of the tsunami can be estimated. The tides and any coseismic deformation must be filtered out from these data in order to see the tsunami, particularly for large-scale tsunamis that can extend through several tidal periods. The observed waves (particularly in shallow water) are also highly dependent on the local bathymetry, and can vary greatly between nearby points. Tide gauges in bays or harbours often register much more wave action than would be seen farther from shore, due to reflections and

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resonant sloshing. To have any hope of properly capturing this numerically it is generally necessary to provide the model with fine-scale local bathymetry. The wave amplitude in the deep ocean cannot be measured by traditional tide gauges, but in recent years a network of gauges have been installed on the ocean floor that measure the water pressure with sufficient sensitivity to estimate the depth. In Section 12 we use data from a DART buoy (Meinig, Stalin, Nakamura, Gonz´ alez and Milburn 2006), which transmits data from a pressure sensor at a depth of more than 4000 m. The DART system was developed by NOAA and originally deployed only along the western coast of the United States. Many other nations have also developed similar buoy systems, and after the 2004 Indian Ocean tsunami the world-wide network was greatly expanded. Real-time and historical data sets are available online via DART Data (www5). Also useful in tsunami modelling is the wealth of data collected by tsunami survey teams that respond after any tsunami event. Attempts are made to map the run-up and inundation along stretches of the affected coast, by examining water marks on buildings, wrack lines, debris lodged in trees, and other markers. This evidence often disappears relatively quickly after the event and the rapid response of scientists and volunteers is critical. The findings are generally published and are valuable sources of data for validation studies. Again it is often necessary to have high-resolution local bathymetry and topography in order to model the great variation in run-up and inundation that are often seen between nearby coastal locations. Survey teams sometimes collect these data as well. For some sample survey results, see for example Gelfenbaum and Jaffe (2003), Liu, Lynett, Fernando, Jaffe and Fritz (2005) and Yeh et al. (2006). Information about past tsunamis can also be gleaned from the study of tsunami deposits (Bourgeois 2009). As a tsunami approaches shore it generally becomes quite turbulent, even forming a bore, and picks up sediment such as sand and marine microorganisms that may be deposited inland as the tsunami decelerates. These deposits can often be identified, either near the surface from a recent tsunami or in the subsurface from prehistoric events, as illustrated in Figure 2.1. In some coastal regions, excavations and core samples reveal more than ten distinct layers of deposits from tsunamis in the past few thousand years. Much of what is known about the frequency of megathrust earthquakes along subduction zones has been learned from studying tsunami deposits, as these deposits are commonly the only remaining evidence of past earthquakes. For example, Figure 2.2 shows the record of 17 sand layers interpreted as tsunami deposits, from the coast of Oregon state, indicating that megathrust events along the Cascadia Subduction Zone (CSZ) occur roughly every 500 years. The CSZ runs from northern California to British Columbia, and the last great earthquake and triggered

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(a)

(b)

Figure 2.1. 2004 and older tsunami deposits in western Thailand (Jankaew et al. 2008). (a) Coastal profile of a part of western Thailand hit by the 2004 Indian Ocean tsunami (simplified from Figure 2 in Jankaew et al. (2008)). (b) Photo and sketch of a trench along this profile, showing the 2004 tsunami deposit and three older tsunami deposits, all younger than about 2500 years ago.

tsunami were on 26 January 1700, as determined from matching Japanese historical records of a tsunami with dated tsunami deposits in the Pacific Northwest of the US (Satake, Shimazaki, Tsuji and Ueda 1996, Satake, Wang and Atwater 2003). An interesting account of this scientific discovery can be found in Atwater et al. (2005). The next such event will have disastrous consequences for many communities in the Pacific Northwest, and the tsunami is expected to cause damage around the Pacific. 2.2. Uses of tsunami modelling There are many reasons to study tsunamis computationally, and ample motivation for developing faster and more accurate numerical methods. Applications include the development of more accurate real-time warning systems, the assessment of potential future hazards to assist in emergency planning, and the investigation of past tsunamis and their sources. In this section we give a brief introduction to some of the issues involved.

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Figure 2.2. An example of long-term records of tsunami deposits interpreted to be from the Cascadia subduction zone: from Bradley Lake on the coast of southern Oregon. Seventeen different sediment deposits were identified and correlated at eight different locations. The far right column shows the approximate age of each set of deposits. From Bourgeois (2009), based on a figure of Kelsey, Nelson, Hemphill-Haley and Witter (2005).

Real-time warning systems rely on numerical models to predict whether an earthquake has produced a dangerous tsunami, and to identify which communities may need to be warned or evacuated. Mistakes in either direction are costly: failing to evacuate can lead to loss of life, but evacuating unnecessarily is not only very expensive but also leads to poor response to future warnings. Real-time prediction is difficult for many reasons: a code is required that will run faster than real time and still provide detailed results, usually for many different locations. Moreover, the source is usually poorly known initially since solving the inverse problem of determining the focal mechanism from seismic signals takes considerable time and consolidation of data from multiple sites. The DART buoys were developed in part to address this problem. By measuring the actual wave at one or more locations near the source, a better estimate of the tsunami can be quickly generated and used to select initial data for real-time prediction, as discussed by Percival et al. (2010).

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Most codes used for studying tsunamis are not designed for real-time warning; this is a specialized and demanding application (Titov et al. 2005). However, there are many other applications where research codes can play a role. For example, hazard assessment and mitigation requires the use of tsunami models to investigate the potential damage from a future tsunami, to locate safe havens and plan evacuation routes, and to assist government agencies in planning for emergency response. For this, information about past tsunamis in a region is valuable both in validating the code and in designing hypothetical tsunami sources for assessing the vulnerability to future tsunamis. A topic of growing interest is the development of probabilistic models that take into account the uncertainty of future earthquakes. Seismologists can often provide information about the likelihood of ruptures of various magnitudes along several fault planes, and tsunami modellers then seek to produce from this a probabilistic assessment of the risk of inundation to varying degrees. Although these simulations do not need to be set up and run in real time, the need to do large numbers of simulations for a probabilistic study is additional motivation for developing fast and accurate techniques that can handle the entire simulation from tsunami generation to detailed modelling of specific distant communities. For more on this topic, see for example Geist and Parsons (2006), Gonz´ alez, Geist, Jaffe, Kanoglu et al. (2009) and Geist, Parsons, ten Brink and Lee (2009). Another use of tsunami modelling is to better understand past tsunamis, and to identify the earthquakes that generated them. Much of what is known about earthquakes that happened before the age of seismic monitoring or historical records has been determined through the study of tsunami deposits, as illustrated in Figures 2.1 and 2.2 and discussed above. Tsunami modelling is often required to assist in solving the inverse problem of determining the most likely earthquake source and magnitude from a given set of deposits. For this it would be desirable to couple the tsunami model to sedimentation equations capable of modelling the suspension of sediments and their transport and deposition, ideally also taking into account the resulting changes in bathymetry and topography that may affect the fluid dynamics. Moreover, tsunami deposits often exhibit layers in which the grain size either increases or decreases with depth, and this grading contains information about how the flow was behaving at this location while the sediment was deposited; e.g., Higman, Gelfenbaum, Lynett, Moore and Jaffe (2007) and Martin et al. (2008). Ideally the model would include multiple grain sizes and accurately simulate the entrainment and sedimentation of each. The development of sufficiently accurate sedimentation models and computational tools adequate to do this type of analysis is an active area of research; see for example Huntington et al. (2007).

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3. The shallow water equations The shallow water equations are the standard governing model used for transoceanic tsunami propagation as well as for local inundation: e.g., Yeh, Liu, Briggs and Synolakis (1994) and Titov and Synolakis (1995, 1998). Because we use shock-capturing methods that can converge to discontinuous weak solutions, we solve the most general form of the equations: a nonlinear system of hyperbolic conservation laws for depth and momentum. In one space dimension these take the form ht + (hu)x = 0, 2

(hu)t + (hu +

2 1 2 gh )x

= −ghBx ,

(3.1a) (3.1b)

where g is the gravitational constant, h(x, t) is the fluid depth, u(x, t) is the vertically averaged horizontal fluid velocity. A drag term −D(h, u)u can be added to the momentum equation and is often important in very shallow water near the shoreline. This is discussed in Section 7. The function B(x) is the bottom surface elevation relative to mean sea level. Where B < 0 this corresponds to submarine bathymetry and where B > 0 to topography. Although in tsunami studies the term bathymetry is commonly used, in much of this paper we will use the term topography to refer to both bathymetry and onshore topography, both for conciseness and because in many other geophysical flows (debris flows, lava flows, etc.) there is only topography. We will also use η(x, t) to denote the water surface elevation, η(x, t) = h(x, t) + B(x, t). We allow the topography to be time-dependent since most tsunamis are generated by motion of the ocean floor resulting from an earthquake or landslide. Figure 3.1 shows a simple sketch of the variables. Note that (3.1) is in fact a ‘balance law’, since variable bottom topography and drag introduce source terms in the momentum equation. The physically relevant form (3.1) introduces some difficulties for numerical solution, particularly with regard to steady state preservation. As mentioned above, this has led to the development of well-balanced schemes for such systems (see e.g. Bale, LeVeque, Mitran and Rossmanith (2002), Bouchut (2004), George (2008), Greenberg and LeRoux (1996), Botta, Klein, Langenberg and L¨ utzenkirchen (2004), Gallardo, Parés and Castro (2007), Gosse (2000), LeVeque (2010) and Noelle, Pankrantz, Puppo and Natvig (2006)). This is sometimes circumvented by using alternative non-conservative forms of the shallow water equations for η(x, t) and u(x, t), but these forms are problematic if discontinuities appear in the inundation regime (bore formation), and conservation of mass is not easily guaranteed.

227

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(B

η

ηs

(B

h

− ηs ) > 0

−η s ) < 0

Figure 3.1. Sketch of the variables of the shallow water equations. The shaded region is the water of depth h(x, t), and the water surface is η(x, t) = B(x, t) + h(x, t). The dashed line shows the mean sea level ηs .

For tsunami modelling we solve the two-dimensional shallow water equations ht + (hu)x + (hv)y = 0,

(3.2a)

+ (huv)y = −ghBx ,

(3.2b)

(hv)t + (huv)x + (hv + 12 gh2 )y = −ghBy ,

(3.2c)

2

(hu)t + (hu +

2 1 2 gh )x 2

where u(x, y, t) and v(x, y, t) are the depth-averaged velocities in the two horizontal directions, B(x, y, t) is the topography. Again a drag term might be added to the momentum equations. For simplicity, we will discuss many issues in the context of the onedimensional shallow water equations (3.1) whenever possible. We also first consider the equations in Cartesian coordinates, with x and y measured in metres, as might be appropriate when modelling local effects of waves on a small portion of the coast or in a wave tank. For transoceanic tsunami propagation it is necessary to propagate on the surface of the earth, as discussed further in Section 6.2. For this it is common to use latitude and longitude coordinates, assuming the earth is a perfect sphere. A more accurate geoid representation of the earth could be used instead. Latitude– longitude coordinates present difficulties for many problems posed on the sphere due to the fact that grid lines coalesce at the poles and cells are much smaller in the polar regions than elsewhere, which can lead to time step restrictions. For tsunamis on the earth we are generally only interested in the mid-latitudes and this is not a problem, but in Section 6.2 we mention an alternative grid that may be useful in other contexts. On a rotating sphere the equations should also include Coriolis terms in the momentum equations. For tsunami modelling these are generally neglected. During propagation across an ocean, the fluid velocities are small and are concentrated within the wave region and Coriolis effects have been shown to be very small (e.g., Kowalik, Knight, Logan and Whitmore

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(2005)). Our own tests have also indicated that Coriolis terms can be safely ignored. On the other hand, they are simple to include numerically along with the drag terms via a fractional step approach, as discussed in Section 7. 3.1. Hyperbolicity and Riemann problems The shallow water equations (3.1) belong to the more general class of hyperbolic systems qt + f (q)x = ψ(q, x),

(3.3)

where q(x, t) is the vector of unknowns, f (q) is the vector of corresponding fluxes, and ψ(q, x) is a vector of source terms: hu h 0 q= , f (q) = , ψ= . (3.4) hu −ghBx hu2 + 12 gh2 We will also introduce the notation µ = hu for the momentum and φ = hu2 + 12 gh2 for the momentum flux, so that µ h . (3.5) , f (q) = q= φ µ The Jacobian matrix f (q) then has the form 0 1 ∂µ/∂h ∂µ/∂µ = f (q) = . ∂φ/∂h ∂φ/∂µ gh − u2 2u

(3.6)

Hyperbolicity requires that the Jacobian matrix be diagonalizable with real eigenvalues and linearly independent eigenvectors. For the shallow water equations the matrix in (3.6) has eigenvalues (3.7) λ1 = u − gh, λ2 = u + gh and corresponding eigenvectors 1√ 1 r = , u − gh

2

r =

u+

1√

gh

.

(3.8)

We will use superscripts to index these eigenvalues and eigenvectors since subscripts corresponding to grid cells will be added later. Note that the eigenvalues are always real for physically relevant depths h ≥ 0. For h > 0 they are distinct and the eigenvectors are linearly independent. Hence the equations are hyperbolic for h > 0, and the solution consists of propagating waves. The eigenvalues correspond to velocities of propagation and the eigenvectors give information about the relation between h and hu in a wave propagating at this √ speed. Note that waves propagate at velocities ± gh relative to the background √ fluid velocity u. The velocity c = gh is the gravity wave speed and is

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analogous to the sound speed for small-amplitude acoustic waves. For twodimensional shallow water equations the theory is somewhat more complicated, since waves can propagate √ in any direction, but the speed of propagation in any direction is again gh relative to the fluid velocity. 2 , but they could Note also that in general√the eigenvalues satisfy λ1 < λ√ both be negative (if u < − gh) or both positive (if u > gh). Such flows are called supercritical and correspond to supersonic flow in gas dynamics. 1 For tsunami modelling, the flow is nearly always subcritical, with √ λ x For the linear hyperbolic problem (3.13), it is easy to see (using the construction of the exact solution described above), that the solution consists of ˆ 1 and two discontinuities propagating away from the point x ¯ at velocities λ 2 ˆ λ . Moreover the jump in q across each of these waves must be proportional to the corresponding eigenvector, and so the solution has the form  ˆ 1 (t − t¯),  if x < x ¯+λ  Q ˆ 1 (t − t¯) < x < x ˆ 2 (t − t¯), (3.18) q(x, t) = Qm if x ¯+λ ¯+λ   2 ˆ if x > x ¯ + λ (t − t¯), Qr where the middle state Qm satisfies Qm = Q + α1 rˆ1 = Qr − α2 rˆ2

(3.19)

for some scalars α1 and α2 . We will denote the waves by W 1 = Qm − Q = α1 rˆ1 ,

W 2 = Qr − Qm = α2 rˆ2 .

(3.20)

232


The weights α1 and α2 can be found as the two components of the vector α by solving the linear system ˆ = Qr − Q . Rα

(3.21)

The solution is easily determined to be ˆ 2 ∆h − ∆µ ˆ 1 ∆h − ∆µ λ −λ , α2 = . (3.22) 2ˆ c 2ˆ c where ∆h = hr −h and ∆µ = µr −µ = hr ur −h u . Note in particular that if u = ur = u ˆ then α1 = α2 = (hr −h )/2, and the initial jump in h resolves into equal-amplitude waves propagating upstream and downstream. For the constant coefficient linear problem the characteristic structure determines the Riemann solution. For variable coefficient or nonlinear problems, the exact solution for general initial data can no longer be computed by characteristics in general, but the Riemann problem can still be solved and is a key tool in analysis and numerics. α1 =

3.5. Varying topography To linearize the shallow water equations in the case of variable topography, it is easiest to work in terms of the surface elevation η(x, t) = B(x) + h(x, t). We will linearize about a flat surface ηˆ and zero velocity u ˆ = 0. We will ˆ define h(x) = ηˆ − B(x), which is no longer constant and may have large variations if the topography B(x) varies. The momentum equation can be rewritten as µt + (hu2 )x + gh(h + B)x = 0, (3.23) and linearizing this gives the equation ˆ ηx = 0 µ ˜t + g h(x)˜

(3.24)

for the perturbation (˜ η, µ ˜) about (ˆ η , 0). Combining this with the already linear continuity equation η˜t + µ ˜x = 0 and dropping tildes gives the variable coefficient linear hyperbolic system 0 1 η 0 η . (3.25) + ˆ = 0 µ t g h(x) 0 µ x If we try to diagonalize these equations, we find that because the eigenvector matrix R now varies with x, the advection equations for the characteristic variables w1 and w2 are coupled together by source terms that only vanish where the bathymetry is flat. Over varying bathymetry a wave in one characteristic family is constantly losing energy into the other family, corresponding to wave reflection from the bathymetry. Nonetheless, we can define a Riemann problem for this variable coefficient ˆ from h ˆ to h ˆ r at x system by allowing a jump in h ¯, along with a jump in the

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data from (η , µ ) to (ηr , µr ). The solution to this Riemann problem conˆ )1/2 and a right-going wave sists of a left-going wave with speed cˆ = −(g h ˆ r )1/2 . Each wave propagates across a region of constant with speed cˆr = (g h topography (B or Br respectively) at the appropriate speed, and hence the jump in (η, µ) across each wave must be an eigenvector corresponding to the coefficient matrix on that side of x ¯: 1 1 1 1 1 2 2 2 2 1 W = α rˆ = α , W = α rˆr = α , (3.26) −ˆ c cˆr The weights α1 and α2 can be determined by solving the linear system 1 1 α1 η r − η ∆η = ≡ , (3.27) 2 µ r − µ −ˆ c cˆr α ∆µ yielding α1 =

cˆr ∆η − ∆µ , c + cr

α2 =

cˆl ∆η + ∆µ . c + cr

(3.28)

ˆ r = h, ˆ ˆ = h Note that in the case when there is no jump in topography, h 1/2 ˆ we find that −c = cr = (g h) , and ∆η = ∆h, so that (3.28) agrees with (3.22). Another way to derive this linearized solution is to linearize the system (3.9) that we obtained by introducing B(x, y) as a new component. Linˆ and u earizing about h ˆ = 0 gives the variable coefficient matrix  

0 1 0 ˆ if x < x h ¯, ˆ ˆ ˆ ˆ , h(x) = ˆ (3.29) A(x) = g h(x) 0 g h(x) ¯. hr if x > x 0 0 0 The Riemann solution consists of three waves, found by decomposing         ∆h 1 1 −1 c  + α2 cˆr  + α0  0 . (3.30) ∆q =  ∆µ  = α1 −ˆ ∆B 1 0 0 From the third equation we find α0 = ∆B, and then α1 and α2 can be found by solving       1 1 ∆h + ∆B  ∆µ  = α1 −ˆ c + α2  cˆ. (3.31) 0 0 0 Since ∆h + ∆B = ∆η, this gives the same system as (3.27), and the same propagating waves as before. We will make use of this Riemann solution for the linearized shallow water equations in developing an approach for the full nonlinear equations in Section 5.

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3.6. Interaction with the continental shelf Often there is a broad and shallow continental shelf that is separated from the deep ocean by a very steep and narrow continental slope (narrow relative to the wavelength of the tsunami, that is). Figure 12.4 shows the continental shelf near Lima, Peru and the refraction of the 27 February 2010 tsunami wave hitting this shelf. In this section we consider an idealized model to help understand the amplification of a tsunami that takes place as it approaches the coast. Consider piecewise constant bathymetry with a jump from an undisturbed depth h to a shallower depth of hr . Figure 3.2 shows an example of a smallamplitude wave interacting with such bathymetry, in this case a step discontinuity 30 km offshore at the location indicated by the dashed line. The undisturbed depths are h = 4000 and hr = 200 m. At time t = 0 a hump of stationary water is introduced with amplitude 0.4 m. This hump splits into left-going and right-going waves of equal amplitude, sufficiently small that propagation is essentially linear on both sides of the discontinuity. A purely positive perturbation of the depth is used here to make the figures clearer, but any small-amplitude waveform would behave in the same manner. We observe in Figure 3.2 that the right-going wave is split into transmitted and reflected waves when it encounters the discontinuity in bathymetry. The transmitted wave has large amplitude, but shorter wavelength, while the reflected wave has smaller amplitude. At later times the right-going wave on the shelf reflects off the right boundary and becomes a left-going wave. In this model problem the shore is simply a solid vertical wall, but a similar reflection would be observed from a beach. This left-going wave reflected from shore later hits the discontinuity in bathymetry and is itself split into a transmitted wave (left-going in the ocean) and a reflected wave (right-going on the shelf). The reflected right-going wave is now a wave of depression, which later reflects off the shore, then off the discontinuity, etc. It is important to note that much of the wave energy is trapped on the continental shelf and reflects multiple times between the discontinuity in bathymetry and the shore. This has practical implications and is partly responsible for the fact that multiple destructive tsunami waves are often observed on the coast. Moreover, the trapped wave continues to radiate energy back into the ocean each time the wave reflects off the discontinuity. This leads to a more complex wave pattern elsewhere in the ocean than would be observed from the initial tsunami alone, or from including only the single reflection that would be seen from a shore with no shelf. This suggests that to accurately simulate tsunamis it may be important to adequately resolve continental shelves, even in regions away from the coastline of primary interest in the simulation. As an example of this, the simulation shown in Figures 12.1–12.4 shows that large-amplitude waves remain trapped on the shelf off Peru long after the main tsunami has passed by.

235

Tsunami modelling Surface at t = 0 seconds

Surface at t = 1400 seconds 0.4 Metres

Metres

0.4 0.2 0.0

0.2 0.0

−0.2

−0.2

−0.4

−0.4

300

200

100

30

Surface at t = 200 seconds

Metres

Metres

0.0

30

0.2 0.0

−0.2

−0.2

−0.4

−0.4

300

200

100

30

300

200

100

30


0.4 Metres

100

0.4

0.2


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0.2 0.0

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200

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300

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30


0.4 Metres

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0.4

0.2 0.0

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0.4 Metres

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0.2 0.0

0.2 0.0

−0.2

−0.2

−0.4

−0.4

300

200

100

30

300

200

100

30

Figure 3.2. An idealized tsunami interacting with a step discontinuity representing a continental shelf. The dashed line indicates the location of the discontinuity, 30 km offshore. See Figure 3.3 for the same solution as a contour plot in the x–t plane.

236


Contours of surface

2 .0

1 .5

Hours

4800 seconds

1 .0

3400 seconds 2800 seconds 2000 seconds

0 .5

1400 seconds 1000 seconds

0 .0

600 seconds 400 seconds 200 seconds 300

200 Kilometres offshore

100

30

Figure 3.3. Contour plot in the x–t plane of an idealized tsunami interacting with a step discontinuity representing a continental shelf. Solid contour lines are at 0.025, 0.05, . . . , 0.35 m. Dashed contour lines are at −0.025, −0.05, −0.1, −0.15 m. This is a different view of the results shown in Figure 3.2, and the times shown there are indicated as horizontal lines.

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Consider the first interaction of the wave shown in Figure 3.2 with the discontinuity. Note that the lower wave speed on the shelf results in a shorter-amplitude wave. To understand this, suppose the initial wave√has wavelength W . The tail of the wave reaches the step at time ∆t = W / gh later than the front of the wave. At this time the front√of the transmitted wave on the shallow side has moved a distance ∆t ghr and so the wavelength observed on the shallow side is Wr = hr /h W < W . The wavelength decreases by the same factor as the decrease in wave speed. On the other hand, the amplitude of the transmitted wave is larger than the amplitude of the original wave by a factor CT > 1, the transmission coefficient, while the reflected wave is smaller by a factor CR < 1, the reflection coefficient. For the idealized step discontinuity, these coefficients are given by CT =

2c , c + cr

CR =

c − cr , c + cr

(3.32)

analogous to the transmission and reflection coefficients of linear acoustics, for example, at an interface between materials with different impedance. For the example shown in Figures 3.2 and 3.3, the coefficients are CT ≈ 1.63 and CR = CT − 1 ≈ 0.63. There are several ways to derive these coefficients. An approach that fits well here is to use the structure of the Riemann solution derived above, as is done for acoustics in LeVeque (2002). Consider a pure right-going wave consisting of a jump discontinuity of magnitude ∆η in depth, that hits the discontinuity in bathymetry at some time t¯. From this time forward we have a Riemann problem in which ∆µ = c ∆η by the jump conditions across a right-going wave in the deep water. The Riemann solution consists of a left-going wave (the reflected wave) and a right-going wave (the transmitted wave) of the form (3.26), and the formulas (3.28) when applied to this particular Riemann data yield directly the coefficients (3.32). A more general waveform can be viewed as a sequence of small step discontinuities approaching the shelf, each of which must have the same relation between ∆η and ∆µ, and so each is split in the same manner into transmitted and reflected waves. Note that if c = cr there is no discontinuity, and in this case CT = 1 while CR = 0. On the other hand, in the limiting case of very shallow water on the right, CT → 2 while CR → 1. This limiting case corresponds to a solid wall boundary condition, and this factor of 2 amplification is apparent at time t = 1000 s in Figure 3.2, when the wave is reflecting off the shore. In general the amplification factor for a wave transmitted into shallower water is between 1 and 2, while the reflection coefficient is between 0 and 1 if c > cr . When a wave is transmitted from shallow water into deeper water (e.g., if c < cr ) then the reflection coefficient in (3.32) is negative,

238


explaining the negation of amplitude seen in Figures 3.2 and 3.3 when the trapped wave reflects off the discontinuity, for example between times 1400 and 2000 seconds in those plots. We can also calculate the fraction of energy that is transmitted and reflected at the shelf. In a pure right-going wave (or a pure left-going wave) the energy is equally distributed between potential and kinetic energy by the equipartition principle. If η(x) is the displacement of the surface from sea level ηs = 0 and u(x) is the velocity of the fluid, then these are given by 1 ρgη 2 (x) dx, Potential energy = 2 (3.33) 1 2 Kinetic energy = ρu (x) dx, 2 where ρ is the density of the water. It is easy to check that these are equal for a wave in a single characteristic family (for the linearized equations about a constant depth h and zero velocity) √ by noting that the form of the eigenvectors (3.8) shows that hu(x) = ± gh η(x) for each x. Let E be the energy in the wave approaching the step. The reflected wave has the same shape but the amplitude of η(x) is reduced by CR everywhere, and 2 E . By conservation of energy, hence the energy in the reflected wave is CR 2 )E . This result can also be the amount of energy transmitted is (1 − CR found by calculating the potential energy of the transmitted wave directly from the integral in (3.33), taking into account both the amplitude of the wave by the factor CT and the reduction in wavelength by hr /h . For the example shown in Figures 3.2 and 3.3, approximately 60% of the energy is transmitted onto the shelf at the first reflection time. At the kth reflection of the wave trapped on the shelf, the energy radiated can be calculated to (k−1) be (1 − CR )2 CR E . The total of the initially reflected energy plus all the radiated energy is given by an infinite series that sums to E .

4. Finite volume methods Before continuing our discussion of Riemann problems for the shallow water equations, we pause to introduce the basic ideas of finite volume methods, both as motivation and in order to see what information will be required from Riemann solutions. Nonlinear hyperbolic systems (3.3) present some well-known difficulties for numerical solution, and a considerable amount of research has been dedicated to the development of suitable numerical methods for them; see LeVeque (2002) for an overview. A class of numerical methods that has been very successful for these problems are the shock-capturing Godunovtype methods: finite volume methods making use of Riemann problems to determine the numerical update.

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In a one-dimensional finite volume method, the numerical solution Qni is an approximation to the average value of the solution in the ith grid cell Ci = [xi−1/2 , xi+1/2 ]: 1 n q(x, tn ) dx, (4.1) Qi ≈ Vi Ci where Vi is the volume of the grid cell (simply the length in one dimension, Vi = xi+1/2 − xi−1/2 ). The wave propagation algorithm updates the numerby solving Riemann problems at xi−1/2 and ical solution from Qni to Qn+1 i xi+1/2 , the boundaries of Ci , and using the resulting wave structure of the Riemann problem to determine the numerical update. For a homogeneous system of conservation laws qt + f (q)x = 0, such methods are often written in conservation form, ∆t n n Qn+1 = Qni − − Fi−1/2 ) (4.2) (F i ∆x i+1/2 n is a numerical flux approximating the time average of the true where Fi−1/2 flux across the left edge of cell Ci over the time interval: tn+1 1 n f (q(xi−1/2 , t)) dt. (4.3) Fi−1/2 ≈ ∆t tn If the method is in conservation form, then no matter how the numerical over all fluxes are chosen the method will be conservative: summing Qn+1 i grid cells gives a cancellation of fluxes except for fluxes at the boundaries. The classical Godunov’s method is obtained by solving the Riemann problem at each cell edge (using x ¯ = xi−1/2 and t¯ = tn in our general description of the Riemann problem, for example) and then evaluating the resulting Riemann solution at xi−1/2 to define the numerical flux, setting n Fi−1/2 = f (Q(xi−1/2 )).

This gives a first-order accurate method that can be viewed as a generalization of the upwind method for scalar advection. For equations (3.3) with a source term, one common approach is to use a fractional step method in which each time step is subdivided into a step on the homogeneous conservation law qt + f (q)x = 0, followed by a step on the source terms alone, solving qt = ψ(q, x). This approach generally works well for the friction or Coriolis terms in the shallow water equations, as discussed further in Section 7, but is not suitable for handling the bathymetry terms. For the steady state solution of the ocean at rest, the bathymetry source term must exactly cancel out the gradient of hydrostatic pressure that appears in the momentum flux. A fractional step method will not achieve this and will generate large spurious waves. Instead these source terms must be incorporated into the Riemann solution directly, as discussed further below.

240


To incorporate source terms, it is no longer possible to use the conservation form (4.2). Instead we will write the method in fluctuation form = Qni − Qn+1 i

∆t + (A ∆Qni−1/2 + A− ∆Qni+1/2 ), ∆x

(4.4)

where the vector A+ ∆Qni−1/2 represents the net effect of all waves propagating into the cell from the left boundary, while A− ∆Qni+1/2 is the net effect of all waves propagating into the cell from the right boundary. For a homogeneous conservation law, this will be conservative if we choose these fluctuations as a flux-difference splitting at each interface, so that for example A− ∆Qni−1/2 + A+ ∆Qni−1/2 = f (Qni ) − f (Qni−1 ).

(4.5)

When source terms are incorporated, the right-hand side of (4.5) must be suitably modified as discussed below. The notation A± ∆Q is motivated by the linear case. If f (q) = Aq, then Godunov’s method is the simple generalization of the scalar upwind method obtained by taking A± ∆Qni−1/2 = A± (Qni − Qni−1 ), where the matrices A± are defined by A± = RΛ± R−1 ,

Λ± =

(4.6)

(λ1 )± 0 , 0 (λ2 )±

(4.7)

where λ+ = max(λ, 0) and λ− = min(λ, 0). For the linearized shallow water equations, note that in the subcritical case these fluctuations are simply ˆ1W 1 ˆ2W 2 , A+ ∆Qi−1/2 = λ . (4.8) A− ∆Qi−1/2 = λ i−1/2

i−1/2

In the supercritical case, one of the fluctuations would be the zero vector ˆpW p while the other is the sum of λ i−1/2 over p = 1, 2, which gives the full jump in the flux difference A(Qni − Qni−1 ). 4.1. Second-order corrections and limiters Godunov’s method is only first-order accurate and introduces a great deal of numerical diffusion into the solution. In particular, steep gradients are badly smeared out. To obtain a high-resolution method , we add additional terms to (4.4) that model the second derivative terms in a Taylor series expansion of q(x, t + ∆t) about q(x, t), and then apply limiters to avoid the nonphysical oscillations that often arise near discontinuities when a dispersive second-order method is used. To maintain conservation, these corrections

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can be expressed in a flux-differencing form, and so we replace (4.4) by = Qni − Qn+1 i

∆t + ∆t ˜ n n (A ∆Qni−1/2 + A− ∆Qni+1/2 ) − (F − F˜i−1/2 ). (4.9) ∆x ∆x i+1/2

For a constant coefficient linear system, second-order accuracy is achieved by taking ∆t 1 n ˜ Fi−1/2 = I− |A| |A|(Qni − Qni−1 ), (4.10) 2 ∆x where |A| = R(Λ+ − Λ− )R−1 . Inserting (4.10) and (4.6) into (4.9) and simplifying reveals that this is simply the Lax–Wendroff method, 1 ∆t 2 2 n n+1 n 1 ∆t n n Qi = Qi − A (Qi+1 −Qni +Qni−1 ). (4.11) A(Qi+1 −Qi−1 )+ 2 ∆x 2 ∆x Although this is second-order accurate on smooth solutions, the dominant term in the error is dispersive, and so non-physical oscillations appear near steep gradients. This can be disastrous, particularly if they lead to negative values of the depth. By viewing the Lax–Wendroff method in the form (4.9), as a modification to the upwind Godunov method, we can apply limiters to produce ‘high-resolution’ results. To do so, note that the correction flux (4.10) can be rewritten in terms of the waves W 1 and W 2 as 2 1 ∆t p p ˜ Fi−1/2 = , 1− |λ | |λp |Wi−1/2 2 ∆x

(4.12)

i=1

where we have dropped the time step index n and the superscript p refers p to the wave family. We introduce limiters by replacing Wi−1/2 by a limited p p p p version W = Φ(θ )W , where θ is a scalar measure of the i−1/2

i−1/2

i−1/2

i−1/2

p relative to the wave in the same family arising strength of the wave Wi−1/2 from a neighbouring Riemann problem, while Φ(θ) is a scalar-valued limiter function that takes values near 1 where the solution appears to be smooth and is typically closer to 0 near perceived discontinuities. See LeVeque (2002) for more details. There is a vast literature on limiter functions and methods with a similar flavour. Often the limiter is applied to the numerical flux function (giving flux-limiter methods) or to slopes in a reconstruction of a piecewise polynomial approximate solution from the cell averages (e.g., slope limiter methods). The above formulation in terms of ‘wave limiters’ has the advantage that it extends very naturally to arbitrary hyperbolic systems of equations, even those that are not in conservation form. This wave propagation approach is the basic method used throughout the Clawpack software. The generalization to two space dimensions is briefly discussed in Section 6.

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4.2. The f-wave formulation Another formulation of the wave propagation algorithms known as the fwave form has been found to be very useful in many contexts, including the incorporation of source terms as discussed below. An approximate Riemann p solver generally produces a set of wave basis vectors ri−1/2 (often as the eigenvectors of some matrix) and then determines the waves by decomposing the vector Qi − Qi−1 as a linear combination of these basis vectors, p p p αi−1/2 ri−1/2 ≡ Wi−1/2 . (4.13) Qi − Qi−1 = p

p

The f-wave approach instead splits the flux difference as a linear combination of these vectors, p p p f (Qi ) − f (Qi−1 ) = βi−1/2 ri−1/2 ≡ Zi−1/2 . (4.14) p

p

From this splitting we can easily define fluctuations A± ∆Qi−1/2 satisfying p (4.5) by assigning the f-waves Zi−1/2 for which the corresponding eigenvalue or approximate wave speed is negative to A− ∆Qi−1/2 , and the remaining f-waves to A+ ∆Qi−1/2 . For the linearized shallow water equations in the subcritical case, this reduces to 1 2 A− ∆Qi−1/2 = Zi−1/2 , A+ ∆Qi−1/2 = Zi−1/2 , 2 1 ∆t ˆ p p , 1− |λ | sgn(λp )Zi−1/2 F˜i−1/2 = 2 ∆x

(4.15)

p=1

p p is a limited version of Zi−1/2 . The f-waves are limited in where Zi−1/2 p exactly the same manner as waves Wi−1/2 would be. One advantage of this formulation is that the requirement (4.5) is satisfied no matter how the eigenvectors r1 and r2 are chosen for the nonlinear case. Another advantage is that source terms are easily included into the Riemann solver in a well-balanced manner.

5. The nonlinear Riemann problem Although linearized equations may be suitable in deep water, as a tsunami approaches shore the nonlinearities cannot be ignored. In the nonlinear equations the characteristic speeds (eigenvalues of the Jacobian matrix) vary with the solution itself. Over flat bathymetry the fluid depth is greater at the peak of a wave than in the trough, so the peak travels faster and can even overtake the trough in water that is shallow relative to the wavelength. This wave breaking is clearly visible for ordinary wind-generated waves on

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(a)

Time 0

Time 0

Time 3.00

Time 0.08

Time 6.00

Time 0.16

(b)

Figure 5.1. Solution to the ‘dam-break’ Riemann problem for the shallow water equations with initial velocity 0. The shading shows a passively advected tracer to help visualize the fluid velocities, compression, and rarefaction. The bathymetry is (a) B = −1 and Br = −0.5, (b) B = −4000 and Br = −200. In both cases, η = 1 and ηr = 0.

the ocean as they move into sufficiently shallow water in the surf zone. In the shallow water equations the depth must remain single-valued and so overturning waves cannot be modelled directly. Instead a shock wave forms, also called a hydraulic jump in shallow water theory. This models a bore, a near-discontinuity in the surface elevation that is often seen at the leading edge of tsunamis as they approach shore or propagate up a river. The nonlinear Riemann problem over flat bathymetry can be solved and consists of two waves moving at constant velocities, though now each wave is generally either a shock wave (if characteristics are converging) or a spreading rarefaction wave (if characteristics are diverging, i.e., the eigenvalue is strictly increasing from left to right across the wave). For details on solving the nonlinear Riemann problem exactly, see for example LeVeque (2002) or Toro (2001). On varying topography we can consider a generalized Riemann problem in which the bathymetry is allowed to be discontinuous at the point x ¯ along with the state variables. The solution to this nonlinear Riemann problem

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generally consists of three waves. In addition to the two propagating waves, which each propagate over flat bathymetry to one side or the other of x ¯ as in the linear case discussed above, there will also be a stationary wave (propagating with speed zero) at x ¯, where the jump in bathymetry leads to a jump in depth h, and also in the surface η if water is flowing across the step. This is illustrated in Figure 5.1. In the linearized model this stationary jump in η does not appear because the jump in the surface at a stationary discontinuity is of order u2 /gh for small perturbations. Figure 5.1(b) shows the solution to the nonlinear Riemann problem with the same jump in the surface η as in Figure 5.1(a), but over much deeper water. The spread of characteristics across the rarefaction wave is so small that it appears as a discontinuity and the fluid velocity is so small that the jump in surface at the stationary discontinuity can not be seen. 5.1. Approximate Riemann solvers For the linearized shallow water equations on flat topography, the exact eigenstructure is known and easily used to compute the exact Riemann solution for any states Q and Qr , as has been done in Section 3.4. For the nonlinear problem, the exact solution is more difficult to compute and generally not worth the effort, since the waves and speeds are used in a finite volume method that introduces errors when computing cell averages in each time step. Since a Riemann problem is solved at every cell interface in each time step, the cost of the Riemann solver often dominates the computational cost of the method and it is important to develop efficient approximate solvers. Moreover, rarefaction waves such as those shown in Figure 5.1(a) are not directly handled by the wave propagation algorithms, which assume each wave is a jump discontinuity. Instead of using the exact Riemann solution, most Godunov-type methods use approximate Riemann solvers. For GeoClaw we use approximate solvers that always return a set of waves (or f-waves) that are simple discontinuities propagating at constant speeds. These must be chosen in a manner that: • gives a good approximation to the nonlinear Riemann solution, • preserves steady states, in particular the ocean at rest, • handles dry states h = 0 or hr = 0, • works well in conjunction with AMR. The Riemann solver used in GeoClaw is rather complicated and will not be described in detail. We will just give a flavour of how it is constructed. Full details can be found in George (2006, 2008), and the dry state problem is discussed further in George (2010).

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The f-wave approach developed in Section 3.5 is expanded to an augmented Riemann solver in which the vector   ∆h  ∆µ    (5.1)  ∆φ  ∆B is decomposed into 4 waves. Note that the first two components of this vector correspond to the jump in q = (h, µ) in the Riemann problem data, while the second and third components together correspond to the jump in flux f (q) = (µ, φ). The jump in h is explicitly included in order to apply techniques that ensure that no negative depths are generated in the Riemann solutions near the shoreline. The equations defining the Riemann problem consist of the equations (3.9) for h, µ, and B, together with an equation for the momentum flux φ derived by differentiating 1 φ = µ/h + gh2 (5.2) 2 with respect to t and using the equations for the time derivatives of h and µ to obtain (5.3) φt + 2(u2 − gh)µx + 2uφx + 2ghuBx = 0. This results in the non-conservative system        0 1 0 0 0 h h gh − u2 0  µ µ 2u 0 gh    =  .   +  0 0 φ gh − u2 2u 2ghu  φ  0 B x B t 0 0 0 0 The eigenvalues of this matrix are λ1 = u − gh, λ2 = u + gh,

λ3 = 2u,

and the corresponding eigenvectors are       1 1 0 1  2      λ λ 2 3    0 r1 =  (λ1 )2 , r = (λ2 )2 , r = 1, 0 0 0

λ0 = 0,

 gh/λ1 λ2   0  r0 =   −gh . 1

(5.4)

(5.5)



(5.6)

Again the eigenvector r0 corresponds to the stationary wave induced by the jump in topography. Note that the first component of r0 can be written as −1/(1 − u2 /gh) and for zero velocity reduces to −1, corresponding to the jump ∆h = −∆B that gives the ocean at rest, ∆η = 0. It is shown in George (2006, 2008) that a well-balanced method for both the ocean at rest and also a flowing steady state is obtained by defining a discrete

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approximation to the steady wave as



 −ρ1  0   W 0 = ∆B  ¯ 2 , −g hρ 1

(5.7)

¯ = (h + hr )/2 and the ratios ρ1 and ρ2 are nearly 1 for small where h velocities: ¯ ¯ gh max(u ur , 0) − g h , ρ = , (5.8) ρ1 = ¯ 2 ¯ gh − u ¯2 u ¯2 − g h where u ¯ = 12 (u + ur ). Subtracting this wave from the vector (5.1) reduces the problem to a system of three equations for the remaining waves. The eigenvalues λ1 and λ2 are replaced by wave speeds s1 and s2 estimated from the Riemann data, and these values are also used in the discrete eigenvectors r1 and r2 . The wave speeds are approximated using a variant of the approach suggested by Einfeldt (1988) in connection with the HLL solver of Harten, Lax and van Leer (1983) to avoid difficulties with the vacuum state in gas dynamics, which is analogous to the dry state problem in shallow water. This HLLE solver is further discussed in Einfeldt, Munz, Roe and Sjogreen (1991) and elsewhere. These HLLE speeds are given by s1 = min(ˆ s1 , u − c ),

s2 = max(ˆ s2 , u r + c r )

(5.9)

where sˆ1 and sˆ2 are the speeds used in the Roe solver for the shallow water equations, sˆ1 = u ¯ − cˆ, sˆ2 = u ¯ + cˆ, (5.10) where

cˆ =

¯ g h,

√ √ u h + ur hr √ u ˆ= √ . h + hr

(5.11)

The wave decomposition is then done by solving the linear system to determine the weights β 1 , β 2 , and β 3 in         1 1 ∆h 0 ∆µ = β 1  s1  + β 2  s2  + β 3 0. (5.12) ∆φ 1 (s1 )2 (s2 )2 Further improvements can be made by replacing the third eigenvector by a different choice in certain situations, as discussed further in George (2008). Finally, the second and third components of these waves are used as fwaves in the algorithm described in Section 4.2 along with the wave speeds s1 , s2 , and s3 = 2ˆ u. This results in a method that conserves mass (and momentum when ∆B = 0), avoids dry states, and is well-balanced. A number of related Riemann solvers and Godunov-type methods have been

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(a)

Time 0

Time 0

Time 3.00

Time 3.00

Time 6.00

Time 6.00

(b)

Figure 5.2. Solution to the Riemann problem for the shallow water equations with a dry state on the right and positive velocity in the left state. The velocity is larger in the case shown in column (b). The shading shows a passively advected tracer to help visualize the fluid velocities, compression, and rarefaction.

proposed in the literature that can also achieve these goals. The approach outlined above that splits the jump in q and in f (q) is also related to the relaxation approaches discussed in Bouchut (2004) and LeVeque and Pelanti (2001). See also Bale et al. (2002), Gosse (2001, 2000) and In (1999). Riemann problems with an initial dry state on one side raise additional issues that we will not discuss in detail here. Figure 5.2 shows two examples to illustrate one aspect of this problem. In each case there is a step discontinuity in bathymetry with the left cell wet and the right cell dry, data of the sort that naturally arise along the shoreline. In the case illustrated in Figure 5.2(a), the velocity in the left state is positive but sufficiently small that the step discontinuity acts as a solid wall and the Riemann solution consists of a left-moving 1-shock, with stationary water to the right of the shock. The case illustrated in Figure 5.2(b) has a larger positive fluid velocity, in which case the flow overtops the step and there is a right-going 1-rarefaction invading the dry cell along with a left-going 1-shock. For more details about the handling of dry states in the Riemann solver used in GeoClaw, see George (2008, 2010).

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6. Algorithms in two space dimensions In two space dimensions, hyperbolic systems such as (3.2) more generally take the form qt + f (q)x + g(q)y = ψ(q, x, y). (6.1) Godunov-type finite volume algorithms can be naturally extended to two dimensions by solving 1D Riemann problems normal to each edge of a finite volume cell, and using the Riemann solution to define an edge flux or a set of waves propagating into the neighbouring cells. High-resolution correction terms can then be added to achieve greater accuracy without spurious oscillations. The methods used in GeoClaw are the standard wave propagation algorithms of Clawpack, which are described in detail in LeVeque (2002). For a logically rectangular quadrilateral grid, the cells can be indexed by (i, j) and each cell has four neighbours. In this case the numerical solution Qnij is an approximation to the average value of the solution over the grid cell Cij , 1 n Qij ≈ q(x, y, tn ) dx dy, (6.2) Vij Cij where Vij is the area of the cell. For a regular Cartesian grid, the cell areas are simply Vij = ∆x∆y, but the methods can also be applied on any quadrilateral grid defined by a mapping of the uniform computational grid. The basic idea of the wave propagation algorithms in two dimensions is illustrated in Figure 6.1, where six quadrilateral grid cells are shown. Figure 6.1(a) shows the left-going and right-going waves that might be generated by solving the Riemann problem normal to the cell edge in the middle of this patch. The shallow water equations are rotationally invariant, and the Riemann problem normal to any edge can easily be solved by rotating the momentum components of the cell averages Q to normal and tangential components. The normal components are used in solving a 1D Riemann problem along with the depth h on either side. The jump in tangential velocity is simply advected by a third wave propagating at the intermediate velocity found from the 1D Riemann solution. Using these waves to update the cell averages in the two cells neighbouring this edge gives the natural generalization of Godunov’s method, which is first-order accurate and stable only for Courant numbers up to 0.5 (because of the waves that also enter the cell from above and below when solving Riemann problems in the orthogonal direction). To increase the accuracy we need to add second-order correction terms that model the next terms in a Taylor series expansion of the solution at the end of the time step about the starting values, requiring an estimate of qtt . In the Lax–Wendroff framework used in the wave propagation algorithms, this is replaced by spatial derivatives by differentiating the original

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system of equations in time. The result involves qxx and qyy and these terms can be incorporated by a direct extension of the one-dimensional correction terms, with limiters used as in one dimension to give high resolution (sharp gradients without overshoots or undershoots). The time derivative qtt also involves mixed derivatives qxy and it is important to include these terms as well, both to achieve full second-order accuracy and also to improve the stability properties of the method. The cross-derivative terms are included by taking the waves propagating normal to the interface shown in Figure 6.1(a) and splitting each wave into up-going and down-going pieces that modify the cell averages above or below. This is accomplished by decomposing the fluctuations A± ∆Q into eigenvectors of the Jacobian matrix in the transverse direction (tangent to the cell interface we started with). The resulting eigen-decomposition is used to split each of the fluctuations into an downgoing part (illustrated in Figure 6.1(b)) and a up-going part (illustrated in Figure 6.1(c)), and is done in the transverse Riemann solver of Clawpack. The triangular portions of these waves that lie in the adjacent row of grid cells can be used to define a flux from the cells in the middle row to the cells in the bottom or top row of cells respectively. The algorithms must of course be modified to take into account the areas swept out by the waves relative to the area of the grid cells in order to properly update cell averages. This approach is described in more detail in LeVeque (2002) and has been successfully used in solving a wide variety of hyperbolic systems in two space dimensions, and also in three dimensions after introducing an additional set of transverse terms (Langseth and LeVeque 2000). See also LeVeque (1996) for a simpler discussion in the context of advection equations. With the addition of these transverse terms, the resulting method is stable up to a Courant number of 1. The methods can be used on an arbitrary logically rectangular grid: the mapping from computational to physical space need not be smooth, an advantage for some applications such as the quadrilateral grid on the sphere used for AMR calculations in Berger, Calhoun, Helzel and LeVeque (2009). 6.1. Ghost cells and boundary conditions Boundary conditions are imposed by introducing an additional two rows of grid cells (called ghost cells) around the edge of the grid. In each time step values of Q are set in these cells in some manner, depending on the physical boundary condition, and then the finite volume method is applied over all cells in the original domain. Updating cells adjacent to the original boundaries will use ghost cell values in determining the update, and in this way the physical boundary conditions indirectly affect the solution. For tsunami modelling we typically take the full domain to be sufficiently large that any waves leaving the domain can be safely ignored; we assume

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(a)

(b)

(c)

Figure 6.1. (a) Six quadrilateral grid cells and the waves moving normal to a cell interface after solving the normal Riemann problem. (b) Down-going portions of these waves resulting from transverse Riemann solve. (c) Up-going portions of these waves resulting from transverse Riemann solve.

they should not later reflect off a physical feature and re-enter the domain. So we require non-reflecting boundary conditions (also called absorbing boundary conditions) that allow outgoing waves to leave the domain without unphysical numerical reflections at the edge of the computational domain. For Godunov-type methods such as the wave propagation methods we employ, a very simple extrapolation method gives a reasonable nonreflecting boundary condition as discussed in LeVeque (2002): in each time step we simply copy the values of Q in the cells adjacent to each boundary into the adjacent ghost cells. Solving a Riemann problem between two identical states results in zero-strength waves and so the Riemann problems at the cell interfaces at the domain boundary give no spurious incoming waves. This is illustrated in Figure 12.1, for example, where the tsunami is seen to leave the computational grid with very little spurious reflection. When adaptive mesh refinement is used, many grid patches will have edges that are within the full computational domain. In this case ghost cell values are filled either from an adjacent grid at the same level of refinement, if such a grid exists, or by interpolating from coarser levels. This is described further in Section 9. It is important to ensure that spurious waves are not generated from internal interfaces between grids at different levels. Again Godunov-type methods seem to handle this quite well, as is also apparent from the results shown in Figure 12.1, for example.

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6.2. Solving on the sphere To properly model the propagation of tsunamis across the ocean, it is necessary to solve the shallow water equations on the surface of the sphere rather than in Cartesian coordinates. This can be done using the approach discussed above and illustrated in Figure 6.1, where now the cell area is calculated as an area on the sphere. The coordinate lines bounding the quadrilaterals are assumed to lie along great circles on the sphere between the corner vertices and so these areas are easily computed. The current implementation in GeoClaw assumes that latitude–longitude coordinates are used on the sphere. This gives some simplification of the Riemann solvers since the cell edges are then orthogonal to one another and the momenta that are stored in the Q vectors are the components of momentum in these two directions. Latitude–longitude grids are generally used for teletsunami modelling since interest is generally focused on the mid-latitudes. To obtain an accurate representation of flow on the sphere, it is necessary to compute the cell volumes Vij using surface area on the sphere. The grid cells are viewed as patches of the sphere obtained by joining the four corners by great circle arcs between the specified latitude and longitude values. The length of the cell edges also come into the finite volume methods and must be calculated using great circle distance. On the full sphere, latitude–longitude coordinates have the problem that grid lines coalesce at the poles. The cells are very small near the poles relative to those near the equator, requiring very small time steps in order to keep the global Courant number below 1. A variety of other grids have been proposed for solving problems on the full sphere, particularly in atmospheric sciences where flow at the poles is an important part of the solution. One approach that fits well with the AMR algorithms described in this paper is discussed in Berger et al. (2009).

7. Source terms for friction Topographic source terms are best incorporated into the Riemann solver, as described in Section 3.1. Additional source terms arise from bottom friction in shallow water, and are particularly important in modelling inundation. Run-up and inundation distance are affected by the roughness of the terrain, and would be much larger on a bare sandy beach than through a mangrove swamp, for example. To model friction, we replace the momentum equations of (3.2) by (hu)t + (hu2 + 12 gh2 )x + (huv)y + ghBx = −D(h, u, v)hu,

(7.1a)

+ ghBy = −D(h, u, v)hv,

(7.1b)

2

(hv)t + (huv)x + (hv +

2 1 2 gh )y

with some frictional drag coefficient D(h, u, v). Various models are available

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in the literature. We generally use the form D(h, u, v) = n2 gh−7/3 u2 + v 2 .

(7.2)

The parameter n is the Manning coefficient and depends on the roughness. If detailed information about the surface is known then this could be a spatially varying parameter, but for generic tsunami modelling a constant value of n = 0.025 is often used. Note that in deep water the friction term in (7.1) is generally negligible, being of magnitude O(|u|2 h−4/3 ), and so we only apply these source terms in coastal regions, e.g., in depths of 100 m or less. In these regions the source term is applied as an update to momentum at the end of each time step. We loop over all grid cells and in shallow regions update the momenta (hu)ij and (hv)ij by −7/3 2, Dij = n2 ghij u2ij + vij (7.3) (hu) = (hu) /(1 + ∆tD ), ij

ij

ij

(hv)ij = (hv)ij /(1 + ∆tDij ), This corresponds to taking a step of a linearized backward Euler method on the ordinary differential equations for momentum obtained from the source alone. By using backward Euler, we ensure that the momentum is driven to zero when ∆tDij is large, rather than potentially changing sign as might happen with forward Euler, for example. A higher-order method could be used, but given the uncertainty in the Manning coefficient (and indeed in the friction model itself), this would be of questionable value. Including friction is particularly important at the shoreline where the depth h approaches zero, and the above procedure helps to stabilize the method and ensure that velocities remain bounded as the shoreline moves while a wave is advancing or retreating.

8. Adaptive mesh refinement In this section we will first describe the general block-structured adaptive mesh refinement (AMR) algorithms that are widely used on structured logically rectangular grids. This approach is discussed in numerous papers including Berger and Oliger (1984) and Berger and Colella (1989). The implementation specific to Clawpack and hence to GeoClaw is described in more detail in Berger and LeVeque (1998). We will summarize the basic approach and then concentrate on some of the challenges that arise when combining AMR with geophysical flow algorithms, in particular in dealing with dry states and with the need for well-balanced algorithms that maintain steady states.

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8.1. AMR overview Block-structured AMR algorithms are designed to solve hyperbolic systems on a hierarchy of logically rectangular grids. A single coarse (level 1) grid comprises the entire domain, while grids at a given level + 1 are finer than the coarser level grids by fixed integer refinement ratios rx and ry in the two spatial directions, ∆x+1 = ∆x /rx ,

∆y +1 = ∆y /ry .

(8.1)

In practice we normally take rx = ry at each level since in this application there is seldom any reason to refine differently in the two spatial directions. The nesting requirements of subgrids are not restrictive, in that a single level ( + 1) grid may overlap several level grids, and may be adjacent to level ( − 1) grids. Since subgrids at a given level can appear and disappear adaptively, the highest grid level present at a given point in the domain changes with time. The subgrid arrangement changes during the process of regridding, which occurs every few time steps. This allows subgrids to essentially ‘move’ with features in the solution. On the current set of grids, the solution on each grid is advanced using the same numerical method that would be used on a single rectangular grid, together with some special procedures at the boundaries of subgrids. The time steps on level + 1 grids are typically smaller than the time step on the level grids by a factor rt . Since Godunov-type explicit methods like the wave propagation method are stable only if the Courant number is bounded by 1, it is common practice to choose the same refinement factor in time as in space, rt = rx = ry , since this usually leads to the same Courant number on the finer grids as on the coarser grid. The Courant number can be thought of as a measure of the fraction of a grid cell that a wave can traverse in one time step, and is given by |smax ∆t/∆x|, where smax is the maximum wave speed over the grid. However, for tsunami applications of the type considered in this paper, it is often desirable to choose rt to be smaller than the spatial refinement factor for the levels corresponding to the finest grids, which are often introduced only near the shoreline in regions where run-up and inundation are to be studied. This √ √ is because the Courant number is based on the wave speed |u ± gh| ≈ gh, which depends on the water depth. For grids that are confined to coastal regions, h is much smaller than on the coarser grids that cover the ocean. If the coarsest grid covers regions where the ocean is 4000 m deep while a fine level is restricted to regions where the depth is at most 40 m, for example, then refining by the same factor in space and time would lead to a Courant number of 0.1 or less on the fine grid and potentially require 10 times as many time steps on the fine grids than

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are necessary for stability. Since the vast majority of grid cells are often associated with fine grids near the shore, this can have a huge impact on the efficiency of the method (and also its accuracy, since solving a hyperbolic equation with very small Courant number introduces additional numerical viscosity and is typically less accurate than if a larger time step is used). We will first give a brief summary of the AMR integration algorithm and the regridding strategy. We then focus on the modifications that are required for tsunami modelling, which are also important in modelling other depth-averaged geophysical flows of the type mentioned in Section 1. 8.2. AMR procedure The basic AMR integrating algorithm applies the following steps recursively, starting with the coarsest grids at level = 1. AMR Integration Strategy. (1) Take a time step of length ∆t on all grids at level . (2) Using the solution at the beginning and end of this time step, perform space–time interpolation to determine ghost cell values for all level +1 grids at the initial time and all rt − 1 intermediate times, for any ghost cells that do not lie in adjacent level + 1 grids. (Where there is an adjacent grid at the same level, values are copied directly into the ghost cells at each intermediate time step.) (3) Take rt time steps on all level + 1 grids to bring these grids up to the same advanced time as the level grids. (4) For any grid cell at level that is covered by a level + 1 grid, replace the solution Q in this cell by an appropriate average (described in Section 9) of the values from the rx ry grid cells on the finer grid that cover this cell. (5) Adjust the coarse cell values adjacent to fine grids to maintain conservation of mass (and of momentum in regions where the source terms vanish). This step is described in more detail in Section 9.4, after discussing the interpolation issues. After each of the level + 1 time steps in step (3) above, the same algorithm is applied recursively to advance even finer grids (levels + 2, . . .). Every few time steps on each level a regridding step is applied (except on the finest allowed level). The frequency depends on how fast the waves are moving, and how wide a buffer region around the grid patches there is. The larger the buffer region, the less frequently regridding needs to be performed. On the other hand a wide buffer region results in more grid cells to integrate on the finer level. We typically use a buffer width of 2 or 3 cells and regrid every 2 or 3 time steps on each level.

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AMR Regridding Algorithm. (1) Flag cells at level that require refinement to level + 1. Our flagging strategy for tsunami modelling is summarized below. (2) Cluster the flagged cells into rectangular patches using the algorithm of Berger and Rigoutsos (1991). This heuristic tries to strike a balance between minimizing the number of grids (to reduce patch overhead), and minimizing the number of unnecessarily refined cells when clustering into rectangles. (3) Initialize the solution on each level + 1 grid. For each cell, either copy the data from an existing level + 1 grid or, if no such grid exists at this point, interpolate from level grids using procedures described in the next section.

8.3. AMR cell flagging criteria Depending on the application, a variety of different criteria might be used for flagging cells. In many applications an error estimation procedure or a feature detection algorithm is applied to all grid points on levels l < Lmax , where Lmax is the maximum number of levels allowed. Cells where a threshold is exceeded are flagged for inclusion in a finer grid patch. A common choice is to compute the spatial gradient of one or more components of the solution vector q. For the simulation of tsunamis, we generally use the elevation of the sea surface relative to sea level, |h + B − ηs |. This is non-zero only in the wave and is a much better flagging indicator than the gradient of h, for example, which can be very large even in regions where the ocean is at rest due to variations in topography. The sheer scale of tsunami modelling makes it necessary to allow much more refinement in some spatio-temporal regions than in others. In particular, the maximum refinement level and refinement ratios may be chosen to allow a very fine resolution of some regions of the coast that are of particular interest, for example a harbour or bay where a detailed inundation map is desired. Other regions of the coast may be of less interest and require less refinement. We may also wish to allow much less refinement away from the coast where the tsunami can be well represented on a much coarser grid. Conversely it is sometimes useful to require refinement up to a given level in certain regions. This is useful, for example, to force some refinement of a region before the wave arrives. These regions of required or allowed refinement may vary with time, since one part of the coast may be of interest at early times and another part of the coast (more distant from the source) of interest at later times. To address this, in the GeoClaw software the user

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can specify a set of space–time regions of the form L1 , L2 , x1 , x2 , y1 , y2 , t1 , t2 to indicate that on the given space–time rectangle [x1 , x2 ] × [y1 , y2 ] × [t1 , t2 ], refinement to at least level L1 is required, and to at most level L2 is allowed.

9. Interpolation strategies for coarsening and refining If the refinement level increases in a region during regridding, the solution in the cells of the finer grid may need to be interpolated from coarser levels in order to initialize the new grids. In the other direction, averaging from fine grids to coarser underlying grids is done in step (4) of the AMR algorithm of Section 8.2. This produces the best possible solution on the coarse grid at each time. When a fine grid disappears in some region during regridding, the remaining coarser grid already contains the averaged solution based on the finer grid, and so no additional work is required to deal with coarsening during the regridding stage. We will first discuss refining and coarsening in the context of a onedimensional problem where it is easier to visualize. The formulas we develop all extend in a natural way to the full two-dimensional case, discussed in Section 9.3. When refining and coarsening it is important to maintain the steady states of an ocean at rest. This is particularly important since refinement often occurs just before the tsunami wave arrives in an undisturbed area of the ocean, and coarsening occurs as waves leave an area and the ocean returns to a steady state. Since the interpolation procedures are intimately tied to the representation of the bathymetry and its interpolation between grids at different levels, we start the discussion there. We consider a cell Ck at some level , and say that a cell Ci+1 at the finer level is a subcell of Ck if it covers a subset of the interval Ck (recall we are still working in one space dimension). The set of indices i for which Ci+1 is a subcell of Ck will be denoted by Γk . We will say that the topography is consistent between the different levels if the topography value Bk in a cell at level is equal to the average of the values Bi+1 in all subcells of Ck at level + 1: 1 +1 Bi . (9.1) Bk = rx i∈Γk

If the cells have non-uniform sizes, for example on a latitude–longitude grid in two dimensions, then this formula generalizes to the requirement that 1 +1 +1 V i Bi . (9.2) Bk = Vk i∈Γk

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Tsunami modelling ηs

ηs

C1 +1 C2 +1 C3 +1 C4 +1 C5 +1 C6 +1

C1

(a)

C2

C3

(b)

Figure 9.1. (a) Level + 1 topography and water depth with a constant sea surface elevation ηs . The dashed lines are the level topography. (b) Level

topography and water depth on the coarse grid.

where the cell volumes (lengths in 1D, areas in 2D) satisfy Vk = Vi+1 .

(9.3)

i∈Γk

Since a discussion of this consistency is most relevant in 2D, we will defer discussion of how we accomplish (9.2) to Section 9.3, and assume that it holds for now. 9.1. Coarsening and refining away from shore We first consider a situation such as illustrated in Figure 9.1, where all the cells are wet on both levels. In this figure and the following figures, the darker region is the earth below the topography Bi and the lighter region is the water between Bi and ηi = Bi + hi . The coarse-grid topography of Figure 9.1(b) (which is also shown as a dashed line in Figures 9.1(a) and (b)) is consistent with the fine grid topography: each coarse-grid value of B is the average of the two fine grid values. The water depths illustrated in Figures 9.1(a) and 9.1(b) are consistent with each other (the total mass of water is the same) and both correspond to an undisturbed ocean with η ≡ ηs . Suppose we are given the solution Q+1 for i = 1, 2, . . . , 6 on the fine i (level + 1) grid shown in Figure 9.1(a) and we wish to coarsen it to obtain Figure 9.1(b). Assume the topography Bi+1 is consistent, so that 1 +1 +1 + B2k ), Bk = (B2k−1 2

k = 1, 2, 3.

(9.4)

To compute the water depth hk in the coarser cells we can simply set 1 hk = (h+1 + h+1 2k ), 2 2k−1

k = 1, 2, 3.

(9.5)

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This preserves the steady state of water at rest Bi+1 + h+1 = ηs for all i i ∈ Γk , since then Bk + hk = ηs as well. More generally, with an arbitrary refinement factor rx and possibly varying cell volumes, we would set 1 +1 +1 Vi hi . (9.6) hk = Vk i∈Γk

The momentum µk can be averaged from level + 1 to level in the same manner, replacing h by µ in (9.6). To go in the other direction, now suppose we are given the coarse-grid solution of Figure 9.1(b) and wish to interpolate to the fine grid, for example after a new grid is created. We would like to obtain Figure 9.1(a) on the fine grid in this case, with the flat water surface preserved. Unfortunately, the standard approach using linear interpolation of the conserved variables in the coarse cell and evaluating them at the centre of each fine grid cell described in Berger and LeVeque (1998) works very well for most conservation laws but fails miserably here. For the data shown in Figure 9.1(b), the depth is decreasing linearly over the three coarse-grid cells. Using this linear function as the interpolant to compute the fluid depth h in the cells Ci+1 on the finer grid would conserve mass but would not preserve the sea surface, because the fine grid bathymetry is not varying linearly. Variation in the sea surface would generate gradients of h + B and hence spurious waves. In tsunami calculations on coarse ocean grids this interpolation strategy can easily generate discontinuities in the surface level on the order of tens or hundreds of metres, destroying all chances of modelling a tsunami. Instead, the interpolation must be based on surface elevation ηk = Bk + hk , which for Figure 9.1(b) would all be equal to ηs . We construct a linear interpolant to these data over each grid cell and evaluate this at the fine cell centres to obtain values ηi+1 , and then set h+1 = ηi+1 − Bi+1 . The interpolant in coarse cell k is i η(x) = ηk + σk (x − xk ),

(9.7)

where xk is the centre of this cell. The slope σk is chosen to be , ηk+1 − ηk )/∆x . σk = minmod(ηk − ηk−1

(9.8)

We generally use the standard minmod function (e.g., LeVeque (2002)), which returns the argument of minimum modulus, or zero if the two arguments have opposite sign. This interpolation strategy prevents the introduction of new extrema in the water surface elevation, preserves a flat sea surface (provided that all depths are positive), and produces Figure 9.1(a) from the data in Figure 9.1(b). In a tsunami wave the sea surface is not flat but is nearly so, and the surface is a smoothly varying function of x even when the topography varies

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rapidly. The approach of interpolating the water surface elevation also works well in this case and produces a second-order accurate approximation to smooth waves. Note that although we switch to the variable η when doing the interpolation, the equations are still being solved in terms of the conserved quantities. To interpolate the momentum, we begin with the standard approach; we determine a linear interpolant to µk and then evaluate this at the fine cell centres. Again we use minmod slopes to prevent the introduction of new local extrema in momentum. However, because we might be interpolating to fine cells that are much shallower than the coarse cell, we ensure that the interpolation does not introduce new extrema in velocities as well. We +1 check the velocities in the fine cells, defined by µ+1 i /hi , for all i ∈ Γk , to see if there are new local extrema that exceed the coarse velocities Ck , Ck−1 and Ck+1 . If so, we redefine the fine cell momenta, for all i ∈ Γk , by µ+1 µk /hk . (9.9) = h+1 i i Note that this still conserves momentum, assuming that (9.6) is satisfied, since +1 +1 Vi+1 µ+1 = µk /hk Vi hi by (9.9). i i∈Γk i∈Γk (9.10) = Vk µk

by (9.6)

While this additional limiting may at first seem unnecessary or overly restrictive, without it the velocities created in shallow regions where fine cells have vanishingly small depths can become unbounded. This makes the interpolation procedures near the shore, at the interface of wet and dry cells, especially difficult. This is the subject of the next section. 9.2. Coarsening and refining near the shore The averaging and interpolation strategies just presented break down near the shoreline where one or more cells is dry. Figure 9.2 illustrates two possible situations. In both cases it is impossible to maintain conservation of mass and also preserve the flat sea surface. In this case we forgo conservation and maintain the flat surface, since otherwise the resulting gradient in sea surface will generate spurious waves near the coast that can easily have larger magnitude than the tsunami itself. In Figure 9.2(b) the middle coarse cell is wet, h2 > 0, while on the refined > 0 but h+1 = 0 in grid only one of the two refined cells is wet, h+1 3 4 Figure 9.2(a). Figures 9.2(c) and 9.2(d) show a case where the middle coarse cell is dry, but on the fine grid one of the underlying fine cells must be wet in order to maintain a constant sea surface. In both cases, the total

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ηs

C1 +1 C2 +1 C3 +1 C4 +1 C5 +1 C6 +1

ηs

C1

(a)

C2

C3

(b)

ηs

C1 +1 C2 +1 C3 +1 C4 +1 C5 +1 C6 +1

(c)

ηs

C1

C2

C3

(d)

Figure 9.2. (a) Level + 1 topography and water depth on a beach where the three rightmost cells are dry. (b) Corresponding level representation with one dry cell. (c) Second example of level + 1 topography and water depth on a beach where the three rightmost cells are dry. (d) The corresponding level representation with two dry cells. Note that refining the middle dry cell leads to one wet cell and one dry cell.

mass of water is not preserved either when going from the coarse to fine or from the fine to coarse grid. The lack of conservation of mass near shorelines is perhaps troubling, but there is no way to avoid this when different resolutions of the topography are used. For ocean-scale tsunami modelling it may easily happen that the entire region of interest along the coast lies within a single grid cell on the coarsest level, and this cell will be dry if the average topography value in this coarse cell is above sea level. Obviously, when this cell is refined as the wave approaches land, water must be introduced on the finer grids in order to properly represent the fine-scale topography and shoreline. Stated more generally, in order to prevent the generation of new sea-surface extrema and hence hydraulic gradients near the shoreline, the coarsening and refining formulas presented in the previous section require additional modifications, which do not conserve mass in general. To mitigate this we try to ensure

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that the shoreline is appropriately refined, using fine topography and a flat sea, before the wave arrives. Then the change in mass does not affect the computed solution at all: exactly the same solution would be computed if the shoreline had been fully resolved from the start of the computation. When coarsening, the simple averaging (9.6) cannot be used near the shoreline unless all fine cells are dry h+1 = 0, in which case the coarse cell i hk = 0 is also dry and is the average. In general, to go from the fine grid values to a coarse-grid value we average over only those subcells that are wet, setting +1 +1 sgn(h+1 i∈Γ Vi i )ηi , (9.11) η˜k = k +1 sgn(h+1 i∈Γ Vi i ) k

and then set hk = max(0, η˜k − Bk ).

(9.12)

Note that sgn(H) is always 0 (if the cell is dry) or 1 (if it is wet) and by assumption at least one subcell is wet, so the denominator of (9.11) is nonzero. If all cells are wet then we will have η˜k > Bk and mass is conserved. In fact the formula (9.12) reduces to (9.6) in this case, and in practice we always use (9.12) for coarsening. Now consider refinement. To interpolate the depth from a coarse cell to the underlying fine cells in a situation such as those shown in Figure 9.2, we first construct a linear interpolant for the surface elevation η, a function of the form Nk (x) = η¯k + σ ¯k (x − xk ), (9.13) Here η¯k is not the usual surface variable η, but is modified to account for dry cells. This will be defined below. Once defined on all coarse cells, σ ¯k is computed again using minmod slopes based on the values of η¯ . We then compute the depth in the subcells using this linear function and the fine grid topography, +1 = max(0, Nk (x+1 ), h+1 i i ) − Bi

(9.14)

for each subcell Ci+1 of the coarse cell Ck . If the cell Ck is wet then the surface value η¯k in (9.13) is taken to be η¯k = ηk = Bk + hk

if hk > 0.

(9.15)

If the coarse cell is dry, we need to determine an appropriate surface elevation η¯k for use in the linear function (9.13). In this case ηk = Bk , and this topography value may be above sea level. Instead of using this value we set η¯k = ηs , the specified sea level, in this case. For interpolating and coarsening momentum near the shore, because mass is not conserved in general, we must treat momentum carefully by adopting

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additional procedures to change the momentum in a way that is consistent with the change in mass. Upon coarsening the momentum, the procedure we use conserves momentum whenever mass is conserved. If mass is lost upon coarsening, such as would occur when coarsening the solutions shown in Figures 9.2(a) and 9.2(c), the momentum associated with the mass that no longer exists is removed. That is, for cells with non-zero mass, we define the coarse momentum by min(Vk hk , i∈Γ Vi+1 h+1 i ) 1 +1 +1 k µk = Vi µ i . (9.16) +1 +1 Vk hi i∈Γ Vi i∈Γk

k

Note that (9.16) reduces to the standard coarsening formula when the mass is conserved, yet when mass is reduced upon coarsening the coarse momentum is multiplied by the ratio of the coarse mass to the mass in the fine subcells. Upon refinement, we begin with the standard procedure used away from the shore: a linear interpolation of momentum is performed, and then the momentum in each fine subcell is checked to see if new extrema in velocities are generated (in determining velocity bounds, we define the velocity to be zero in dry neighbouring coarse cells), in which case we resort to (9.9) for all subcells i ∈ γk . This certainly includes the case where a dry (h+1 = 0) i subcell has non-zero momentum and hence an infinite velocity. That is, when velocity bounds are violated, each fine-cell momentum becomes the product of the fine-cell depth and the coarse velocity (for coarse cells that are dry (hk = 0), the velocity is defined to be zero). Note that this procedure alone does not conserve momentum if mass is not conserved: rather, if mass is altered on the fine grid, the momentum would be altered by the ratio of fine subcells’ mass to the coarse cell’s mass. To prevent the addition of momentum to the system purely through refinement, we modify (9.9) to µ+1 = h+1 i i

max hk , V1 k

µ k i∈Γk

Vi+1 h+1 i

.

(9.17)

Note that (9.17) implies that momentum is conserved even when mass has been added, since in that case i∈Γk

Vi+1 µ+1 = i

max

hk , V1 k

µ k

+1 +1 hi i∈Γk Vi

Vi+1 h+1 = Vk µk . i

i∈Γk

(9.18) When mass is lost, (9.17) implies that the momentum is multiplied by the ratio of the remaining fine-cell mass and coarse mass, essentially removing momentum associated with the lost mass.

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All of the formulas in this section reduce to those of the previous section when mass is conserved. Therefore we can implement the coarsening and refining strategy in a uniform manner for all cases. While the formulas may seem overly complicated, they ensure the following properties upon regridding. • • • •

Mass is conserved except possibly near the shore. Mass conservation implies momentum conservation. If mass is gained, momentum is conserved. If mass with non-zero momentum is lost, the momentum associated with that mass is removed as well. • New extrema in surface elevation and hence hydraulic gradients are not created. • New extrema in water velocity are not created. In general, coarsening and refinement near the shoreline should ideally happen just prior to the arrival of waves, while the shoreline is still at a steady state. In this case, all of the specialized procedures described in this section produce the same solution on the fine grids as would exist if the fine grids had been initialized with a constant sea level long before the tsunami arrival. Therefore, although mass and momentum are not necessarily conserved upon refinement, the adaptive solution is ideally close to the solution that would exist if fixed (non-AMR) grids were used, yet at a much reduced computational expense. 9.3. Extension to two dimensions Interpolation and averaging All of the interpolation and averaging strategies described above extend naturally to two dimensions. In fact, if we continue to let a single index represent grid cells, e.g., i ∈ Γk represents the index of a level + 1 rectangular subcell Ci+1 within a level rectangular cell Ck , then most of the formulas described above need only minor modification. The length ratio rx becomes an area ratio rx ry in the case of a Cartesian grid. More generally we continue to use Vk to represent the area of cell k on level . For interpolation, we simply extend the linear interpolants (9.7) and (9.13) to two dimensions, f (x, y) = fk + (σ x )k (x − xk ) + (σ y )k (y − yk ).

(9.19)

Here, (xk , yk ) is the centre of this cell, and the slopes (σ x )k and (σ y )k are the minmod limited slopes in the x and y directions respectively. Lastly, when considering neighbouring cells to determine if new velocity extrema are generated in Ci+1 , we consider all nine coarse cells including and surrounding Ck .

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Consistent computational topography The topographic data used in a computation are often specified by several different rectangular gridded digital elevation models (DEMs) that are at different resolutions. For example, over the entire ocean 10-minute data may be sufficient, while in the region near the earthquake source or along a coastal region of interest one or more finer-scale DEMs must be provided. The DEMs also do not necessarily align with the finite volume computational grids, and so the consistency property (9.2) for topography on different grid levels requires careful consideration. We accomplish (9.2) in the following manner. The topography data sets are ordered in terms of their spatial resolution (if two data sets have the same resolution they are arbitrarily ordered). We define the topographic surface B(x, y) as the piecewise bilinear function that interpolates the topography data set of the highest resolution DEM at any given point (x, y) in the domain. Away from boundaries of DEMs, this function is continuous and defined within each rectangle of the DEM grid using bilinear interpolation between the four corner points. Where a fine DEM grid is overlaid on top of a coarser one, there are potentially discontinuities in B(x, y) across the outer boundaries of the finer DEM. This procedure defines a unique piecewise bilinear function B(x, y) based only on the DEM grids, independent of the computational grid(s). When a new computational grid is created, either at the start of a computation or when regridding, the computational topography in each finite volume cell is defined by integrating B(x, y) over the cell: 1 Bk = B(x, y) dx dy. (9.20) Vi Ck Note that each integral may span several DEM cells if it overlaps a DEM boundary, but since it is a piecewise bilinear function the integral can be computed exactly. Since these topography values are based on exact integrals of the same surface, at all refinement levels, the consistency property (9.2) will always be satisfied. 9.4. Maintaining conservation at grid interfaces The discussion above focused on maintaining constant sea level and conserving mass and momentum when grid cells are coarsened and refined. We now turn to step (5) in the AMR Integration Strategy of Section 8.2. We wish to ensure that the method conserves mass away from the shoreline at least, and also momentum in the case without source terms. The final solution will not be strictly conservative due to the source terms of topography and friction, and due to the shoreline algorithms which favour maintaining a constant sea level over maintaining conservation of mass, but the underlying method should be conservative to reduce these effects.

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Recall that in step (4) of the algorithm in Section 8.2 the value of Q in any coarse-grid cell that is covered by fine grids is replaced by the appropriate weighted average of the more accurate fine-grid values. This potentially causes a conservation error since the fine grid cells at the boundary of the patch were updated using fluxes from ghost cells rather than from the neighbouring coarse cells directly. The standard fix for this, applied to finite volume methods written in flux-differencing form, is to adjust the adjacent coarse-grid values by the difference between the coarse grid flux at the patch boundary (originally used to compute the value in this cell) and the weighted average of fine grid fluxes that was instead used interior to the patch (Berger and Colella 1989). This restores global conservation and presumably also improves the value in these coarse-grid cells by using a more accurate approximation of the flux, as determined on the fine cells. We use the f-wave propagation algorithm instead of flux-differencing since this allows the development of a well-balanced method in the non-conservative form described above. This requires a modification of the flux-based fix-up procedure that is described in detail in Berger and LeVeque (1998) and implemented more generally in the AMR algorithms of Clawpack. This modification works in general for wave propagation algorithms based on fluctuations rather than fluxes. Note that since the fine grid typically takes many time steps between each coarse-grid step, performing this fix-up involves saving the fluctuations around the perimeter of each fine grid at each intermediate step, which is easily done. The harder computation is the modification of the coarse cell values based on these fluctuations, since the coarse cells affected are generally interior to a coarse grid and appear in an irregular manner. Each coarse grid keeps a linked list of cells needing this correction, and saves the fluctuations on the edge adjacent to a fine grid. For example, if the coarse cell is to the left of the fine grid, the left-going fluctuation is needed. One additional correction step is needed for conservation when using the wave propagation approach. A Riemann problem between a coarse-grid cell and the fine grid ghost cell needs to be accounted for to maintain conservation. This leads to an additional Riemann problem at the boundary of each fine grid cell at each intermediate time, as discussed by Berger and LeVeque (1998). In the absence of source terms and away from the shoreline, these two steps ensure that conservation is maintained in spite of the fact that the method does not explicitly calculate fluxes.

10. Verification, validation, and reproducibilty Verification and validation (V&V) is an important aspect of research in computational science, and often poses a large-scale challenge of its own for complex applications: see Roache (1998) for a general discussion. Our

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goal in this paper is not to provide detailed V&V studies of the GeoClaw software, but only to give a flavour of some of the issues that arise and approaches one might take in relation to tsunami models. For some other discussions of this topic and possible test problems, see for example Synolakis and Bernard (2006) or Synolakis, Bernard, Titov, Kâno˘ glu and Gonz´ alez (2008). Verification in the present context consists of verifying that the computational algorithms and software can give a sufficiently accurate solution to the shallow water equations they purportedly solve, with the specified topography and initial conditions. In particular, this requires checking that the adaptive refinement algorithms provide accurate results in the regions of interest even when much coarser grids are used elsewhere, without generating spurious reflections at grid interfaces, for example. Exact solutions to the shallow water equations over topography are difficult to come by, but a few solutions are known that are useful, in particular as tests of the shoreline algorithms. The paper of Carrier, Wu and Yeh (2003) provides the exact solution for a one-dimensional wave on a beach that is suggested as a verification problem in Synolakis et al. (2008) and was one of the benchmark problems discussed by many authors in Liu, Yeh and Synolakis (2008). The paper of Thacker (1981) presents some other exact solutions, including water sloshing in a parabolic bowl, which has often been used as a test problem for numerical methods, for example by Gallardo et al. (2007) and in the GeoClaw test suite. In Section 11 we illustrate another technique for testing whether a tsunami code accurately solves the shallow water equations in the necessary regimes for modelling both transoceanic propagation and local inundation. No finite set of tests will prove that the program always gives correct solutions, and of course no numerical method will: the accuracy depends on the grid resolution used and other factors. However, by exercising the code on problems where an exact or highly accurate reference solution is available, it is possible to gain a useful appreciation for the accuracy and limitations of a code. Validation of a code is generally more difficult, since this concerns the question of whether the computational results provide a useful approximation to reality under a certain range of conditions. The depth-averaged shallow water equations provide only an approximation to the full threedimensional Navier–Stokes equations, so even the exact solution to these equations will only be an approximation to the real flow. Several assumptions are made in deriving the shallow water equations, in particular that the wavelength of the waves of interest is long relative to the fluid depth. This is often true for tsunamis generated by megathrust events, at least for the transoceanic propagation phase. It is less clear that this assumption holds as tsunamis move into shallower water and interact with small-scale local features. A great deal of effort has gone into validation studies for

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the shallow water equations and into the development and testing of other model equations that may give a better representation of reality. Validation studies of tsunami models take many forms. Comparison with actual tsunami events similar to ones the code is designed to model is the best form of validation in many ways. Due to the recent frequency of large tsunamis there is a wealth of data now available, far beyond what was available 10 years ago. These data have been used in numerous validation studies of tsunami models, for example Grilli et al. (2007) and Wang and Liu (2007). These studies are seldom clear-cut, however, due to the wide range of unknowns concerning the earthquake source structure, the resulting seafloor deformation, the proper drag coefficient to use in friction terms, and various other factors. To perform more controlled experiments, large-scale wave tanks are used to simulate tsunami inundation, with scaled-down versions of coastal features and precisely controlled sources from wave generators. The resulting flow and inundation can then be accurately measured with tools such as depth gauges, flow meters, and high-speed cameras. By running a tsunami code on the wave tank topography with the same source, careful comparisons between numerical results and the actual flow can be performed. Some standard test problems are described in Liu et al. (2008) and Synolakis et al. (2008). A problem with this, of course, is that this can only validate the code relative to the wave tank, which is itself a scaled-down model of real topography. There is still the question of how well this flow corresponds to reality. Reproducibility of computational experiments is an issue of growing concern in computational science; see for example Fomel and Claerbout (2009), Merali (2010) and Quirk (2003). By this we mean the performance of computational experiments in a controlled and documented manner that can potentially be reproduced by other scientists. While this is a standard part of the scientific method in laboratory sciences, in computational science the culture has put little emphasis on this. Many publications contain numerical experiments where neither the method used or the test problem itself are described in sufficient detail for others to verify the results or to perform meaningful comparisons against competing methods. Part of our goal with GeoClaw, and with the Clawpack project more generally, is to facilitate the specification, sharing, and archiving of computational experiments (LeVeque 2009). The codes for all the experiments in the next sections can be found on the webpage for this paper (www13), along with pointers to additional codes such as those used for the experiments presented in Berger et al. (2010).

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11. The radial ocean As a verification test that the shallow water equations are solved correctly on the surface of the sphere, and that the wetting and drying algorithms give similar results regardless of the orientation of the shoreline relative to the grid, it is useful to test the code on synthetic problems where comparisons are easy to perform. We illustrate this with an example taken from Berger et al. (2010), which should be consulted for more details on the bathymetry and a related test problem. The domain consists of a radially symmetric ocean with a radius of 1645 km on the surface of a sphere of radius comparable to the earth’s radius, R = 6367.5 km, centred at 40◦ N. The extent of the ocean in latitude–longitude space is shown in Figure 11.1. The bathymetry is flat at −4000 m up to a 1500 km, and then is followed by a smooth continental slope and continental shelf with a depth of 100 m, and finally a linear beach. The initial conditions for the tsunami consist of a Gaussian hump of water at the centre, given by η(r) = A0 exp −2r2 /109 , (11.1) where r is the great-circle distance from the centre, measured in metres. The amplitude A0 is varied to illustrate the effect of different size tsunamis. At some location along the shelf we place a circular island with a radius of roughly 10 km, centred 45 km offshore. In theory the flow around the island should be identical regardless of where it is placed, though numerically this will not be true. We compare the results for two different locations as a test of consistency. Cross-section through island centre

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Figure 11.1(a) shows the ocean (which does not look circular in latitude– longitude coordinates). The outer solid curve is the position of the shoreline, with constant distance from the centre when measured on the surface of a sphere. The dashed line shows the extent of the continental shelf. The boxes labelled Test 1 and Test 2 are regions where the island is located in the tests presented in the following figures. The small circle near the centre shows roughly the extent of the hump of water used as initial data. Figure 11.1(b) shows a cross-section of the bathymetry through the island. Figure 11.2 shows zoomed views of the two boxes labelled Test 1 and Test 2 in Figure 11.1(a), with contours of the bathymetry. The solid contour lines are shoreline (B = 0) and the dashed contour lines are at elevations

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B = −40, −80, −120, −160 m. Note that the continental shelf has a uniform depth of −100 m. In this figure we also indicate the locations of four gauges where the surface elevation as a function of time is recorded as the computation progresses. Specifying gauges is a standard feature in GeoClaw and at each time step the finest grid available near each gauge location is used to interpolate the surface elevation to the gauge location. Figure 11.4 shows results at these gauges for several different tests, described below. We solve using five levels of AMR with the same parameters as in Berger et al. (2010): a 40 × 40 level 1 grid over the full domain (1◦ on a side), and factor of 4 refinement in each subsequent level (a cumulative factor 256 refinement on the finest level). Levels 4 and 5 are used only near the island. We show results for three different values of the amplitude in (11.1): A0 = 0.005, 5.0 and 150.0. Figure 11.4 shows the surface elevation measured at the four gauges shown in Figure 11.2. For any fixed amplitude A0 , the gauge responses should be the same in Test 1 and Test 2, since the island and gauges are simply rotated together to a new position. This is illustrated in Figure 11.4, where the solid curve is from Test 1 and the dashed curve is from Test 2. The good agreement indicates that propagation is handled properly on the surface of the sphere and that the topography of the island and shore are well approximated on the grid, regardless of orientation. The tsunami propagation over the deep ocean is essentially linear for all of these amplitudes. For the two smaller-amplitude cases the propagation on the shelf and around the island also shows nearly linear dependence on the data, especially for gauges 1 and 4, where the undisturbed ocean depth is 100 m. Gauges 2 and 3 are at locations where the depth is about 11 m, and some nonlinear effects can be observed. Compare the gauge plots in Figure 11.4(a,b) (for A0 = 0.005) with those in Figure 11.4(c,d) (for A0 = 5), and note that the vertical axis has been rescaled by a factor of 100. Also note that for the A0 = 0.005 case, the maximum amplitude seen at any of the gauges is below 1 cm. Before hitting the continental shelf, this tsunami had even smaller amplitude. This test illustrates that it is possible to accurately capture even very small-amplitude tsunamis. Figure 11.4(e,f) shows a much larger-amplitude tsunami, using the same initial data (11.1) but with A0 = 150. Propagation across the ocean is still essentially linear and so the arrival time is nearly the same, but the wave amplitude is large enough that steepening occurs on the shelf. The nonlinear effects are evident in these gauge plots, which are no longer simply scaled up linearly from the first two cases. In Figure 11.5 we show surface plots of the run-up at different times, comparing the two computations with the island in different locations. Figure 11.3 shows contour plots for the first of these times, t = 10000 s. Four gauge locations are shown in Figure 11.2 and the surface elevation at these gauges is shown in Figure 11.4. The agreement is quite good.

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Figure 11.4. Comparison of gauge output from Test 1 and Test 2, showing the surface elevation as a function of time (in seconds) for the gauges shown in Figure 11.2. In each figure, the solid curve is from Test 1 and the dashed curve is from Test 2. (a,b) Very small-amplitude tsunami, with A0 = 0.005 in (11.1). (c,d) A0 = 0.5. (e,f) A0 = 150, giving the large-amplitude tsunami seen in Figure 11.5. Note the difference in vertical scale in each set of figures (metres in each case).

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R. J. LeVeque, D. L. George and M. J. Berger Time = 10000

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Figure 11.5. Surface plots of the tsunami interacting with the island for Test 1 and Test 2, in the case A0 = 150 in (11.1). At time t = 10000 seconds the tsunami is just wrapping around the island, as shown also in Figure 11.3. At time 13500 the reflected wave from the mainland has run up the lee side of the island and is flowing back down. For an animation of these results, see the webpage for this paper (www13).

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12. The 27 February 2010 Chile tsunami As an illustration of the use of adaptive mesh refinement to explore realworld tsunamis, we will show some results obtained using GeoClaw for tsunamis similar to the one generated by the 27 February 2010 earthquake near Maule, Chile. Some computations of the 2004 Indian Ocean tsunami calculated with this software can be found in LeVeque and George (2004) and George and LeVeque (2006). We will use relatively coarse grids so that the computations can be easily repeated by the interested reader using the source code available on the webpage for this paper (www13). Several source mechanisms have been proposed for this event. Here we use a simple source computed by applying the Okada model to the fault parameters given by USGS earthquake data (www12). We use bathymetry at a resolution of 10 minutes (1/6 degree) in latitude and longitude, obtained from the ETOPO2 data set at the National Geophysical Data Center (NGDC) GEODAS Grid Translator (www9). Figures 12.1 and 12.2 show comparisons of results obtained with two simulations using different AMR strategies. In both cases, the level 1 grid has a 2◦ resolution in each direction (roughly 222 km in latitude), and grid cells of this grid can be seen on the South American continent. The finest grid level also has the same resolution in both cases, a factor of 20 smaller in each direction. For the calculation in Figures 12.1(a) and 12.2(a), only two AMR levels are used, with a refinement factor of rx1 = ry1 = 20 in each direction and flagging all cells where the water is disturbed from sea level. The level 2 grid grows as the tsunami propagates until it covers the full domain at 5 hours. (It is split into 4 level 2 grids at this point because of restrictions in the software on the size of any single grid to reduce the memory overhead.) Results of this calculation agree exactly with what would be obtained with a single grid with a uniform grid size of 0.1◦ . For the calculation in Figures 12.1(b) and 12.2(b), three AMR levels are used and cells are only flagged for refinement if the deviation from sea level is greater than 0.1 m. Moreover, after t = 3 hours, we flag points only if the latitude is greater than −25◦ S. The refinement factors are rx1 = ry1 = 4 from level 1 to level 2 and rx2 = ry2 = 5 from level 2 to level 3, so the finest grid has the same resolution as in (a). Ideally the results in the regions covered by the finest grid (the smallest rectangles in the figure) would be identical to those in (a). Visually they agree quite well. In particular, it should be noted that there is no apparent difficulty with spurious wave generation at the interfaces between patches at different refinement levels. Also note that in both calculations the wave leaves the computational domain cleanly with little spurious reflection at the open boundaries.

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Figure 12.1. The 27 February 2010 tsunami as computed using GeoClaw. The rectangles show the edges of refinement patches. (a) Two levels of AMR, with refinement everywhere around and behind the wave. (b) Three levels of AMR, with the same finest grid resolution but refinement in limited regions. The colour scale is the same as in Figure 1.1, ranging from −0.2 to 0.2 m.

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Figure 12.2. Continuation of Figure 12.1. Note the reflection from the Galapagos on the equator at 6 hours, and that elsewhere the wave leaves the computational domain cleanly. For an animation of these results, see the webpage for this paper (www13).

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For a more quantitative comparison of these results, Figure 12.3(b) shows a comparison of the computed surface elevation as a function of time at the location of DART buoy 32412. The solid line is from the level 2 (uniform fine grid) computation, while the dashed line is from the level 3 computation in Figures 12.1(b) and 12.2(b). The agreement is very good up to about 5 hours, after which the DART buoy is in a region that is not refined. Note from Figure 12.2 that at 8 hours there is still wave action visible on the coast of Peru to the northeast of the DART buoy. This is a region where the continental shelf is very wide and shallow, and waves become trapped in this region due to the slow propagation speed and reflections for the steep shelf slope. In Figure 12.4 we show a zoomed view of these regions from another computation in which a fourth level of AMR has been added, refining by an additional factor of 4. (We have also used finer bathymetry

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Figure 12.4. Tsunami interaction with the broad continental shelf off the coast of Peru. A fourth level of refinement has been added beyond the levels shown on Figure 12.2. Grid patch edges are not shown and grid lines are shown only for levels 1–3 on land. For an animation of these results, see the webpage for this paper (www13).

in this region, 4-minute data from (www9).) In these figures one can clearly see the fast and broad tsunami sweeping northwards at times 4 and 4.5 hours, and the manner in which this wave is refracted at the continental slope to become a narrower wave of larger amplitude moving towards the coast. These waves continue to propagate up and down the coast in this region for more than 24 hours after the tsunami has passed. 12.1. Inundation of Hilo Some bays have the dubious distinction of being tsunami magnets due to local bathymetry that tends to focus and amplify tsunamis. Notable examples on the US coastline are Crescent City, CA and Hilo, HI. In this section

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we use adaptive refinement to track the 27 February 2010 tsunami originating in Chile across the Pacific Ocean and then add several additional levels of refinement to resolve the region near Hilo Harbor. The two simulations described in this section had the same computational parameters such as refinement criteria, but with two different source mechanisms. The computational domain for these simulations spanned 120 degrees of latitude and 100 degrees of longitude, from the source region near Maule, Chile in the southeastern corner of the domain to the Hawaiian Islands in the northwestern corner. We used five levels of grids, using refinement ratios of 8, 4, 16 and 32. The coarsest level consisted of a 60 × 50 grid with 2◦ grid cells, yielding a very coarse grid over the ocean at rest. Transoceanic propagating waves were resolved in level 2 grids. Level 3 grids were allowed only near the Hawaiian island chain, with the refinement ratio chosen to roughly match the resolution of bathymetric data used (1 minute). Level 4 grids were allowed surrounding the Big Island of Hawaii, where Hilo is located. In this region we used 3-arcsecond bathymetry from (www8). Finally, level 5 grids were allowed only near Hilo Harbor, where 1/3-arcsecond data from the same source were used. Figure 12.5 shows the the domain of the simulation at four times, as the waves propagate across the Pacific. The outlines of level 3–5 grids can be seen in the final frame, and appear just as the waves reach Hawaii. Figure 12.6 shows a close-up of the grids surrounding Hilo. The finest fifth-level grids were sufficient to resolve the small-scale features necessary to model inundation, such as shoreline structures and a sea wall in Hilo Harbor. The computational grids on the fifth level had grid cells with roughly 10 m grid resolution. Note that the finest grids are refined by a factor 214 = 16 384 in each coordinate direction relative to level 1 grids. Each grid cell on the coarsest level would contain roughly 286 million grid cells if a uniform fine grid were used, far more than the total number of grid cells used at any one time with adaptive refinement. The two source mechanisms used were identical except for the magnitude of slip. We first modelled the actual 27 February 2010 event described in Section 12, by applying the Okada model to fault parameters given by (www12). USGS earthquake data (www12). For the second source mechanism we used the same fault parameters, but increased each subfault dislocation by a factor of 10. The motivation for this second source model is twofold. First, the actual 27 February 2010 tsunami produced little or no inundation in Hilo, so to demonstrate inundation modelling with GeoClaw we created a much larger hypothetical tsunami. Second, amplifying the dislocation while keeping all other parameters fixed allows us to examine linearity in the off-shore region versus nonlinearity for tsunami inundation. The waves produced in Hilo Harbor by the larger hypothetical source are comparable to those arising from more tsunamigenic events in the past (for

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Figure 12.5. The 27 February 2010 tsunami propagates toward the Hawaiian Islands on level 1 and 2 grids. Higher-level grids appear around Hawaii as the waves arrive.

(a)

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Figure 12.6. Close-up of the higher-level grids appearing as the tsunami reaches the Hawaiian Islands. (a) All levels of grids, with grid lines omitted from levels 3–5. (b) Close-up of the Island of Hawaii, where an outline of level 5 grids surrounding the city of Hilo can be seen on the east coast of Hawaii. Grid lines are omitted from levels 4–5.

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Figure 12.7. Computed inundation maps of Hilo, HI, based on two source mechanisms. (a) Inundation using an actual USGS fault model for the February 2010 event. (b) Hypothetical source mechanism, with the dislocations amplified by a factor of 10, in order to show inundation.

example, the 1960 Chile quake and the 1964 Alaska event). Figure 12.7 shows the two different inundation patterns from the original and amplified source. The location of simulated water level gauges are indicated in the figures. The output of the gauges is shown in Figure 12.8. Gauge 1 is in Hilo Harbor and shows the incoming waves. Note that the waves in the harbour are still close to linear with respect to the source dislocation, several thousand kilometres away. However, in the nearer-shore and onshore regions inundation becomes strongly nonlinear, and determining the patterns of inundation cannot be done by a linear scaling of solutions from different source models.

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(a)

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Figure 12.8. Time series of the surface elevation at three different simulated gauge locations near Hilo, HI. The locations of the gauges are shown in Figure 12.7. The results for the ‘amplified fault model’ were obtained by increasing the slip displacement at the source by a factor of 10. The solid curve in (a) shows the solution for the amplified source mechanism multiplied by 0.10. This lies nearly on top of the curve from the original source model, indicating that the response is nearly linear in the harbour. The inland gauges in (b,c) exhibit nonlinear behaviour: they remain dry for the smaller tsunami but show large inundation waves for the amplified fault model.

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13. Final remarks In this paper we have focused on tsunami modelling using Godunov-type finite volume methods, and the issues that arise when simulating the propagation of a very small-amplitude wave across the ocean, followed by modelling the nonlinear run-up and inundation of specific regions remote from the initial event. The scale of these problems makes the use of adaptive mesh refinement crucial. The same techniques are applicable to a variety of other geophysical flow problems such as those listed in Section 1. The GeoClaw software has recently been applied to other problems, such as modelling the failure of the Malpasset dam in 1959 (George 2010). This has often been used as a test problem for validation of codes modelling dam breaks. Storm surge associated with tropical storms is another application where shallow water equations are often used, and some preliminary results on this topic have been obtained by Mandli (2010) with the GeoClaw code. For many problems the shallow water equations are insufficient and other depth-averaged models must be developed. Even in the context of tsunami modelling, there are some situations where it may be important to include dispersive terms (Gonz´ alez and Kulikov 1993, Saito, Matsuzawa, Obara and Baba 2010), particularly for shorter-wavelength waves arising from submarine landslides, as discussed for example by Watts et al. (2003) or Lynett and Liu (2002). A variety of dispersive terms might be added; see Bona, Chen and Saut (2002) for a recent survey of Boussinesq models and other alternatives. Dispersive terms generally arise in the form of third-order derivatives in the equation, generally requiring implicit algorithms in order to obtain stable results with reasonable time steps. This adds significant complication in the context of AMR and this option is not yet available in GeoClaw. An alternative is to use a dispersive numerical method, tuned to mimic the true dispersion; see for example Burwell, Tolkova and Chawla (2007). This seems problematic in the context of AMR. Other flows require the use of more complex rheologies than water, for example landslides, debris flows, or lava flows. We are currently extending GeoClaw to handle debris flows using a variant of the models of Denlinger and Iverson (2004a, 2004b). For some related work with similar finite volume algorithms, see Pelanti, Bouchut and Mangeney (2008, 2011) and Costa and Macedonio (2005). A variety of numerical approaches have been used for modelling tsunamis and other depth-averaged flows in recent years. There is a large literature on topics such as well-balanced methods and wetting-and-drying algorithms, not only for finite volume methods but also for finite difference, finite element, discontinuous Galerkin, and other methodologies, on both structured and unstructured grids. We have not attempted a full literature survey in this paper, either on numerical methods or tsunami science, but hope that

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the references provided give the reader some entry into this field. There are a host of challenging problems remaining in the quest to better understand and protect against these hazards.

Acknowledgements The GeoClaw software is an extension and generalization of the TsunamiClaw software originally developed from Clawpack (www3) as part of the PhD thesis of one of the authors (George 2006). Our work on tsunami modelling has benefited greatly from the generous help of many researchers in this community, in particular Harry Yeh, who first encouraged us to work on this problem, members of the tsunami sedimentology group at the University of Washington, and members of the NOAA Center for Tsunami Research in Seattle. Several people provided valuable feedback on this paper, including B. Atwater, J. Bourgeois, G. Gelfenbaum, F. Gonz´ alez, B. MacInnes, K. Mandli, and M. Martin. Thanks to Jody Bourgeois for providing Figures 2.1 and 2.2. This work was supported in part by NSF grants CMS-0245206, DMS0106511, DMS-0609661 and DMS-0914942, ONR grant N00014-09-1-0649, DOE grant DE-FG02-88ER25053, AFOSR grant FA9550-06-1-0203, the Founders Term Professorship in Applied Mathematics at the University of Washington, and a USGS Mendenhall Postdoctoral Fellowship.

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Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs∗ Christoph Schwab and Claude Jeffrey Gittelson Seminar for Applied Mathematics, ETH Z¨ urich, R¨ amistrasse 101, CH-8092 Z¨ urich, Switzerland E-mail: [email protected], [email protected]

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions’ laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N -term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

∗

Work partially supported by the European Research Council under grant number ERC AdG 247277-STAHDPDE and by the Swiss National Science Foundation under grant number SNF 200021-120290/1.

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C. Schwab and C. J. Gittelson

CONTENTS Introduction 1 Operator equations with stochastic data 2 Stochastic Galerkin discretization 3 Optimal convergence rates 4 Sparse tensor discretizations Appendix A Review of probability B Review of Hilbert spaces C Review of Gaussian measures on Hilbert spaces References

292 296 332 367 394 419 428 439 461

Introduction The numerical solution of partial differential equation models in science and engineering has today reached a certain maturity, after several decades of progress in numerical analysis, mathematical modelling and scientific computing. While there certainly remain numerous mathematical and algorithmic challenges, for many ‘routine’ problems of engineering interest, today numerical solution methods exist which are mathematically understood and ‘operational’ in the sense that a number of implementations exist, both academic and commercial, which realize, in the best case, algorithms of provably optimal complexity in a wide range of applications. As a rule, the numerical analysis and the numerical solution methods behind such algorithms suppose that a model of the system of interest is described by a well-posed (in the sense of Hadamard) partial differential equation (PDE), and that the PDE is to be solved numerically to prescribed accuracy for one given set of input data. With the availability of highly accurate numerical solution algorithms for a PDE of interest and one prescribed set of exact input data (such as source terms, constitutive laws and material parameters) there has been increasing awareness of the limited significance of such single, highly accurate ‘forward’ solves. Assuming, as we will throughout this article, that the PDE model of the physical system of interest is correct, this trend is due to two reasons: randomness and uncertainty of input data and the need for efficient prediction of system responses on high-dimensional parameter spaces. First, the assumption of availability of exact input data is not realistic: often, the simulation’s input parameters are obtained from measurements or from sampling a large, but finite number of specimens or system snapshots which are incomplete or stochastic. This is of increasing importance in classical engineering disciplines, but even more so in emerging models in

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the life sciences and social sciences. Rather than producing efficiently accurate answers for single instances of exact input data, increasingly the goal of computation in numerical simulations is to efficiently process statistical information on uncertain input data for the PDE of interest. While mathematical formulations of PDEs with random inputs have been developed with an eye towards uncorrelated, or white noise inputs (see, e.g., Holden, Oksendal, Uboe and Zhang (1996), Da Prato and Zabczyk (1992), Da Prato (2006), Lototsky and Rozovskii (2006), Prévôt and Röckner (2007), Dalang, Khoshnevisan, Mueller, Nualart and Xiao (2009) and the references therein), PDEs with random inputs in numerical simulation in science and engineering are of interest in particular in the case of so-called correlated inputs (or ‘coloured noise’). Second, in the context of optimization, or of risk and sensitivity analysis for complex systems with random inputs, the interest is in computing the systems’ responses efficiently given dependence on several, possibly countably many parameters, thereby leading to the challenge of numerical simulation of deterministic PDEs on high-dimensional parameter spaces. Often, the only feasible approach in numerical simulation towards these two problems is to solve the forward problem for many instances, or samples, of the PDE’s input parameters; for random inputs, this amounts to Monte Carlo-type sampling of the noisy inputs, and for parametric PDEs, responses of the system are interpolated from forward solves at judiciously chosen combinations of input parameters. With the cost of one ‘sample’ being the numerical solution of a PDE, it is immediate that, in particular for transient problems in three spatial dimensions with solutions that exhibit multiple spatial and temporal length scales, the computational cost of uniformly sampling the PDE solution on the parameter space (resp. the probability space) is prohibitive. Responding to this by massive parallelism may alleviate this problem, but ultimately, the low convergence rate 1/2 of Monte Carlo (MC) sampling, respectively the so-called ‘curse of dimensionality’ of standard interpolation schemes in high-dimensional parameter spaces, requires advances at the mathematical core of the numerical PDE solution methods: the development of novel mathematical formulations of PDEs with random inputs, the study of the regularity of their solutions is of interest, both with respect to the physical variables and with respect to parameters, and the development of novel discretizations and solution methods of these formulations. Importantly, the parameters may take values in possibly infinite-dimensional parameter spaces: for example, in connection with Karhunen–Loève expansions of spatially inhomogeneous and correlated inputs. The present article surveys recent contributions to the above questions. Our focus is on linear PDEs with random inputs; we present various formulations, new results on the regularity of their solutions and, based on these

294


regularity results, we design, formulate and analyse discretization schemes which allow one to ‘sweep’ the entire, possibly infinite-dimensional input parameter space approximately in a single computation. We also establish, for the algorithms proposed here, bounds on their efficiency (understood as accuracy versus the number of degrees of freedom) that do not deteriorate with respect to increasing dimension of the computational parameter domain, i.e., that are free from the curse of dimensionality. The algorithms proposed here are variants and refinements of the recently proposed stochastic Galerkin and stochastic collocation discretizations (see, e.g., Xiu (2009) and Matthies and Keese (2005) and the references therein for an account of these developments). We exhibit assumptions on the inputs’ correlations which ensure an efficiency of these algorithms which is superior to that of MC sampling. One insight that emerges from the numerical analysis of recently proposed methods is that the numerical resolution in physical space need not be high uniformly on the entire parameter space. The use of ‘polynomial chaos’ type spectral representations (and their generalizations) of the laws of input and output random fields allows a theory of regularity of the random solutions and, based on this, the optimization of numerical methods for their resolution. Here, we have in mind discretizations in physical space and time as well as in stochastic or parameter space, aiming at achieving a prespecified accuracy with minimal computational work. From this broad view, the recently proposed multi-level Monte Carlo methods can also be interpreted as sparse tensor discretizations. Accordingly, we present in this article an error analysis of single- and multi-level MC methods for elliptic problems with random inputs. As this article’s title suggests, the notion of sparse tensor products of operators and hierarchical sequences of finite-dimensional subspaces pervades our view of numerical analysis of high-dimensional problems. Sparsity in connection with tensorization has become significant in several areas of scientific computing in recent years: in approximation theory as hyperbolic cross approximations (see, e.g., Temlyakov (1993)) and, in finite element and finite difference discretizations, the so-called sparse grids (see Bungartz and Griebel (2004) and the references therein) are particular instances of this concept. We note in passing that the range of applicability of sparse tensor discretizations extends well beyond stochastic and parametric problems (see, e.g., Schwab (2002), Hoang and Schwab (2004/05) and Schwab and Stevenson (2008) for applications to multiscale problems). On the level of numerical linear algebra, the currently emerging hierarchical low-rank matrix formats, which were inspired by developments in computational chemistry, are closely related to some of the techniques developed here. The present article extends these concepts in several directions. First, on the level of mathematical formulation of PDEs with random inputs: we present deterministic tensorized operator equations for two- and k-point


295

correlation functions of the the random system responses. Such equations also arise in the context of moment closures of kinetic models in atomisticto-continuum transitions. Discretizations for their efficient, deterministic numerical solution may therefore be of interest in their own right. For the spectral discretizations, we review the polynomial chaos representation of random fields and the Wiener–Itô chaos decomposition of probability spaces and of random fields into tensorized Hermite polynomials of a countable number of Gaussians. The spectral representation of random outputs of PDEs allows for a regularity theory of the laws of random fields which goes substantially beyond the mere existence of moments. According to the particular application, in this article sparsity in tensor discretizations appears in roughly three forms. First, we use sparse tensor products of multi-level finite element spaces in the physical domain D ⊂ Rd to build efficient schemes for the Galerkin approximation of tensorized equations for k-point correlation functions. Second, we consider heterogeneous sparse tensor product discretizations of multi-level finite element, finite volume and finite difference discretizations in the physical domain with hierarchical polynomial chaos bases in the probability space. As we will show, the use of multi-level discretizations in physical space actually leads to substantial efficiency gains in MC methods; nevertheless, the resulting multi-level MC methods are of comparable efficiency as sparse tensor discretizations for random outputs with finite second moments. However, as soon as the outputs have additional summability properties (and the examples presented here suggest that this is so in many cases), adaptive sparse tensor discretizations outperform MLMC methods. The outline of the article is as follows. We first derive tensorized operator equations for deterministic, linear equations with random data. We establish the well-posedness of these tensorized operator equations, and introduce sparse tensor Galerkin discretizations based on multi-level, wavelettype finite element spaces in the physical domain. We prove, in particular, stability of sparse tensor discretizations in the case of indefinite operators such as those arising in acoustic or electromagnetic scattering. We also give an error analysis of MC discretizations which indicates the dependence of its convergence rate on the degree of summability of the random solution. Section 2 is devoted to stochastic Galerkin formulations of PDEs with random coefficients. Using polynomial chaos representations of the random inputs, for example in a Karhunen–Loève expansion, we give a reformulation of the random PDEs of interest as deterministic PDEs which are posed on infinite-dimensional parameter spaces. While the numerical solution of these PDEs with standard tools from numerical analysis is foiled by the curse of dimensionality (the raison d’être for the use of sampling methods on the stochastic formulation), we review recent regularity results for these problems which indicate that sparse, adaptive tensorization of discretizations

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in probability and physical space can indeed produce solutions whose accuracy, as a function of work, is independent of the dimension of the parameter space. We cover both affine dependence, as is typical in Karhunen–Loève representations of the random inputs, as well as log-normal dependence in inputs. We focus on Gaussian and on uniform measures, where ‘polynomial chaos’ representations use Hermite and Legendre polynomials, respectively (other probability measures give rise to other polynomial systems: see, e.g., Schoutens (2000) and Xiu and Karniadakis (2002b)). Section 3 addresses the regularity of the random solutions in these polynomial chaos representations by an analysis of the associated parametric, deterministic PDE for their laws. The analysis allows us to deduce best N -term convergence rates of polynomial chaos semidiscretizations of the random solutions’ laws. Section 4 combines the results from the preceding sections with space and time discretizations in the physical domain. The error analysis of fully discrete algorithms reveals that it is crucial for efficiency that the level of spatial and temporal resolution be allowed to depend on the stochastic mode being discretized. Our analysis shows that, in fact, a highly non-uniform level of resolution in physical space should be adopted in order to achieve algorithms that scale favourably with respect to the dimension of the space of stochastic parameters. As this article and the subject matter draw on tools from numerical analysis, from functional analysis and from probability theory, we provide some background reference material on the latter two items in the Appendix. This is done in order to fix the notation used in the main body of the text, and to serve as a reference for readers with a numerical analysis background. Naturally, the selection of the background material is biased towards the subject matter of the main text. It does not claim to be a reference on these subjects. For a more thorough introduction to tools from probability and stochastic analysis we refer the reader to Bauer (1996), Da Prato (2006), Da Prato and Zabczyk (1992), Prévôt and Röckner (2007) and the references therein.

1. Sparse tensor FEM for operator equations with stochastic data For the variational setting of linear operator equations with deterministic, boundedly invertible operators, we assume that X, Y are separable Hilbert spaces over R with duals X and Y , respectively, and A ∈ L(X, Y ) a linear, boundedly invertible deterministic operator. We denote its associated bilinear form by a(u, v) := Y Au, vX : X × Y → R.

(1.1)


297

Here, and throughout, for w ∈ Y and v ∈ X the bilinear form Y w, vX denotes the Y ×X duality pairing. As is well known (see, e.g., Theorem C.20) the operator A from X onto Y is boundedly invertible if and only if a(·, ·) satisfies the following conditions. (i) a(·, ·) is continuous: there exists C1 < ∞ such that ∀w ∈ X, v ∈ Y :

|a(w, v)| ≤ C1 w X v Y .

(1.2)

(ii) a(·, ·) is coercive: there exists C2 > 0 such that inf

sup

0=w∈X 0=v∈Y

a(w, v) ≥ C2 > 0.

w X v Y

(1.3)

sup a(w, v) > 0.

(1.4)

(iii) a(·, ·) is injective: ∀0 = v ∈ Y :

0=w∈X

If (1.2)–(1.4) hold, then for every f ∈ Y the linear operator equation u∈X:

a(u, v) = Y f, vX

∀v ∈ Y

(1.5)

admits a unique solution u ∈ X such that

u X ≤ C2−1 f Y .

(1.6)

We consider equation (1.5) with stochastic data: to this end, let (Ω, F, P) be a probability space and let f : Ω → Y be a random field, i.e., a measurable map from (Ω, F, P) into Y which is Gaussian (see Appendix C for the definition of Gaussian random fields). Analogous to the characterization of Gaussian random variables by their mean and their (co)variance, a Gaussian random field f ∈ L2 (Ω, F, P; Y ) is characterized by its mean af ∈ Y and its covariance operator Qf ∈ L+ 1 (Y ). We use the following linear operator equation with Gaussian data: given f ∈ L2 (Ω, F, P; Y ), find u ∈ L2 (Ω, F, P; X) such that Au = f

in L2 (Ω, F, P; Y )

(1.7)

admits a unique solution u ∈ L2 (Ω, F, P; X) if and only if A satisfies (1.2)– (1.4). By Theorem C.31, the unique random solution u ∈ L2 (Ω, F, P; X) of (1.7) is Gaussian with associated Gaussian measure Nau ,Qu on X which, in turn, is characterized by the solution’s mean, au = mean(u) = A−1 af ,

(1.8)

and the solution’s covariance operator Qu ∈ L+ 1 (X), which satisfies the (deterministic) equation AQu A∗ = Qf

in L(Y , Y ).

(1.9)

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In the Gaussian case, therefore, solving the stochastic problem (1.7) can be reduced to solving the two deterministic problems (1.8) and (1.9). Whereas the mean-field problem (1.8) is one instance of the operator equation (1.7), the covariance equation (1.8) is an equation for the operator Qu ∈ L+ 1 (X). As we show in Theorem C.31, this operator is characterized by the socalled covariance kernel Cu , which satisfies, in terms of the corresponding covariance kernel Cf of the data, the covariance equation (see (C.50)) (A ⊗ A)Cu = Cf ,

(1.10)

which is understood to hold in the sense of (Y ⊗ Y ) Y ⊗ Y . One approach to the numerical treatment of operator equations Au = f , where the data f are random fields, i.e., measurable maps from a probability space (Ω, F, P) into the set Y of admissible data for the operator A, is via tensorized equations such as (1.10) for their statistical moments. The simplest approach to the numerical solution of the linear operator equation Au = f with random input f is Monte Carlo (MC) simulation, i.e., generating a large number M of i.i.d. data samples fj and solving, possibly in parallel, for the corresponding solution ensemble {uj = A−1 fj ; j = 1, . . . , M }. Statistical moments and probabilities of the random solution u are then estimated from {uj }. As we will prove, convergence of the MC method as the number M of samples increases is ensured (for suitable sampling) by the central limit theorem. We shall see that the MC method allows in general only the convergence rate O(M −1/2 ). If statistical moments, i.e., mean-field and higher-order moments of the random solution u, are of interest, one can exploit the linearity of the equation Au = f to derive a deterministic equation for the kth moment of the random solution, similar to the second-moment equation (1.10); this derivation is done in Section 1.1. For the Laplace equation with stochastic data, this approach is due to I. Babuˇska (1961). We then address the numerical computation of the moments of the solution by either Monte Carlo or by direct, deterministic finite element computation. If the physical problem is posed in a domain D ⊂ Rd , the kth moment of the random solution is defined in the domain Dk ⊂ Rkd ; standard finite element (FE) approximations will therefore be inadequate for the efficient numerical approximation of the kth moments of the random solution. The efficient deterministic equation and its FE approximation were investigated in Schwab and Todor (2003a, 2003b) in the case where A is an elliptic partial differential operator. It was shown that the kth moment of the solution could be computed in a complexity comparable to that of an FE solution for the mean-field problem by the use of sparse tensor products of standard FE spaces for which a hierarchical basis is available. The use of sparse tensor product approximations is a well-known device in high-dimensional numerical integration going back to Smolyak (1963), in


299

multivariate approximation (Temlyakov 1993), and in complexity theory; see Wasilkowski and Wo´zniakowski (1995) and the references therein. In the present section, we address the case when A is a non-local operator, such as a strongly elliptic pseudodifferential operator, as arises in the boundary reduction of boundary value problems for strongly elliptic partial differential equations. In this case, efficient numerical solution methods require, in addition to Galerkin discretizations of the operator equation, some form of matrix compression (such as the fast multipole method or wavelet-based matrix compression) which introduces additional errors into the Galerkin solution that will also affect the accuracy of second and higher moments. We briefly present the numerical analysis of the impact of matrix compressions on the efficient computation of second and higher moments of the random solution. Therefore, the present section will also apply to strongly elliptic boundary integral equations obtained by reduction to the boundary manifold D = ∂D of elliptic boundary value problems in a bounded domain D ⊂ Rd+1 , as is frequently done in acoustic and electromagnetic scattering. For such problems with stochastic data, the boundary integral formulation leads to an operator equation Au = f , where A is an integral operator or, more generally, a pseudodifferential operator acting on function spaces on ∂D. The linearity of the operator equation allows, without any closure hypothesis, formulation of a deterministic tensor equation for the k-point correlation function of the random solution u = A−1 f . We show that, as in the case of differential operators, sparse tensor products of standard FE spaces allow deterministic approximation of the kth moment of the random solution u with relatively few degrees of freedom. To achieve computational complexity which scales log-linearly in the number of degrees of freedom in a Galerkin discretization of the mean-field problem, however, the Galerkin matrix for the operator A must be compressed. Accordingly, one purpose of this section is the design and numerical analysis of deterministic and stochastic solution algorithms to obtain the kth moment of the random solution of possibly non-local operator equations with random data in log-linear complexity in the number N of degrees of freedom for the mean-field problem. We illustrate the sparse tensor product Galerkin methods for the numerical solution of Dirichlet and Neumann problems for the Laplace or Helmholtz equation with stochastic data. Using a wavelet Galerkin finite element discretization allows straightforward construction of sparse tensor products of the trial spaces, and yields well-conditioned, sparse representations of stiffness matrices for the operator A as well as for its k-fold tensor product, which is the operator arising in the kth-moment problem. We analyse the impact of the operator compression on the accuracy of functionals of the Galerkin solution, such as far-field evaluations of the random potential in a point. For example, means and variances of the potential

300


in a point can be computed with accuracy O(N −p ) for any fixed order p, for random boundary data with known second moments in O(N ) complexity, where N denotes the number of degrees of freedom on the boundary. The outline of this section is as follows. In Section 1.1, we describe the operator equations considered here and derive the deterministic problems for the higher moments, generalizing Schwab and Todor (2003b). We establish the Fredholm property for the tensor product operator and regularity estimates for the statistical moments in anisotropic Sobolev spaces with mixed highest derivative. Section 1.2 addresses the numerical solution of the moment equations, in particular the impact of various matrix compressions on the accuracy of the approximated moments, the preconditioning of the product operator and the solution algorithm. In Section 1.4, we discuss the implementation of the sparse Galerkin and sparse MC methods and estimate their asymptotic complexity. Section 1.5 contains some examples from finite and boundary element methods.

1.1. Operator equations with stochastic data Linear operator equations We specialize the general setting (1.1) to the case X = Y = V , and consider the operator equation Au = f,

(1.11)

where A is a bounded linear operator from the separable Hilbert space V into its dual V . The operator A is a differential or pseudodifferential operator of order on a bounded d-dimensional manifold D, which may be closed or have ˜ s (D) := H s (D) a boundary. Here, for a closed manifold and for s ≥ 0, H denotes the usual Sobolev space. For s < 0, we define the spaces H s (D) ˜ s (D) by duality. For a manifold D with boundary we assume that and H ˜ and define this manifold can be extended to a closed manifold D, ˜ u| ˜ = 0} ˜ s (D) := { u| ; u ∈ H s (D), H D D\D ˜ := Rd . with the induced norm. If D is a bounded domain in Rd we use D /2 (D). In the case when A is a second-order We now assume that V = H differential operator, this means that we have Dirichlet boundary conditions (other boundary conditions can be treated in an analogous way). The manifold D may be smooth, but we also consider the case when D is a polyhedron in Rd , or the boundary of a polyhedron in Rd+1 , or part of the boundary of a polyhedron. For the deterministic operator A in (1.11), we assume strong ellipticity in the sense that there exists α > 0 and a compact operator T : V → V such


that the G˚ arding inequality ∀v ∈ V :

(A + T ) v, v ≥ α v 2V

301

(1.12)

holds. For the deterministic algorithm in Section 1.4 we need the slightly stronger assumption that T is smoothing with respect to a scale of smoothness spaces (see (1.63) below). Here and in what follows, ·, · denotes the V × V duality pairing. We assume also that A is injective, i.e., that ker A = {0},

(1.13)

,

which implies that for every f ∈ V (1.11) admits a unique solution u ∈ V and, moreover, that A−1 : V → V is continuous, i.e., there exists CA > 0 such that, for all f ∈ V ,

u V = A−1 f V ≤ CA f V .

(1.14)

Here CA = C2−1 with the constant C2 as in (1.3). We shall consider (1.11) in particular for data f , which are Gaussian random fields on the data space V . By the linearity of the operator equation (1.11), then the solution v ∈ V is a Gaussian random field as well. Throughout, we assume that V and V are separable Hilbert spaces. Random data A Gaussian random field f with values in a separable Hilbert space X is a mapping f : Ω → X which maps events E ∈ Σ to Borel sets in X, and such that the image measure f# P on X is Gaussian. In the following, we allow more general random fields. Of particular interest will be their summability properties. We say that a random field u : Ω → X is in the Bochner space 1 L (Ω; X) if ω → u(ω) X is measurable and integrable so that u L1 (Ω;X) := Ω u(ω) X P(dω) is finite. In particular, then the ‘ensemble average’ u(ω)P(dω) ∈ X Eu := Ω

exists as a Bochner integral of X-valued functions, and it satisfies

Eu X ≤ u L1 (Ω;X) .

(1.15)

Let k ≥ 1. We say that a random field u : Ω → X is in the Bochner space Lk (Ω; X) if u kLk (Ω;X) = Ω u(ω) kX P(dω) is finite. Note that ω →

u(ω) kX is measurable due to the measurability of u and the continuity of the norm · X on X. Also, Lk (Ω; X) ⊃ Ll (Ω; X) for k < l. Let B ∈ L(X, Y ) denote a continuous linear mapping from X to another separable Hilbert space Y . For a random field u ∈ Lk (Ω; X), this mapping defines a random variable v(ω) = Bu(ω) taking values in Y . Moreover, v ∈ Lk (Ω; Y ) and we have

Bu Lk (Ω;Y ) ≤ C u Lk (Ω;X) ,

(1.16)

302


where the constant C is given by C = B L(X,Y ) . In addition, we have u P(dω) = Bu P(dω). (1.17) B Ω

Ω

MC estimation of statistical moments We are interested in statistics of the random solution u of (1.11) and, in particular, in statistical moments. To define them, for a separable Hilbert space X and for any k ∈ N we define the k-fold tensor product space X (k) = X ⊗ · · · ⊗ X , k times

and equip it with the natural cross-norm · X (k) . The significance of a cross-norm was emphasized by Schatten. The cross-norm has the property that, for every u1 , . . . , uk ∈ X,

u1 ⊗ · · · ⊗ uk X (k) = u1 X · · · uk X

(1.18)

(see Light and Cheney (1985) and the references therein for more on crossnorms on tensor product spaces). The k-fold tensor products of, for example, X are denoted analogously by (X )(k) . For u ∈ Lk (Ω; X) we now consider the random field u(k) defined by u(ω) ⊗ · · · ⊗ u(ω). By Lemma C.9, u(k) = u ⊗ · · · ⊗ u ∈ L1 (Ω; X (k) ), and we have the isometry (k) (1.19)

u L1 (Ω;X (k) ) = u(ω) ⊗ · · · ⊗ u(ω) X (k) P(dω) Ω = u(ω) X · · · u(ω) X P(dω) = u kLk (Ω;X) . Ω

We define the moment Mk u as the expectation of u ⊗ · · · ⊗ u. Definition 1.1. For u ∈ Lk (Ω; X), for some integer k ≥ 1, the kth moment of u(ω) is defined by

k · · ⊗ u = u(ω) ⊗ · · · ⊗ u(ω) P(dω) ∈ X (k) . (1.20) M u = E u ⊗ · ω∈Ω k times

k times

Note that (1.15) and (1.18) give, with Jensen’s inequality and the convexity of the norm · V → R, the bound

Mk u X (k) = Eu(k) X (k) ≤ E u(k) X (k) = E u kX = u kLk (Ω;X) .

(1.21)

Deterministic equation for statistical moments We now consider the operator equation Au = f , where f ∈ Lk (Ω; V ) is given with k ≥ 1. Since A−1 : V → V is continuous, we obtain, using


303

(1.16), (1.14) and (1.21), that u ∈ Lk (Ω; V ), and that we have the a priori estimate k

Mk u V (k) ≤ u kLk (Ω;V ) ≤ CA

f kLk (Ω;V ) .

(1.22)

Remark 1.2. One example of a probability measure P on X is a Gaussian measure; we refer to, e.g., Vakhania, Tarieladze and Chobanyan (1987) and Ledoux and Talagrand (1991) for general probability measures over Banach spaces X and, in particular, to Bogachev (1998) and Janson (1997) for a general exposition of Gaussian measures on function spaces. Since A−1 : V → V in (1.11) is bijective, by (1.12) and (1.13), it induces := A−1 P on the space V of solutions to (1.11). If P is Gaussian a measure P # is Gaussian over V by Theorem C.18. over V and A in (1.11) is linear, then P We recall that a Gaussian measure is completely determined by its mean and covariance, and hence only Mk u for k = 1, 2 are of interest in this case. We now consider the tensor product operator A(k) = A⊗· · ·⊗A (k times). This operator maps V (k) to (V )(k) . For v ∈ V and g := Av, we obtain that A(k) v ⊗ · · · ⊗ v = g ⊗ · · · ⊗ g. Consider a random field u ∈ Lk (Ω; V ) and let f := Au ∈ Lk (Ω; V ). Then the tensor product u(k) = u ⊗ · · · ⊗ u (k times) belongs to the space L1 (Ω; V (k) ), and we obtain from (1.17) with B = A(k) that the k-point correlations u(k) satisfy P-a.s. the tensor equation A(k) u(k) = f (k) , where f (k) ∈ L1 (Ω; (V )(k) ). Now (1.17) implies for linear and deterministic operators A that the k-point correlation functions of the random solutions, i.e., the expectations Mk u = E[u(k) ], are solutions of the tensorized equations A(k) Mk u = Mk f.

(1.23)

In the case k = 1 this is just the equation AEu = Ef for the mean field. Note that this equation provides a way to compute the moments Mk u of the random solution in a deterministic fashion, for example by Galerkin discretization. As mentioned before, with the operator A acting on function spaces X, Y in the domain D ⊂ Rd , the tensor equation (1.23) will require discretization in Dk , the k-fold Cartesian product of D with itself. Using tensor products of, for instance, finite element spaces in D, we find for k > 1 a reduction of efficiency in terms of accuracy versus number of degrees of freedom due to the ‘curse of dimensionality’. This mandates sparse tensor product constructions. We will investigate the numerical approximation of the tensor equation (1.23) in Section 1.4. The direct approximation of (1.23) by, for example, Galerkin discretization is an alternative to the Monte Carlo approximation of the moments which will be considered in Section 1.3.

304


In the deterministic approach, explicit knowledge of all joint probability densities of f (i.e., the law of f ) with respect to the probability measure P is not required to determine the order-k statistics of the random solution u from order-k statistics of f . Remark 1.3. For nonlinear operator equations, associated systems of moment equations require a closure hypothesis, which must be additionally imposed and verified. For the linear operator equation (1.11), however, a closure hypothesis is not necessary, as (1.23) holds. For solvability of (1.23), we consider the tensor product operator A1 ⊗ A2 ⊗ · · · ⊗ Ak for operators Ai ∈ L(Vi , Vi ), i = 1, . . . , k. Proposition 1.4. For integer k > 1, let Vi , i = 1, . . . , k be Hilbert spaces with duals Vi , and let Ai ∈ L(Vi , Vi ) be injective and satisfy a G˚ arding inequality, i.e., there are compact Ti ∈ L(Vi , Vi ) and αi > 0 such that (Ai + Ti ) v, v ≥ αi v 2Vi , (1.24) ∀v ∈ Vi : where ·, · denotes the Vi × Vi duality pairing. Then the product operator A = A1 ⊗ A2 ⊗ · · · ⊗ Ak ∈ L(V, V ), where V = V1 ⊗ V2 ⊗ · · · ⊗ Vk and V = (V1 ⊗ V2 ⊗ · · · ⊗ Vk ) ∼ = V1 ⊗ V2 ⊗ · · · ⊗ Vk , is injective, and for every f ∈ V , the problem Au = f admits a unique solution u with

u V ≤ C f V . Proof. The injectivity and the G˚ arding inequality (1.24) imply the bounded invertibility of Ai for each i. This implies the bounded invertibility of A on V → V since we can write A = (A1 ⊗ I k−1 ) ◦ (I ⊗ A2 ⊗ I (k−2) ) ◦ · · · ◦ (I (k−1) ⊗ Ak ), where I (j) denotes the j-fold tensor product of the identity operator on the appropriate Vi . Note that each factor in the composition is invertible. To apply this result to (1.23), we require the special case (k) (k) (k) (k) A(k) := A · · · ⊗ A ∈ L(V , (V ) ) = L(V , (V ) ). ⊗A⊗

(1.25)

k times

Theorem 1.5. If A in (1.11) satisfies (1.12) and (1.13), then for every k > 1 the operator A(k) ∈ L(V (k) , (V )(k) ) is injective on V (k) , and for every f ∈ Lk (Ω; V ), the equation A(k) Z = Mk f has a unique solution Z ∈ V (k) .

(1.26)


305

This solution coincides with the kth moment Mk u of the random field in (1.20): Z = Mk u. Proof. By (1.21), the assumption f ∈ Lk (Ω; V ) ensures that Mk f ∈ (V )(k) . The unique solvability of (1.26) follows immediately from Proposition 1.4 and the assumptions (1.12) and (1.13). The identity Z = Mk u follows from (1.23) and the uniqueness of the solution of (1.26). Regularity The numerical analysis of approximation schemes for (1.26) will require a regularity theory for (1.26). To this end we introduce a smoothness scale (Ys )s≥0 for the data f with Y0 = V and Ys ⊂ Yt for s > t. We assume that we have a corresponding scale (Xs )s≥0 of ‘smoothness spaces’ for the solutions with X0 = V and Xs ⊂ Xt for s > t, so that A−1 : Ys → Xs is continuous. When D is a smooth closed manifold of dimension d embedded into Euclidean space Rd+1 , we choose Ys = H −/2+s (D) and Xs = H /2+s (D). The case of differential operators with smooth coefficients in a manifold D with smooth boundary is also covered within this framework by the choices /2 ∩ H /2+s (D). Note that in other cases (a Ys = H −/2+s (D) and Xs = H pseudodifferential operator on a manifold with boundary, or a differential operator on a domain with non-smooth boundary), the spaces Xs can be chosen as weighted Sobolev spaces which contain functions that are singular at the boundary. Theorem 1.6. Assume (1.12) and (1.13), and that there is an s∗ > 0 such that A−1 : Ys → Xs is continuous for 0 ≤ s ≤ s∗ . Then we have for all k ≥ 1 and for 0 ≤ s ≤ s∗ some constant C(k, s) such that

Mk u X (k) ≤ C Mk f Y (k) = C f kLk (Ω;Ys ) . s

Proof.

s

(1.27)

If (1.12) and (1.13) hold, then the operator A(k) is invertible, and Mk u = (A(k) )−1 Mk f = (A−1 )(k) Mk f.

Since

A−1 f Xs ≤ Cs f Ys ,

0 ≤ s ≤ s∗ ,

it follows that

Mk u X (k) = (A−1 )(k) Mk f X (k) ≤ Csk Mk f Y (k) , s

s

s

0 ≤ s ≤ s∗ .

1.2. Finite element discretization In order to obtain a finite-dimensional problem, we need to discretize in both Ω and D. For D we will use a nested family of finite element spaces V ⊂ V , = 0, 1, . . . .

306


Nested finite element spaces The Galerkin approximation of (1.11) is based on a sequence {V }∞ =0 of subspaces of V of dimension N = dim V < ∞ which are dense in V , i.e., V = ≥0 V , and nested, i.e., V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ V ⊂ V+1 ⊂ · · · ⊂ V.

(1.28)

We assume that for functions u in the smoothness spaces Xs with s ≥ 0 we have the asymptotic approximation rate −s/d

inf u − v V ≤ CN

v∈V

u Xs .

(1.29)

Finite elements with uniform mesh refinement We will now describe examples for the subspaces V which satisfy the assumptions of Section 1.2. We briefly sketch the construction of finite element spaces which are only continuous across element boundaries; see Braess (2007), Brenner and Scott (2002) and Ciarlet (1978) for presentations of the mathematical foundations of finite element methods. These elements are suitable for operators of order < 3. Throughout, we denote by Pp (K) the linear space of polynomials of total degree ≤ p on a set K. Let us first consider the case of a bounded polyhedron D ⊂ Rd . Let T0 be a regular partition of D into simplices K. Let {T }∞ =0 be the sequence of regular partitions of D obtained from T0 by uniform subdivision: for example, if d = 2, we bisect all edges of the triangulation T and obtain a new, regular partition of the domain D into possibly curved triangles which belong to finitely many congruency classes. We set V = S p (D, T ) = {u ∈ C 0 (D) ; u|K ∈ Pp (K) ∀K ∈ T } and let h = max {diam(K) ; K ∈ T }. Then N = dim V = O(h−d ) as /2 /2+s ˜ (D), standard finite element → ∞. With V = H (D) and Xs = H approximation results imply that (1.29) holds for s ∈ [0, p + 1 − /2], i.e., −s/d

inf u − v V ≤ CN

v∈V

u Xs .

For the case when D is the boundary D = ∂D of a polyhedron D ⊂ Rd+1 we define finite element spaces on D in the same way as above, but now in local coordinates on D, and obtain the same convergence rates (see, e.g., Sauter and Schwab (2010)): for a d-dimensional domain D ⊂ Rd with a smooth boundary we can first divide D into pieces DJ , which can be mapped to a simplex S by smooth mappings ΦJ : DJ → S (which must be C 0 -compatible where two pieces DJ , DJ touch). Then we can define on D finite element functions which on DJ are of the form g ◦ ΦJ , where g is a polynomial.


307

For a d-dimensional smooth surface D ⊂ Rd+1 we can similarly divide D into pieces which can be mapped to simplices in Rd , and again define finite elements using these mappings. Finite element wavelet basis for V To facilitate the accurate numerical approximation of moments of order k ≥ 2 of the random solution and for the efficient numerical solution of the partial differential equations, we use a hierarchical basis for the nested finite element (FE) spaces V0 ⊂ · · · ⊂ VL . To this end, we start with a basis {ψj0 }j=1,...,N0 for the finite element space V0 on the coarsest triangulation. We represent on the finer meshes T the corresponding FE spaces V , with > 0 as a direct sum V = V−1 ⊕W . Since the subspaces are nested and finite-dimensional, this is possible with a suitable space W for any hierarchy of FE spaces. We assume, in addition, that we are explicitly given basis functions {ψj }j=1,...,M of W . Iterating with respect to , we have that VL = V0 ⊕ W1 ⊕ · · · ⊕ WL , and {ψj ; = 0, . . . , L, j = 1, . . . , M } is a hierarchical basis for VL , where M0 := N0 . (W1) Hierarchical basis. VL = span{ψj ; 1 ≤ j ≤ ML , 0 ≤ ≤ L}. Let us define N := dim V and N−1 := 0; then we have M := N − N−1 for = 0, 1, 2, . . . , L. The hierarchical basis property (W1) is in principle sufficient for the formulation and implementation of the sparse MC–Galerkin method and the deterministic sparse Galerkin method. In order to obtain algorithms of log-linear complexity for integrodifferential equations, impose on the hierarchical basis the additional properties (W2)–(W5) of a wavelet basis. This will allow us to perform matrix compression for non-local operators, and to obtain optimal preconditioning for the iterative linear system solver. (W2) Small support. diam supp(ψj ) = O(2− ). (W3) Energy norm stability. There is a constant CB > 0 independent of L ∈ N ∪ {∞}, such that, for all L ∈ N ∪ {∞} and all L

v =

M L

vj ψj (x) ∈ VL ,

=0 j=1

we have −1 CB

M L

=0 j=1

|vj |2 ≤ v L 2V ≤ CB

M L

|vj |2 .

=0 j=1

Here, in the case L = ∞ it is understood that VL = V .

(1.30)

308


(W4) Wavelets ψj with ≥ 0 have vanishing moments up to order p0 ≥ p− ψj (x) xα dx = 0, 0 ≤ |α| ≤ p0 , (1.31) except possibly for wavelets where the closure of the support intersects the boundary ∂D or the boundaries of the coarsest mesh. In the case of mapped finite elements we require the vanishing moments for the polynomial function ψj ◦ Φ−1 J . (W4) Decay of coefficients for ‘smooth’ functions in Xs . There exists C > 0 independent of L such that, for every u ∈ Xs and every L, M L

0 for 0 ≤ s < p + 1 − /2, 2 2s ν 2 |uj | 2 ≤ CL u Xs , ν = 1 for s = p + 1 − /2. =0 j=1 (1.32) By property (W3), wavelets constitute Riesz bases: every function u ∈ V M has a unique wavelet expansion u = ∞ =0 j=1 uj ψj . We define the projection PL : V → VL by truncating this wavelet expansion of u at level L, i.e., PL u :=

M L

uj ψj .

(1.33)

=0 j=1

Because of the stability (W3) and the approximation property (1.29), we obtain immediately that the wavelet projection PL is quasi-optimal: with (1.29), for 0 ≤ s ≤ s∗ and u ∈ Xs , −s/d

u − PL u V NL

u Xs .

(1.34)

We remark in passing that the appearance of the factor 1/d in the convergence rate s/d in (1.34), when expressed in terms of NL , the total number of degrees of freedom, indicates a reduction of the convergence rate as the dimension d of the computational domain increases. This reduction of the convergence rate with increasing dimension is commonly referred to as the ‘curse of dimensionality’; as long as d = 1, 2, 3, this is not severe and, in fact, shared by almost all discretizations. If the dimension of the computational domain increases, however, this reduction becomes a severe obstacle to the construction of efficient discretizations. In the context of stochastic and parametric PDEs, the dimension of the computational domain can, in principle, become arbitrarily large, as we shall next explain.


309

Full and sparse tensor product spaces To compute an approximation for Mk u ∈ V (k) := V · · ⊗ V ⊗ · k times

we need a suitable finite-dimensional subspace of V (k) . The simplest choice (k) is the tensor product space VL ⊗ · · · ⊗ VL = VL . However, this full tensor product space has dimension (k)

dim(VL ) = NLk = (dim(VL ))k ,

(1.35)

which is not practical for k > 1. A reduction in cost is possible by sparse (k) tensor products of VL . The k-fold sparse tensor product space VL is defined by

(k) V ⊗ · · · ⊗ V , (1.36) V = 1

L

k

∈Nk0 ||≤L

where we denote by the vector (1 , . . . , k ) ∈ Nk0 and its length by || = 1 + · · · + k . The sum in (1.36) is not direct in general. However, since the (k) V are finite-dimensional, we can write VL as a direct sum in terms of the complement spaces Wl : (k) W1 ⊗ · · · ⊗ Wk . (1.37) VL = ∈Nk0 ||≤L

If a hierarchical basis of the subspaces V (i.e., satisfying hypothesis (W1)) is available, we can define a sparse tensor quasi-interpolation op(k) (k) erator PL : V (k) → VL by a suitable truncation of the tensor product wavelet expansion: for every x1 , . . . , xk ∈ D,

(k) ···k 1 vj11···j ψj1 (x1 ) · · · ψjkk (xk ). (1.38) (PL v)(x) := k 0≤1 +···+k ≤L 1≤jν ≤Mν ,ν=1,...,k

(k) If a hierarchical basis is not explicitly available, we can still express PL in terms of the projections Q := P − P−1 for = 0, 1, . . . , and with the convention P−1 := 0 as

(k) Q1 ⊗ · · · ⊗ Qk . (1.39) PL = 0≤1 +···+k ≤L (k) We also note that the dimension of VL is

L = dim(V (k) ) = O(NL (log2 NL )k−1 ), N L

(1.40)

310


that is, it is a log-linear function of the number NL of the degrees of freedom used for approximation of the first moment. Given that the sparse (k) tensor product space VL is substantially coarser, one wonders whether its approximation properties are substantially worse than that of the full ten(k) sor product space VL . The basis for the use of the sparse tensor product (k) (k) spaces VL is the next result, which indicates that VL achieves, up to logarithmic terms, the same asymptotic rate of convergence, in terms of powers of the mesh width, as the full tensor product space. The approximation (k) property of sparse grid spaces VL was established, for example, in Schwab and Todor (2003b, Proposition 4.2), Griebel, Oswald and Schiekofer (1999), von Petersdorff and Schwab (2004) and Todor (2009). Proposition 1.7. inf U − v V (k) ≤ C(k)

−s/d

NL

U

(k)

Xs −s/d ν(k)

U X (k) NL L s

if 0 ≤ s < p + 1 − /2,

if s = p + 1 − /2. (1.41) Here, the exponent ν(k) = (k − 1)/2 is best possible on account of the V -orthogonality of the V best approximation. (k)

v∈VL

Remark 1.8. The exponent ν(k) of the logarithmic terms in the sparse tensor approximation rates stated in Proposition 1.7 is best possible for the approximation in the sparse tensor product spaces V (k) given the regularity (k) U ∈ Xs . In general, these logarithmic terms in the convergence estimate are unavoidable. Removal of all logarithmic terms in the convergence rate (k) estimate as well as in the dimension estimate of VL is possible only if either (a) the norm ◦ V (k) on the left-hand side of (1.41) is weakened, or if (b) the (k) norm Xs on the right-hand side of (1.41) is strengthened. For example, in the context of sparse tensor FEM for the Laplacian in (0, 1)d , it was shown by von Petersdorff and Schwab (2004) and Bungartz and Griebel (2004) that all logarithmic terms can be removed; this is due to the observation that the H 1 ((0, 1)d ) norm is strictly weaker than the corresponding tensorized norm H 1 (0, 1)(d) which appears in the error bound (1.41) in the case of d-point correlations of a random field taking values in H01 (0, 1). The same effect allows us to slightly coarsen the sparse tensor product (k) space VL . This was exploited, for example, by Bungartz and Griebel (2004) and Todor (2009). (k) The error bound (1.41) is for the best approximation of U ∈ Xs from (k) VL . To achieve the exponent ν(k) = (k − 1)/2 in (1.41) for a sparse tensor quasi-interpolant such as (1.38), the multi-level basis ψj of V must be V -orthogonal between successive levels . This V -orthogonality of the

311


multi-level basis can be achieved in V ⊂ H 1 (D), for example, by using so-called spline prewavelets. Let us also remark that it is even possible to construct L2 (D) orthonormal piecewise polynomial wavelet bases satisfying (W1)–(W5). We refer to Donovan, Geronimo and Hardin (1996) for details. The stability property (W3) implies the following result (see, e.g., von Petersdorff and Schwab (2004)). (k) Lemma 1.9. (on the sparse tensor quasi-interpolant PL ) Assume (k) (W1)–(W5) and that the component spaces V of VL are V -orthogonal between scales and have the approximation property (1.29). Then the sparse (k) tensor projection PL is stable: there exists C > 0 (depending on k but independent of L) such that, for all for U ∈ V (k) ,

PL U V (k) ≤ C U V (k) . (k)

(1.42)

For U ∈ Xs and 0 ≤ s ≤ s∗ , if the basis functions ψj satisfy (W1)–(W5) and are V -orthogonal between different levels of mesh refinement, we obtain (k) quasi-optimal convergence of the sparse tensor quasi-interpolant PL U in (1.38): (k)

(k) −s/d

U − PL U V (k) ≤ C(k)NL (log NL )(k−1)/2 U X (k) . s

(1.43)

(k) Remark 1.10. The convergence rate (1.43) of the approximation PL U from the sparse tensor subspace is, up to logarithmic terms, equal to the rate obtained for the best approximation of the mean field, i.e., in the case k = 1. We observe, however, that the regularity of U required to achieve this convergence rate is quite high: the function U must belong (k) to an anisotropic smoothness class Xs which, in the context of ordinary Sobolev spaces, is a space of functions whose (weak ) mixed derivatives of order s belong to V . Evidently, this mixed smoothness regularity requirement becomes stronger as the number k of moments increases. By Theorem 1.6, the k-point correlations Mk u of the random solution u naturally satisfy such regularity.

Galerkin discretization We first consider the discretization of the problem Au(ω) = f (ω) for a single realization ω, bearing in mind that in the Monte Carlo method this problem will have to be approximately solved for many realizations of ω ∈ Ω. The Galerkin discretization of (1.11) reads: find uL (ω) ∈ VL such that vL , AuL (ω) = vL , f (ω)

∀vL ∈ VL ,

P-a.e. ω ∈ Ω,

(1.44)

where ‘P-a.e.’ stands for ‘P almost everywhere’. It is well known that the

312


injectivity (1.13) of A, the G˚ arding inequality (1.12) and the density in V of the subspace sequence {V }∞ =0 imply that there exists L0 > 0 such that, for L ≥ L0 , problem (1.44) admits a unique solution uL (ω). Furthermore, we have the uniform inf-sup condition (see, e.g., Hildebrandt and Wienholtz (1964)): there exists a discretization level L0 and a stability constant γ > 0 such that, for all L ≥ L0 , inf

sup

0=u∈VL 0=v∈VL

Au, v 1 ≥ > 0.

u V v V γ

(1.45)

The inf-sup condition (1.45) implies quasi-optimality of the approximations uL (ω) for L ≥ L0 (see, e.g., Babuˇska (1970/71)): there exist C > 0 and L0 > 0 such that ∀L ≥ L0 :

u(ω) − uL (ω) V ≤ C inf u(ω) − v V v∈VL

P-a.e. ω ∈ Ω. (1.46)

From (1.46) and (1.29), we obtain the asymptotic error estimate: define σ := min{s∗ , p + 1 − /2}. Then there exists C > 0 such that for 0 < s ≤ σ ∀L ≥ L0 :

−s/d

u(ω) − uL (ω) V ≤ CNL

u Xs

P-a.e. ω ∈ Ω.

(1.47)

1.3. Sparse tensor Monte Carlo Galerkin FEM We next review basic convergence results of the Monte Carlo method for the approximation of expectations of random variables taking values in a separable Hilbert space. As our exposition aims at the solution of operator equations with stochastic data, we shall first consider the MC method without discretization of the operator equation, and show convergence estimates of the statistical error incurred by the MC sampling. Subsequently, we turn to the Galerkin approximation of the operator equation and, in particular, the sparse tensor approximation of the two- and k-point correlation functions of the random solution. Monte Carlo error for continuous problems For a random variable Y , let Y1 (ω), . . . , YM (ω) denote M ∈ N copies of Y , i.e., the Yi are random variables which are mutually independent and identically distributed to Y (ω) on the same common probability space (Ω, Σ, P). M Then the arithmetic average Y (ω), 1 M Y1 (ω) + · · · + YM (ω) , Y (ω) := M is a random variable on (Ω, Σ, P) as well. The simplest approach to the numerical solution of (1.11) for f ∈ L1 (Ω; V ) is MC simulation. Let us first consider the situation without discretization of V . We generate M draws f (ωj ), j = 1, 2, . . . , M , of f (ω) and find the


313

solutions u(ωj ) ∈ V of the problems Au(ωj ) = f (ωj ),

j = 1, . . . , M.

(1.48)

¯ M [u(k) ] We then approximate the kth moment Mk u with the sample mean E of u(ωj ) ⊗ · · · ⊗ u(ωj ): M

¯ M [u(k) ] := u ⊗ · · · ⊗ uM = 1 u(ωj ) ⊗ · · · ⊗ u(ωj ). E M

(1.49)

j=1

It is well known that the Monte Carlo error decreases as M −1/2 in a probabilistic sense provided the variance of u(k) exists. By (1.18), this is the case for u ∈ L2k (Ω; V ). We have the following convergence estimate. Theorem 1.11. Let k ≥ 1 and assume that in the operator equation (1.11) f ∈ L2k (Ω; V ). Then, for any M ∈ N of samples for the MC estimator (1.49), we have the error bound k −1/2 ¯ M [u(k) ] 2 CA f L2k (Ω;V ) . (1.50)

Mk u − E L (Ω;V (k) ) ≤ M Proof. We observe that f ∈ L2k (Ω; V ) implies with (1.22) that u(k) ∈ L2 (Ω; V (k) ). For i = 1, . . . , M we denote by u i (ω) the M i.i.d. copies of the random variable u(ω) = A−1 f (ω), which corresponds to the M many MC samples u i = A−1 fi . Using that the u i are independent and identically distributed, we infer that, for each value of i, u i (ω) ∈ L2k (Ω; V ). Therefore (k) ¯ M [u(k) ] 2 2 ¯ M [u(k) ] 2 (k) = E

E[u ] − E

E[u(k) ] − E (k) L (Ω;V V ) M 2 1 (k) (k) =E u i (k) E[u ] − M V i=1 M M

(k)

1 1 (k) u i , E[u(k) ] − u j = E E[u(k) ] − M M i=1

=

1 M2

M

j=1

(k) (k) E E[u(k) ] − u i , E[u(k) ] − u j

i,j=1

M 1 (k) ( ui (ω) independent) E E[u(k) ] − u i 2V (k) = 2 M i=1 1 (k) ( ui (ω) identically distributed) = E u − E[u(k) ] 2V (k) M 1 (k) E u − E[u(k) ], u(k) − E[u(k) ] = M

314


1 (k) E u − E[u(k) ], E[u(k) ] + E u(k) − E[u(k) ], u(k) M 1 1 (k) 2 E u V (k) −

E[u(k) ] 2V (k) = M M ≤ M −1 u(k) 2L2 (Ω;V (k) ) = M −1 u 2k L2k (Ω;V ) . =

Taking square roots on both sides completes the proof. The previous theorem required that u(k) ∈ L2 (Ω; V (k) ) or (equivalently by (1.18)) that u ∈ L2k (Ω; V ) (resp. f ∈ L2k (Ω; V )) in order to obtain the convergence rate M −1/2 of the MC estimates (1.49), in L2 (Ω; v). In the case of weaker summability of u, the next estimate shows that the MC method converges in L1 (Ω; V (k) ) and at a rate that is possibly lower than 1/2, as determined by the summability of u. We only state the result here and refer to von Petersdorff and Schwab (2006) for the proof. Theorem 1.12. Let k ≥ 1. Assume that f ∈ Lαk (Ω; V ) for some α ∈ ¯ M [u(k) ] as in (1.49). (1, 2]. For M ≥ 1 samples we define the sample mean E Then there exists C such that, for every M ≥ 1 and every 0 < < 1, ¯ M [u(k) ] (k) ≤ C P M u − E V k

f kLαk (Ω;V )

1/α M 1−1/α

≥ 1 − .

(1.51)

The previous results show that one can obtain a rate of up to M −1/2 in a probabilistic sense for the Monte Carlo method. Convergence rates beyond 1/2 are not possible, in general, by the MC method, as is shown by the central limit theorem; in this sense, the rate 1/2 is sharp. So far, we have obtained the convergence rate 1/2 of the MC method essentially in L1 (Ω, V (k) ) and in L2 (Ω, V (k) ). A P-a.s convergence estimate of the MC method can be obtained using the separability of the Hilbert space of realizations and the law of the iterated logarithm; see, e.g., Strassen (1964) and Ledoux and Talagrand (1991, Chapter 8) and the references therein for the vector-valued case. Lemma 1.13. Assume that H is a separable Hilbert space and that X ∈ L2 (Ω; H). Then, with probability 1, M

X − E(X) H lim sup ≤ X − E(X) L2 (Ω;H) . −1 log log M )1/2 M →∞ (2M

(1.52)

For the proof, we refer to von Petersdorff and Schwab (2006). Applying Lemma 1.13 to X = u(k) = u⊗· · ·⊗u and with V (k) in place of H gives (with 2k f k CA as in (1.14)) u ⊗ · · · ⊗ u L2 (Ω;V (k) ) = u kL2k (Ω;V ) ≤ CA , L2k (Ω;V ) whence the following result.


Theorem 1.14.

Let f ∈ L2k (Ω; V ). Then, with probability 1,

lim sup M →∞

315

¯ M [u(k) ] (k)

Mk u − E V ≤ C(k) f kL2k (Ω;V ) . (2M −1 log log M )1/2

(1.53)

Sparse Monte Carlo Galerkin moment estimation We now use Galerkin discretization with the subspaces VL ⊂ V to solve (1.48) approximately and to obtain, for each draw of the load function, Galerkin approximations uL (ωj ). The resulting sample mean approximation of the kth moment Mk u equals M 1 M (k) ¯ E [uL ] := uL (ωj ) ⊗ · · · ⊗ uL (ωj ). M

(1.54)

j=1

This yields a first MC estimation for the k-point correlation function of Mk u. The complexity of forming (1.54) is, however, prohibitive: to form in (1.54) the k-fold tensor product of the Galerkin approximations for the M data samples, one needs O(NLk ) memory and O(M NLk ) operations to compute this mean, which implies loss of linear complexity for k > 1. Therefore we propose using the sparse approximation ¯ M [P(k) u(k) ], M [u(k) ] := P(k) E ¯ M [u(k) ] = E E L L

(1.55)

which requires O(NL (log NL )k−1 ) memory and operations. ¯ M,L [P(k) Mk u] we proceed as follows. First, we generate M To compute E L data samples f (ωj ), j = 1, . . . , M and the corresponding Galerkin approximations uL (ωj ) ∈ VL as in (1.44). Choosing a wavelet basis of VL that satisfies (W1) ((W2)–(W5) are not required at this stage), by (1.33), uL (ωj ) can then be represented as uL (ωj ) =

M L

uk (ωj )ψk ,

(1.56)

=0 k=1

= where {ψk },k denotes the dual wavelet basis to with {ψk },k (Cohen 2003). Based on the representation (1.56), we can compute the sparse tensor product MC estimate of Mk u with the projection operators (k) PL in (1.38) as follows: uk (ωj )

uL (ωj ), ψk ,

M 1 (k) (k) M (k) PL uL (ωj ) ⊗ · · · ⊗ uL (ωj ) ∈ VL . E [uL ] = M

(1.57)

j=1

This quantity can be computed in O(M NL (log NL )k−1 ) operations since, for each data sample f (ωj ), j = 1, . . . , M , the projection PL onto the (k) sparse tensor product space VL of the Galerkin approximation uL (ωj ) is

316


(k) given by PL [uL (ωj ) ⊗ · · · ⊗ uL (ωj )]. This projection can be computed in O(NL (log2 NL )k−1 ) operations as follows. For each j we first compute (k) uL (ωj ) in the wavelet basis and then form PL [uL (ωj ) ⊗ · · · ⊗ uL (ωj )] using the formula

(k) PL (v ⊗ · · · ⊗ v) = vjl11 · · · vjlkk ψjl11 · · · ψjlkk . (1.58) 0≤1 +···+k ≤L 1≤jν ≤Mν ,ν=1,...,k

The following result addresses the convergence of the sparse MC–Galerkin approximation of Mk u. Recall σ := min{s∗ , p + 1 − /2} with s∗ as in Theorem 1.6. Theorem 1.15. Assume that f ∈ Lk (Ω; Ys ) ∩ Lαk (Ω; V ) for some α ∈ (1, 2] and some s ∈ (0, σ]. Then there exists C(k) > 0 such that, for all M ≥ 1, L ≥ L0 and all 0 < < 1, M,L [Mk u] (k) < λ ≥ 1 −

P Mk u − E V with

−s/d −1 λ = C(k) NL (log NL )(k−1)/2 f kLk (Ω;Ys ) + −1/α M −(1−α ) f kLαk (Ω;V ) .

Proof.

We estimate

M,L [Mk u] − Mk u (k)

E V M 1 (k) PL [uL (ωj ) ⊗ · · · ⊗ uL (ωj )] − E(u ⊗ · · · ⊗ u) = M j=1

V (k)

(k) ≤ PL [uL (ωj ) ⊗ · · · ⊗ uL (ωj ) − u(ωj ) ⊗ · · · ⊗ u(ωj )] V (k) M 1 (k) (k) PL [u(ωj ) ⊗ · · · ⊗ u(ωj )] − E(PL [u ⊗ · · · ⊗ u]) + M j=1

+ (I −

V (k)

(k) PL )Mk u V (k) .

The last term is estimated with (1.43), Theorem 1.6, for 0 ≤ s ≤ s∗ by (k) −s/d

(I − PL )Mk u V (k) ≤ C(k)NL (log NL )(k−1)/2 Mk f Y (k) . s

For the first term, we use (1.42) and (1.47) with a tensor product argument. For the second term, the statistical error, by (1.42) it suffices to bound M

1 ¯ M [u(k) ] (k) , [u(ωj ) ⊗ · · · ⊗ u(ωj )] = Mk u − E E([u ⊗ · · · ⊗ u]) − V (k) M j=1

which was estimated in Theorem 1.12.

V


317

Remark 1.16. All results in this section also hold in the case of a stochastic operator A(ω). Specifically, let X now denote the space of bounded linear mappings V → V . Assume that A : Ω → X is measurable (with respect to Borel sets of X) and that there exists C, α > 0 and a compact T : V → V such that

A(ω) V ≤ C (A(ω) + T )u, u ≥

α u 2V

almost everywhere,

(1.59)

almost everywhere.

(1.60)

Let k ≥ 1. Then f ∈ Lk (Ω; V ) implies u = A−1 f ∈ Lk (Ω; V ) and Mk u ∈ (k) V (k) . Also f ∈ Lk (Ω; Ys ) implies u = A−1 f ∈ Lk (Ω; Xs ) and Mk u ∈ Xs . All proofs on the convergence of MC methods in this section still apply to that case. However, as we shall explain below, substantial computational efficiency for MCM can be gained by coupling the multi-level structure (1.28) of the Galerkin discretizations with a level-dependent sample size. 1.4. Deterministic Galerkin approximation of moments Sparse Galerkin approximation of Mk u We now describe and analyse the deterministic computation of the k-point correlation function Mk u of the random solution u by Galerkin discretization (1.26). If we use in the Galerkin discretization the full tensor product (k) (k) space VL , the inf-sup condition of the discrete operator on VL follows directly for L ≥ L0 from the discrete inf-sup condition (1.45) of the ‘meanfield’ operator A by a tensor product argument. The (anisotropic) regularity estimate for the kth moment Mk u,

Mk u X (k) ≤ Ck,s Mk f Y (k) , 0 ≤ s ≤ s∗ , s

s

k ≥ 1,

(1.61)

which was shown in Theorem 1.6, then allows us to obtain convergence rates. However, this ‘full tensor product Galerkin’ approach is prohibitively expensive: with NL degrees of freedom in the physical domain D, it requires the set-up and solution of a linear system with NLk unknowns. We reduce (k) this complexity by using in place of VL the sparse tensor product space (k) VL . The sparse Galerkin approximation ZL of Mk u is then obtained as follows: L ∈ V (k) find Z L

L , v = Mk f, v such that A(k) Z

(k) ∀v ∈ VL .

(1.62)

We first consider the case where the operator A is coercive, i.e., (1.12) holds with T = 0. Then A(k) : V (k) → (V )(k) is also coercive, and the (k) (k) stability of the Galerkin method with VL follows directly from VL ⊂ V (k) . In the case of T = 0 the stability of the Galerkin FEM on the sparse tensor (k) product space VL is not obvious: we know that (A + T ) ⊗ · · · ⊗ (A + T )

318


is coercive for sufficiently fine meshes (i.e., for sufficiently large L), but (A + T ) ⊗ · · · ⊗ (A + T ) − A ⊗ · · · ⊗ A is not compact. Therefore we require some additional assumptions. We assume that (1.12) holds with the additional requirement that T : V → V is smoothing with respect to the scale of spaces Xs , Ys , and we also assume that the adjoint operator A : V → V satisfies a regularity property: we assume that there exists δ > 0 such that T : V = X0 → Yδ is continuous, −1

(A )

(1.63)

: Yδ → Xδ is continuous.

(1.64)

Due to the indefiniteness of A we have to modify the sparse grid space: Let (k) (k) (k) (k) L0 ≥ 0 and L ≥ L0 . We define a space VL,L0 with VL ⊂ VL,L0 ⊂ VL+(k−1)L0 as follows. 1 k := {0, . . . , L}. For k ≥ 2, let SL,L be the set Definition 1.17. Let SL,L 0 0 k of indices l ∈ N0 satisfying the following conditions:

(li1 , . . . , lik−1 ) ∈

l1 k−1 SL,L0

+ · · · + lk ≤ L + (k − 1)L0 ,

(1.65)

if i1 , . . . , ik−1 are different indices in {1, . . . , k}. (1.66)

Then we define (k) VL,L0 :=

k l∈SL,L

W l1 ⊗ · · · ⊗ W lk .

(1.67)

0

k has the following Let JL0 := {0, 1, . . . , L0 }. Then the index set SL,L 0 subsets:

JLk0 ,

1 JLk−1 × SL,L , 0 0

2 JLk−2 × SL,L , 0 0

...,

k−1 JL0 × SL,L . 0

(k) Therefore, VL,L0 contains the following subspaces: (k)

VL0 ,

(k−1)

VL0

(1) ⊗ VL,L0 ,

(k−2)

VL0

(2) ⊗ VL,L0 ,

...,

(k−1) VL0 ⊗ VL,L0 .

(1.68)

To achieve stability of sparse tensor discretizations in the presence of possible indefiniteness of the operator A(ω) in (1.12), we introduce a certain (k) fixed L0 > 0 of mesh refinement and consider the sequence of spaces VL,L0 with L tending to infinity. Since (k) (k) (k) VL ⊂ VL,L0 ⊂ VL+(k−1)L0 , (k) (k) we see that dim VL,L0 grows with the same rate as dim VL as L → ∞. We then have the following discrete stability property.


319

Theorem 1.18. Assume that A and T satisfy (1.12), (1.63) and (1.64). Then there exists L0 ∈ N and γ > 0 such that, for all L ≥ L0 , inf

sup

(k)

0=u∈VL,L

0

(k) 0=v∈VL,L

0

A(k) u, v 1 ≥ > 0.

u V (k) v V (k) γ

(1.69)

In the positive definite case T = 0, this holds with L0 = 0, whereas in the indefinite case, L0 > 0 is necessary in general. For the proof, we refer to von Petersdorff and Schwab (2006). As is by now classical (e.g., Babuˇska (1970/71)), the discrete inf-sup condition (1.69) implies quasi-optimal convergence and therefore the convergence rate is given by the rate of best approximation. The following result from von Petersdorff and Schwab (2006) makes this precise. Theorem 1.19.

Assume (1.12) and (1.13).

Lk (Ω; V ).

(a) Let f ∈ Then with L0 ≥ 0 as in Theorem 1.18 (in particular, L0 = 0 is admissible when T = 0 in (1.12)) such that, for all L ≥ L0 , L ∈ V (k) of Mk u is uniquely the sparse Galerkin approximation Z L,L0 defined and converges quasi-optimally, i.e., there exists C > 0 such that, for all L ≥ L0 ,

Mk u − ZL (k) ≤ C inf Mk u − v (k) → 0 as L → ∞. V

V

(k)

v∈VL,L

0

(b) Assume that f ∈ Lk (Ω; Ys ) and the approximation property (1.29). Then, for 0 ≤ s ≤ σ := min{s∗ , p + 1 − /2}, L (k) ≤ C(k)N −s/d (log NL )(k−1)/2 f k .

Mk u − Z Ys V L

(1.70)

Matrix compression When A is a differential operator, the number of non-zero entries in the stiffness matrix for the standard FEM basis is O(N ) due to the local support assumption (W2) on the basis ψj . This implies that we can compute a matrix– vector product arising typically in iterative solvers with O(N ) operations. In the case of an integral or pseudodifferential equation, the operator A is non-local and all entries of the stiffness matrix are non-vanishing, in general. Then the cost of a matrix–vector product is O(N 2 ), which implies a loss of linear complexity of the algorithm. For boundary integral operators, it is well known that one can improve the complexity to O(N (log N )c ) by using matrix compression techniques. Several approaches to this end are available: either fast multipole methods (e.g., Beatson and Greengard (1997) and the references therein), multiresolution methods (such as wavelets; see, e.g., Schneider (1998), Harbrecht (2001), Dahmen, Harbrecht and Schneider (2006), Dahmen (1997)), or low-rank matrix approximation techniques (e.g., Bebendorf and Hackbusch (2003)).

320


These matrix compression methods have in common that they reduce complexity of the matrix–vector multiplication from O(N 2 ) to O(N (logN )b ) for some (small) non-negative number b. This complexity reduction comes, however, at the expense of being realized only approximately. We will elaborate on the effect of matrix compression on the accuracy of sparse tensor Galerkin approximations in this and in the following section. In the compression step, we replace most of the entries AJJ of the stiffness L . The matrix AL with zeros, yielding an approximate stiffness matrix A L induce mappings from stiffness matrix AL and its compressed variant A VL to (VL ) , which we denote by AL and AL , respectively. We will require AL and AL to be close in the following sense: for certain values s, s ∈ [0, σ] with σ = p + 1 − /2 and for u ∈ Xs , v ∈ Xs , we have (AL − AL ) PL u, PL v ≤ c(s, s )N −(s+s )/d (log NL )q(s,s ) u Xs v X L s (1.71) with c(s, s ) > 0 and q(s, s ) ≥ 0 independent of L. The following result collects some properties of the corresponding approximate solutions. Proposition 1.20.

Assume (1.12) and (1.13).

(a) If (1.71) holds for (s, s ) = (0, 0) with q(0, 0) = 0 and c(0, 0) sufficiently small, then there is an L0 > 0 such that, for every L ≥ L0 , (AL )−1 exists and is uniformly bounded, i.e. ∀L ≥ L0 :

(AL )−1 (VL ) →VL ≤ C

(1.72)

for some C independent of L. (b) If, in addition to the assumptions in (a), (1.71) holds with (s, s ) = (σ, 0), then −σ/d

(A−1 − (AL )−1 ) f V ≤ CNL (log NL )q(σ,0) f Yσ .

(1.73)

(c) Let g ∈ V be such that the solution ϕ ∈ V of A ϕ = g belongs to Xσ . If, in addition to the assumptions in (a) and (b), (1.71) also holds for (s, s ) = (0, σ) and for (s, s ) = (σ, σ), then g, (A−1 − (AL )−1 ) f ≤ CN −2σ/d · (log NL )max{q(0,σ)+q(σ,0),q(σ,σ)} L (1.74) where C = C(f, g). Proof. (a) The G˚ arding inequality (1.12), the injectivity (1.13) and the density in V of the subspace sequence {V L }L imply the discrete inf-sup condition (1.45). Using (1.71) with vL ∈ VL and (s, s ) = (0, 0), we obtain with (1.45)

AL vL (VL ) ≥ AvL (VL ) − (A−AL )vL (VL ) ≥ c−1 s vL V −Cc(0, 0) vL V .

321


This implies that for c(0, 0) < 1/(2Cγ) there is an L0 > 0 such that, for all L ≥ L0 , cs

AL vL (VL ) ∀vL ∈ VL ,

vL V ≤ (1.75) 2 whence we obtain (1.72). L = (AL )−1 f for L ≥ L0 . We have (b) Let f ∈ Yσ and u = A−1 f , u L V .

u − u L V ≤ u − PL u V + PL u − u Using (1.45) and AL uL , vL = Au, vL for all vL ∈ VL , we get

PL u − u L V ≤ C AL (PL u − u L ) (VL ) = C AL PL u − Au (VL ) , which yields the error estimate

u − u L V ≤ u − PL u V + C A(u − PL u) (VL ) + C (A − AL ) PL u (VL ) . (1.76) Here, the first two terms are estimated using the V -stability (W3) and (1.33) of the wavelet basis, which imply

u − PL u V ≤ C inf u − v V ≤ C(NL )−σ/d u Xσ , v∈VL

(1.77)

and the continuity A : V → V . The third term in (1.76) is estimated with (1.71) for (s, s ) = (σ, 0) and PL vL = vL for all vL ∈ VL : (A − AL ) PL u, vL c(σ, 0)N −σ/d (log NL )q(σ,0) u Xσ v V . (1.78) L (c) To show (1.74), we let ϕL := PL ϕ for ϕ = (A )−1 g ∈ Xσ and u = A−1 f , u L = (AL )−1 f for L ≥ L0 . Then g, u− u L ) ≤ A(u− u L ), ϕL . L ), ϕ−ϕL + A(u− u L = ϕ, A(u− u We estimate the first term by C u − u L V ϕ − P L ϕ V , which implies the bound (1.74) using (1.73) and (1.77). The second term is bounded as follows: A(u − u L ), ϕL = (AL − A) u L , ϕL = (AL − A)( uL − PL u), PL ϕ + (AL − A) PL u, PL ϕ . Here we estimate the second term by (1.71) with (s, s ) = (σ, σ). For the first term, we use (1.71) with (s, s ) = (0, σ) to obtain (AL − A) PL ( uL − PL u), PL ϕ −σ/d

(log NL )q(0,σ) uL − PL u V ϕ Xσ

−σ/d

(log NL )q(0,σ) ( uL − u V + u − PL u V ) ϕ Xσ .

NL NL

Using (1.73) and (1.77), we complete the proof.

322


Wavelet compression L , which We next describe how to obtain an approximate stiffness matrix A on the one hand has O(NL (log NL )a ) non-zero entries (out of NL2 ), and on the other hand satisfies the consistency condition (1.71). Here we assume that the operator A is given in terms of its Schwartz kernel k(x, y) by (Aϕ)(x) = k(x, y) ϕ(y) dS(y) (1.79) y∈Γ

C0∞ (Γ),

for ϕ ∈ estimates

where Γ = ∂D and k(x, z) satisfies the Calderón–Zygmund

|Dxα Dyβ k(x, y)| ≤ Cαβ |x − y|−(d++|α|+|β|) ,

x = y ∈ Γ.

(1.80)

In the following, we combine the indices (, j) into a multi-index J = (, j) to simplify notation, and write ψJ , ψJ , etc. Due to moment property (1.31) of the basis {ψJ }, the entries the vanishing AL of the moment matrix AL with respect to the basis {ψJ } = Aψ , ψ J J JJ show fast decay (Schneider 1998, Dahmen et al. 2006). Let SJ = supp(ψJ ), SJ = supp(ψJ ). Then we have the following decay estimate for the matrix entries AJJ (see Schneider (1998, Lemma 8.2.1) and Dahmen et al. (2006)). Proposition 1.21. If the wavelets ψJ , ψJ satisfy the moment condition (1.31) and A satisfies (1.79) and (1.80), then AψJ , ψJ ≤ C dist(SJ , SJ )−γ 2−γ(+ )/2 , (1.81) where γ := + d + 2 + 2(p∗ + 1) > 0. L with This can be exploited to truncate AL to obtain a sparse matrix A 2 at most O(NL (log NL ) ) non-zero entries and such that (1.71) is true with c(0, 0) as small as desired, independent of L, q(0, 0) = 0, and q(0, σ) = q(σ, 0) ≤ 32 , q(σ, σ) ≤ 3; see von Petersdorff and Schwab (1996), Schneider (1998), Harbrecht (2001) and Dahmen et al. (2006), for example. The L ), in the block A L of the compressed number of non-zero entries, nnz((A , , L Galerkin stiffness matrix A is bounded by L ) ≤ C(min(, ) + 1)d 2d max(, ) . nnz(A ,

(1.82)

Remark 1.22. For integral operators A an alternative approach for the efficient computation of matrix–vector products with the stiffness matrix AL is given by the cluster of fast multipole approximation. For these approximations, one additionally assumes for the operator (1.79) that the kernel k(x, z) is analytic for x = y, and the size of its domain of analyticity is proportional to |x − y|. Then one can replace k(x, y) in (1.79) for |x − y| sufficiently large by a cluster of fast multipole approximation with degenerate kernels which are obtained by either truncated multipole expansions

323


or polynomial interpolants of order log NL , allowing us to apply the block L to a vector in at most A ,

C(log NL )d 2d max(, ) ,

0 ≤ , ≤ L,

(1.83)

operations. See Schmidlin, Lage and Schwab (2003) for details on this work estimate. Error analysis for sparse Galerkin with matrix compression L and the corresponding operBased on the compressed stiffness matrix A ator AL : VL → (VL ) induced by it, we define the sparse tensor product approximation of Mk u with matrix compression analogous to (1.62) as fol(k) (k) lows: find ZLk ∈ VL,L0 such that, for all v ∈ VL,L0 , (k) k k AL Z L , v = M f, v.

(1.84)

We prove bounds for the error ZLk − Mk u. Lemma 1.23. Assume (1.12) and (1.13), and that the spaces VL as in Example 1.2 admit a hierarchical basis {ψj }≥0 satisfying (W1)–(W5). Assume further that the operator AL in (1.84) satisfies the consistency estimate (1.71) for s = s = 0, q(0, 0) = 0, and with sufficiently small c(0, 0). Then there exists L0 > 0 such that, for all L ≥ L0 , the kth-moment problem with matrix compression, (1.84), admits a unique solution and we have the error estimate k M u − ZLk (k) (1.85) V (k) (k) |(AL − AL )v, w| ≤ C inf

Mk u − v V (k) + sup . (k)

w V (k) (k) v∈V 0=w∈V L

L

Proof. We show unique solvability of (1.84) for sufficiently large L. By Theorem 1.18 we have that (1.69) holds. To show unique solvability of (1.84), we write for k ≥ 3 (k) (k−1) A(k) − AL = (A − AL ) ⊗ A(k−1) + AL ⊗ A(k−1) − AL = (A − AL ) ⊗ A(k−1) + AL ⊗ (A − AL ) ⊗ A(k−2) (2) (k−2) , + AL ⊗ A(k−2) − AL and obtain, after iteration, (k)

A

(k) − AL = (A − AL ) ⊗ A(k−1) +

k−2

(ν) AL ⊗ (A − AL ) ⊗ A(k−ν−1)

ν=1

+

(k−1) AL

⊗ (A − AL )

(1.86)

324


(where the sum is omitted if k = 2). We get from (1.69) that for any (k) (k) u ∈ VL there exists v ∈ VL such that

(k) (k) AL u, v = A(k) u, v + (AL − A(k) )u, v (1.87) (k) (AL − A(k) )w, w ≥ γ −1 − sup sup

u V (k) v V (k) . w V (k) w

V (k) (k) (k) w∈V w∈ V L

L

(k) (k) To obtain an upper bound for the supremum, we admit w, w ∈ VL,L0 ⊇ VL , use (1.86) and (1.71) with s = s = 0 and q(0, 0) = 0 to get

AL VL →(VL ) ≤ A V →V +c(0, 0), cA (k)

and therefore estimate for any w, w ∈ VL,L0 " !k−2

(k) ν k−ν−1 k−1 (k) A − A )w, w cA + c(0, 0) c c(0, 0) c + A

L

A

ν=1

k−1

+ cA + c(0, 0)

w V (k) w

V (k)

c(0, 0) w V (k) w

V (k) .

(1.88)

(k) If c(0, 0) is sufficiently small, this implies with (1.85) the stability of AL (k) on VL,L0 : there exists L0 > 0 such that

AL u, v 1 > 0, ≥

u V (k) v V (k) 2γ (k)

inf

sup

(k)

0=u∈VL,L

0

(k) 0=v∈VL,L

0

(1.89)

for all L ≥ L0 , and hence the unique solvability of (1.84) for these L follows. To prove (1.85), we follow the proof of the first lemma of Strang (e.g., Ciarlet (1978)). We now use this lemma to obtain the following convergence result. Theorem 1.24. Assume (1.12) and (1.13), V = H /2 (Γ), and that the subspaces {V }∞ =0 are as in Example 1.2, and that in the smoothness spaces Xs = H /2+s (Γ), s ≥ 0, the operator A : Xs → Ys is bijective for 0 ≤ s ≤ s∗ with some s∗ > 0. Assume further that a compression strategy for the matrix AL in the hierarchical basis {ψj } satisfying (W1)–(W5) is available with (1.71) for s = 0, 0 ≤ s ≤ σ = p + 1 − /2, q(0, 0) = 0 and

325


with c(0, 0) sufficiently small, independent of L for L ≥ L0 . Then, with δ = min{p + 1 − /2, s}/d, 0 ≤ s ≤ s∗ , we have the error estimate

Mk u − ZLk V (k) ≤ C(log NL )min{(k−1)/2,q(s,0)} NL−δ Mk f Y (k) . s

(1.90)

(k) We use (1.85) with the choice v = PL,L0 and, for Mk u − v V (k) , (k) (k) apply the approximation result (1.6). We express the difference AL − A˜L using (1.86). Then we obtain a sum of terms, each of which can be bounded (k) (k) using (1.71) and the continuity of AL and A˜L . This yields the following error bound:

Mk u − ZLk V (k) ≤ C (log NL )(k−1)/2 NL−δ −s/d

Mk u X (k) . + c(s, 0)(log NL )q(s,0) NL

Proof.

s

k in the ‘energy’ norm Theorem 1.24 addressed only the convergence of Z L V (k) . In the applications which we have in mind, however, functionals of the solution Mk u are also of interest, which we assume are given in the k form G, M u for some G ∈ (V (k) ) . We approximate such functionals by G, ZLk . Theorem 1.25. With all assumptions as in Theorem 1.24, and in addition that the adjoint problem (A(k) ) Ψ = G

(1.91)

admits a solution Ψ ∈ Xs for some 0 < s ≤ σ, and that the compression L of the stiffness matrix AL satisfies (1.71) with s = s = σ, we have A G, Mk u − G, ZLk ≤ C(log NL )min{k−1,q(s,s )} N −(δ+δ ) Mk f (k) , L Y (k)

s

where δ = min{p + 1 − /2, s}/d,

δ

= min{p + 1 −

/2, s }/d.

The proof is analogous to that of Proposition 1.20(c), using the sparse approximation property (1.41) in place of (1.29). Iterative solution of the linear system We solve the linear system (1.84) using iterative solvers and denote the (k) . We will consider three different methods. matrix of this system by A L

(k) (M1) If A is self-adjoint and (1.12) holds with T = 0, then the matrix A L is Hermitian positive definite, and we use the conjugate gradient algorithm which requires one matrix–vector multiplication by the matrix (k) per iteration. A L

326


(M2) If A is not necessarily self-adjoint, but satisfies (1.12) with T = 0, then we can use the GMRES algorithm with restarts every µ iterations. (k) + (A (k) )H is positive definite. This requires two In this case A L L (k) and one matrix–vector multiplications per iteration, one with A L (k) H with (A ) . L

(M3) In the general case where (1.12) is satisfied with some operator T , we (k) )H and can then apply multiply the linear system by the matrix (A L the conjugate gradient algorithm. This requires one matrix–vector (k) and one matrix–vector multiplication with multiplication with A L (k) H (AL ) per iteration. In order to achieve log-linear complexity, it is essential that we never ex L for (k) . Instead, we only store the matrix A plicitly form the matrix A L the mean-field problem. We can then compute a matrix–vector product (k) (or (A (k) )H ) by an algorithm which multiplies parts of the coefwith A L L L : see Algorithm 5.10 in Schwab and ficient vector with submatrices of A Todor (2003b). This requires O (log NL )kd+2k−2 NL operations (Schwab and Todor 2003b, Theorem 5.12). Let us explain the algorithm in the case k = 2 and L0 = 0. In this case a coefficient vector u has components ulj jl , where l, l are the levels used for (2) V (i.e., l, l ∈ {0, . . . , L} such that l + l ≤ L + L0 ) and L

j ∈ {1, . . . , Ml },

j ∈ {1, . . . , Ml }.

L1 denote the submatrix of A L corresponding to levels l, l ≤ L1 . We Let A (k) u as follows, where we can then compute the coefficients of the vector A L overwrite at each step the current components with the result of a matrix– vector product. • For l = 0, . . . , L, j = 1, . . . , Ml : multiply the column vector with L−l . components (ulj jl )l =0···L−l by the matrix A •

j =0···Ml For = 0, . . . , L, j = 1, . . . , Ml : multiply the L−l . components (ulj jl )l=0···L−l by the matrix A j=0···Ml

l

column vector with

We now analyse the convergence of the iterative solvers. The stability assumptions for the wavelet basis, the continuous and discrete operators (k) . imply the following results about the approximate stiffness matrix A L

Proposition 1.26. pendent of L.

Assume the basis

{ψj }

satisfies (1.30) with cB inde-

327


(a) Assume that AL satisfies (1.71) for q(0, 0) = 0 with sufficiently small c(0, 0). Then there are constants C1 , C2 such that, for all L, the matrix (k) of the problem (1.84) satisfies A L

(k) ≤ C2 < ∞. A L 2

(1.92)

(b) Assume, in addition to the assumptions of (a), that (1.12) holds with T = 0. Then ## (k) H $ $ (k) + A λmin A (1.93) /2 ≥ C1 > 0. L L (c) Assume the discrete inf-sup condition (1.69) holds, and that we have for some constant C independent of L (k) )−1 2 ≤ Cγ.

(A L

(1.94)

Proof. Because of (1.30) the norm vL V (k) of vL ∈ VL,L0 is equivalent to the 2-vector norm v 2 of the coefficient vector v. For (a) we obtain an arbitrarily small upper bound for the bilinear form with the operator A− A˜L with respect to the norm vL V (k) . Since A is continuous we get an upper bound for the norm of A˜ and therefore for the corresponding 2-matrix-norm. In (b), the bilinear form Av, v corresponds to the symmetric part of the matrix, and the lower bound corresponds to the smallest eigenvalue of the matrix. Since the norm of A − A˜ is arbitrarily small we also get the lower bound for the compressed matrix. In (c), the inf-sup condition (1.69) states that for L ≥ L0 , the solution (k) (k) operator mapping (VL,L0 ) to VL,L0 is bounded by γ. Because of the norm equivalence (1.30), this implies (k)

(k) )−1 2 ≤ Cγ.

(A L For method (M1) with a self-adjoint positive definite operator A, we have that λmax /λmin ≤ C2 /C1 =: κ is bounded independently of L, and obtain for the conjugate gradient iterates error estimates m 2 .

u(m) − u 2 ≤ c 1 − 1/2 κ +1 For method (M2) we obtain

u

(m)

1 − u 2 ≤ c 1 − κ

m

for the GMRES from Eisenstat, Elman and Schultz (1983) for the restarted GMRES method (e.g., with restart µ = 1).

328


For method (M3) we use the conjugate gradient method with the matrix (k) (k) )H A B := (A L L and need the largest and smallest eigenvalue of this matrix. Now (1.94) states that λmin (B) ≥ (Cγ)−2 > 0. Therefore we have with κ := C22 (Cγ)2 that m 2 (m) − u 2 ≤ c 1 − 1/2 .

u κ +1 Note that the 2-vector norm u 2 of the coefficient vector is equivalent to the norm u V (k) of the corresponding function on D × · · · × D. If we start with initial guess zero we therefore need a number M of iterations proportional to L to have an iteration error which is less than the Galerkin error. However, if we start on the coarsest mesh with initial guess zero, perform M iterations, use this as the starting value on the next-finer mesh, use M iterations, etc., we can avoid this additional factor L. Therefore we have the following complexity result. Proposition 1.27. We can compute an approximation ZLk for Mk u using a fixed number m0 of iterations such that −s/d β

ZLk − Mk V (k) ≤ CNL

L

where β = (k − 1)/2 for a differential operator, β = min{(k − 1)/2, q(s, 0)} with q(s, 0) from (1.90). The total number of operations is O(N (log N )k−1 ) in the case of a differential operator. In the case of an integral operator we need at most O(N (log N )k+1 ) operations. 1.5. Examples: FEM and BEM for the Helmholtz equation To illustrate the above concepts for an indefinite elliptic operator equation, we now consider the Helmholtz equation in a domain G ⊂ Rn with n ≥ 2 and Lipschitz boundary Γ := ∂G. We discuss two ways to solve this equation with stochastic data. First we use the finite element approximation of the differential equation and apply our results for D = G, which is of dimension d = n. Secondly, we consider the boundary integral formulation, which is an integral equation on the boundary Γ. We discretize this equation and then apply our results for D = Γ, which is of dimension d = n − 1. In this case we can also allow exterior domains G as the computation is done on the bounded manifold Γ. To keep the presentation simple we will just consider smooth boundaries and one type of boundary condition (Dirichlet condition for finite elements, Neumann condition for boundary elements). Other boundary conditions and operators can be treated in a similar way.

329


Finite element methods Let G ⊂ Rn be a bounded domain with smooth boundary. We consider the boundary value problem (−∆ − κ2 )u(ω) = f (ω) in G,

u|Γ = 0.

Here we have V = H 1 (G), V = H −1 (G), and the operator A : V → V is defined by Au, v = ∇u · ∇v − κ2 uv dx, G

and obviously satisfies the G˚ arding inequality Au, u ≥ u 2V − (κ2 + 1) u 2L2 (G) . The operator −∆ : V → V has eigenvalues 0 < λ1 < λ2 < · · · which converge to ∞. We need to assume that κ2 is not one of the eigenvalues λj , so that condition (1.13) is satisfied. The spaces for smooth data for s > 0 are Ys = H −1+s (G); the corresponding solution spaces are Xs = H 1+s (G). We assume that the stochastic right-hand side function f (ω) satisfies f ∈ Lk (Ω; Ys ) = Lk (Ω; H −1+s (G)) for some s > 0. Ld The space VL has NL = O(h−d L ) = O(2 ) degrees of freedom and the (k) sparse tensor product space VL,L0 has O(NL (log NL )(k−1) ) degrees of free(k) for dom. For k ≥ 1 the sparse grid Galerkin approximation Z k ∈ V L

L,L0

Mk u using V -orthogonal wavelets, using a total of O(NL (log NL )(k−1) ) operations, satisfies the error estimate (see Remark 1.8 regarding the exponent of the logarithmic terms)

ZLk − Mk u V (k) ≤ chpL | log hL |(k−1)/2 f kLk (Ω;Yp ) provided that f ∈ Lk (Ω; Yp ). Boundary element methods We illustrate the preceding abstract results with the boundary reduction of the stochastic Neumann problem to a boundary integral equation of the first kind. In a bounded domain D ⊂ Rd with Lipschitz boundary Γ = ∂D, we consider (1.95a) (∆ + κ2 )U = 0 in D with wave number κ ∈ C subject to Neumann boundary conditions γ1 U = n · (∇U )|Γ = σ on Γ, where σ ∈ Lk (Ω; H

− 12

(1.95b)

(Γ)) with integer k ≥ 1 are given random boundary

330


data, n is the exterior unit normal to Γ, and H s (Γ), |s| ≤ 1, denotes the usual Sobolev spaces on Γ: see, e.g., McLean (2000). We assume in (1.95b) that P-a.s. σ, 1 = 0 (1.96) and, if d = 2, in (1.95a) that diam(D) < 1.

(1.97)

Then problem (1.95) admits a unique solution U ∈ Lk (Ω; H 1 (D)) (Schwab and Todor 2003a, 2003b). For the boundary reduction, we define for v ∈ H 1/2 (Γ) the boundary integral operator ∂ ∂ (W v)(x) = − e(x, y) v(y) dsy (1.98) ∂nx Γ ∂ny with e(x, y) denoting the fundamental solution of −∆ − κ2 . The integral operator W is continuous (e.g., McLean (2000)): 1

1

W : H 2 (Γ) → H − 2 (Γ).

(1.99)

To reduce the stochastic Neumann problem (1.95) to a boundary integral 1 equation with σ ∈ Lk (Ω; H − 2 (Γ)) satisfying (1.96) a.s., we use a representation as double-layer potential R2 : ∂ e(x, y) ϑ(y, ω) dsy , (1.100) U (x, ω) = (R2 ϑ)(x, ω) := − ∂n y y∈Γ where Eϑ satisfies the BIE W1 Eϑ = Eσ ,

(1.101)

with the hypersingular boundary integral operator W1 u := W u + u, 1. We see that the mean field M1 U can be obtained by solving the deterministic boundary integral equation (1.101). Based on the compression error analysis in Section 1.4, we obtain an approximate solution EϑL ∈ V L in O(NL (log NL )2 ) operations and memory with error bound −(p+1/2)

Eϑ − EϑL H 1/2 (Γ) NL

(log NL )3/2 σ L1 (Ω;H p+1 (Γ)) .

To determine the variance of the random solution U , second moments of ϑ are required. To derive boundary integral equations for them, we use that by Fubini’s theorem, the operator M2 and the layer potential R2 commute. 1 For (1.95) with σ ∈ L2 (Ω; H − 2 (Γ)), we obtain that Cϑ satisfies the BIE 1 1

(W1 ⊗ W1 ) Cϑ = Cσ in H 2 , 2 (Γ × Γ).

(1.102)

Here, the ‘energy’ space V equals H 1/2 (Γ) and A = W1 . The unique solvability of the BIE (1.102) is ensured by the following result.

331


Proposition 1.28. If k = 0, the integral operator W1 ⊗ W1 is coercive, i.e., there exists γ > 0 such that 1 1

∀Cϑ ∈ H 2 , 2 (Γ × Γ) :

(W1 ⊗ W1 )Cϑ , Cϑ ≥ γ Cϑ 2

1 1

H 2 , 2 (Γ×Γ)

.

(1.103)

Proof. We prove (1.103). The operator W1 is self-adjoint and coercive in H 1/2 (Γ) (e.g., Nédélec and Planchard (1973), Hsiao and Wendland (1977), 1/2 (Γ)-orthonormal base in McLean (2000)). Let {ui }∞ i=1 denote an H 1/2 H (Γ) consisting of eigenfunctions of W1 . Then, {ui ⊗ uj }∞ i,j=1 is an or1 1

1 1

, , 2 2 (Γ×Γ) and we may represent any Cϑ ∈ H 2 2 (Γ×Γ) thonormal base in H ∞ M in the form Cϑ = i,j=1 cij ui ⊗ uj . For any M < ∞, consider Cϑ = M i,j=1 cij ui ⊗ uj . Then we calculate & % M M

M M cij ui ⊗ uj , c i j u i ⊗ u j (W1 ⊗ W1 )Cϑ , Cϑ = (W1 ⊗ W1 ) i ,j =1

i,j=1

=

M

i,j=1

λi λj c2ij

≥

λ21

M

c2ij = λ21 CϑM 2H 1/2,1/2 (Γ×Γ) .

i,j=1

Passing to the limit M → ∞, we obtain (1.103) with γ = λ21 . We remark that the preceding proof shows that the continuity constant k in the a priori estimate (1.22) is sharp: in general the conditioning of CA the tensorized operator A(k) increases exponentially with k. In the case κ = 0, we use that the integral operator W satisfies a G˚ arding inequality in H 1/2 (Γ) and obtain the unique solvability of the BIE (1.102) for Cϑ from Theorem 1.4, provided that W is injective, i.e., that κ is not a resonance frequency of the interior Dirichlet problem. To compute the second moments of the random solution U (x, ω) at an interior point x ∈ D, we tensorize the representation formula (1.100) to yield (M2 U )(x, x) = M2 (R2 ϑ) = (R2 ⊗ R2 )(M2 ϑ). (1.104) Then we obtain from Theorem 1.25 and from the sparse tensor Galerkin approximation (with spline wavelets which are V -orthogonal between lev2 of M2 ϑ in O(NL (log NL )3 ) operations and memory an approxiels) Z L mation of (M2 U )(x, x) which satisfies, for smooth boundary Γ and data σ ∈ L2 (Ω; Yp+1/2 ) = L2 (Ω; H p+1 (Γ)), at any interior point x ∈ D the error bound (M2 U )(x, x) − R2 ⊗ R2 , ZL2 ≤ c(x)(log NL )3 N −2(p+1/2) σ 2 2 . p+1 L

L (Ω;H

(Γ))

So far, we have considered the discretization of the boundary integral equation (1.102) by multi-level finite elements on the boundary surface Γ where convergence was achieved by mesh refinement.

332


We conclude this section with remarks on further developments in the analysis of sparse tensor discretizations of operator equations with random inputs. The results presented in this section are all based on hierarchies of subspaces {V }∞ =0 of V consisting of piecewise polynomial functions of a fixed polynomial degree on a sequence {T }∞ =0 of triangulations of the physical domain. Alternatively, spectral Galerkin discretizations based on sequences of polynomial or trigonometric functions of increasing order can also be considered. For such spaces there are analogous sparse tensor constructions known as ‘hyperbolic cross’ spaces: see, e.g., Temlyakov (1993) for the approximation theory of such spaces and Chernov and Schwab (2009) for an application to Galerkin approximations of boundary integral equations. Also, in the above presentation, we did not consider adaptive refinements in the sparse tensor discretizations. There is, however, a theory of sparse, adaptive tensor discretizations for subspace families satisfying axioms (W1)–(W5) of the present section available in Schwab and Stevenson (2008). As indicated at the beginning of this section, efficient discretization schemes for the tensorized equations (1.23) are also of interest in their own right, as such equations also arise in other models such as turbulence and transport equations. There is, however, one essential difference from (1.23): the equations (1.23) are exact, whereas in the two mentioned applications such equations can only be derived from additional moment closure hypotheses. For example, for PDEs with random operators, a suitable closure hypothesis could be smallness of fluctuations about the inputs’ mean, and neglecting solution fluctuations beyond first order in the inputs’ perturbation amplitude. This so-called first-order second-moment approach was first proposed in Dettinger and Wilson (1981) and was developed, in the context of random domains, in Harbrecht, Schneider and Schwab (2008). The use of quasi-Monte Carlo (QMC) methods for the discretization of stochastic PDEs promises a rate of convergence higher than M −1/2 , which we proved here for MC methods. The numerical analysis of QMC for such problems is currently emerging. We refer to Graham, Kuo, Nuyens, Scheichl and Sloan (2010) and the references therein for algorithms and numerical experiments, as well as for references on QMC.

2. Stochastic Galerkin discretization In Section 1 we considered Galerkin discretizations of k-point correlations of random fields in finite-dimensional spaces which were constructed from sparse tensor products of hierarchies of subspaces used for the approximations of single draws of u. The significance of this approach is twofold. First, for linear operator equations with stochastic data, we showed that k-point correlation functions of the random solutions are in fact solutions of highdimensional deterministic equations for tensorized operators. We showed


333

that these tensorized operator equations naturally afford anistropic regularity results in scales of smoothness spaces of Sobolev or Besov–Triebel– Lizorkin type, so that the efficiency of sparse tensor approximations of kth moments will not incur the curse of dimensionality. In effect, with the formulation of deterministic equations for two- and k-point correlation functions of random solutions, we trade randomness for high-dimensionality. The observation that k-point correlations of random fields satisfy deterministic tensorized operator equations is not new: it has been used frequently, for example in the derivation of moment closures in turbulence modelling or in the transition from atomistic-to-continuum models. For such nonlinear problems, however, there are generally no exact deterministic equations for the k-point correlation functions of the random solution: in general, a ‘closure hypothesis’ in some form is required. Due to the linear dependence of the random solution u on the random input f in (1.11), no closure hypothesis was required in the previous section. While efficient computation of moments of order k ≥ 2 of random solutions may be useful (e.g., if the unknown random field is known a priori to be Gaussian), it is well known that even the knowledge of all k-point correlations will not characterize the law of the random solution if the moment problem is not solvable. In the present section, we therefore address a second approach to the deterministic computation of random fields. It is based on parametrizing the random solution of a PDE with random data in terms of polynomials in a suitable coordinate representation of the random input. After having been pioneered by N. Wiener (1938), who established the representation of functionals of Wiener processes in terms of Hermite polynomials of a countable number of standard normal random variables, this ‘spectral’ view of random fields was shown to be generally applicable to any random field with finite second moments by Cameron and Martin (1947). Its use as a computational tool was pioneered in engineering applications in the 1990s. We mention only the book by Ghanem and Spanos (2007) and the series of papers by Xiu and Karniadakis (2002a), Xiu and Hesthaven (2005), Oden, Babuˇska, Nobile, Feng and Tempone (2005), Babuˇska, Nobile and Tempone (2005, 2007a, 2007b), Nobile, Tempone and Webster (2008a, 2008b), Nobile and Tempone (2009) and the references therein. In these works, it was in particular observed that the expansion in Hermite polynomials of Gaussians originally proposed by Wiener (1938) and by Cameron and Martin (1947) is not always best suited for efficient computations. Often, other (bi)orthogonal function systems are better suited to the law of the random inputs, and are preferable in terms of computational efficiency. This has led to a formal generalization of Wiener’s original polynomial chaos representations for computational purposes by G. E. Karniadakis and collaborators. We refer to Xiu and Karniadakis (2002a) and

334


to the survey by Xiu (2009) for further references on various applications of such ‘generalized polynomial chaos’ (GPC) methods. The GPC approaches have received substantial attention in the past few years, both among numerical analysts and among computational scientists and engineers. From a computational point of view, the spectral representation once more renders the stochastic problem deterministic: rather than focusing on computation of spatiotemporal two- and k-point correlation functions, the law of the unknown random solution is approximated and computed in a parametric form. As we shall see shortly, in this context questions of approximation and computation of deterministic quantities in infinite dimensions arise naturally. In this section, we present the mathematical formulation of generalized polynomial chaos representations of the laws of random solutions, starting with expansions into Hermite polynomials of Gaussians. Since the abovementioned pioneering works of Wiener and of Cameron and Martin, these expansions have found numerous applications in stochastic analysis. We then focus on elliptic problems with random diffusion coefficients, where expansions into polynomials of countably many non-Gaussian random variables are of interest. We present several particular instances of such generalized polynomials chaos expansions, in particular the Wiener–Itˆ o decomposition of random fields, and the Karhunen–Loève expansions, and review recent work on the regularity of solutions of infinite-dimensional parametric, deterministic equations in Section 3. Despite the formally infinite-dimensional setting, we review recent results that indicate that solutions of the infinite-dimensional parametric, deterministic equations for the laws of solutions of sPDEs exhibit regularity properties that allow finite-dimensional approximations free from the curse of dimensionality. We then address several adaptive strategies for the concrete construction of numerical approximations of such finitedimensional approximations. We exhibit in particular sufficient conditions for convergence rates which are larger than the rate 1/2 proved in Theorem 1.11 above for MC methods. 2.1. Hermite chaos Let H be a separable Hilbert space over R and let µ = NQ be a nondegenerate centred Gaussian measure on H. For convenience, we assume that H is infinite-dimensional; all of the following also applies in the finitedimensional setting. Product structure of Gaussian measures Let (em )m∈N be an orthonormal basis of H such that Qem = λm em ,

m ∈ N,

(2.1)

335


for a positive decreasing sequence (λm )m∈N . Note that (2.1) implies em ∈ Q1/2 (H). We define a sequence of random variables on (H, µ) by x, em H , Ym (x) := Wem (x) = x, Q−1/2 em H = λ−1/2 m

m ∈ N.

(2.2)

Lemma 2.1. The random variables (Ym )m∈N on (H, µ) are independent and identically distributed. The distribution of each Ym is the standard Gaussian measure N1 on R. Proof. This is a consequence of Proposition C.36. We give, in addition, a direct proof. Let I be an arbitrary finite subset of N with n := #I, and YI : H → Rn ,

x → (Ym (x))m∈I .

By the change of variables formula for Gaussian measures, the distribution of YI is the centred Gaussian measure (YI )# µ = NYI QYI∗ on Rn , where YI∗ : Rn → H is the adjoint of YI given by

YI∗ (ξ) = ξm Q−1/2 em , ξ = (ξm )m∈I ∈ Rn . m∈I

Since (em )m∈N is an orthonormal basis of H, for all ξ = (ξm )m∈I ∈ Rn , ! " ! "

∗ −1/2 1/2 ξm Q em = Y I ξm Q em YI QYI (ξ) = YI Q ! =

m∈I

m∈I

"

ξm Q1/2 em , Q−1/2 em H

= ξ. m ∈I

m∈I

Therefore, (YI )# µ = NI n =

'

N1 ,

m∈I

where I n is the identity matrix on Rn . By Lemma 2.1, the distribution of the injective map Y : H → R∞ ,

x → (Ym (x))m∈N

is the countable product measure γ := Y# µ =

'

N1

(2.3)

(2.4)

m∈N

on the Borel σ-algebra B(R∞ ). Proposition 2.2.

The pullback

Y ∗ : L2 (R∞ , B(R∞ ), γ) → L2 (H, µ),

f → Y ∗ f = f ◦ Y

(2.5)

336


is an isometric isomorphism of Hilbert spaces. Since γ = Y# µ, for all f ∈ L2 (R∞ , B(R∞ ), γ), ∗ 2 2 f (Y (x)) µ(dx) = f (y)γ(dy) = f 2L2 (R∞ ,B(R∞ ),γ) ,

Y f L2 (H,µ) =

Proof.

R∞

H

so Y ∗ is an isometry. To show surjectivity, note that the Borel σ-algebra B(H) is generated by Y since H is separable and the topology of H is generated by Y . Therefore, the Doob–Dynkin lemma implies that any g ∈ L2 (H, µ) is of the form g = f ◦ Y for a B(R∞ )-measurable function f on R∞ . The above computation implies f ∈ L2 (R∞ , B(R∞ ), γ). The Hermite polynomial basis Consider the analytic function 1 2 +tξ

F (t, ξ) := e− 2 t

t, ξ ∈ R.

,

(2.6)

We define the Hermite polynomials (Hn )n∈N0 through the power series representation of F (·, ξ) around t = 0, ∞

tn √ Hn (ξ), F (t, ξ) = n! n=0

t, ξ ∈ R.

(2.7)

Lemma 2.3. For all n ∈ N0 and ξ ∈ R, if H−1 (ξ) := 0, 1 2 (−1)n 1 2 Hn (ξ) = √ e 2 ξ Dξn e− 2 ξ , n! √ √ ξHn (ξ) = n + 1Hn+1 (ξ) + nHn−1 (ξ), √ Dξ Hn (ξ) = nHn−1 (ξ),

−Dξ2 Hn (ξ) + ξDξ Hn (ξ) = nHn (ξ). Proof.

1 2

1

2

1 2

∞ n

t n=0

1 2

= e2ξ

∞ n

t n=0

n!

(2.9) (2.10) (2.11)

Equation (2.8) follows by Taylor expansion, F (t, ξ) = e 2 ξ e− 2 (t−ξ) = e 2 ξ

(2.8)

n!

1

Dtn |t=0 e− 2 (t−ξ)

2

1 2

(−1)n Dξn e− 2 ξ ,

and comparison with (2.7). To show (2.9), we note that ∞ √ ∞

n tn−1 tn √ ( √ Hn (ξ) = n + 1Hn+1 (ξ). Dt F (t, ξ) = n! (n − 1)! n=1 n=0

337


Also, by (2.6), Dt F (t, ξ) = (ξ − t)F (t, ξ) =

∞ ∞

tn tn √ √ ξHn (ξ) − √ nHn−1 (ξ). n! n! n=0 n=0

Similarly, (2.10) follows by comparing two representations of Dξ F (t, ξ), ∞ ∞

tn tn √ √ Dξ Hn (ξ) = Dξ F (t, ξ) = tF (t, ξ) = √ nHn−1 (ξ). n! n! n=0 n=0

Finally, (2.11) is a consequence of (2.9) and (2.10), √ Dξ2 Hn (ξ) − ξDξ Hn (ξ) = n(Dξ Hn−1 (ξ) − ξHn−1 (ξ)) = −nHn (ξ). In particular, Hn is a polynomial of degree n. The first few Hermite polynomials are H0 (ξ) = 1, Proposition 2.4. Proof.

H1 (ξ) = ξ,

1 H2 (ξ) = √ (ξ 2 − 1). 2

(2.12)

(Hn )n∈N0 is an orthonormal basis of L2 (R, N1 ).

We first show orthonormality. Note that for ξ, s, t ∈ R, 1

e− 2 (t

2 +s2 )+ξ(t+s)

= F (t, ξ)F (s, ξ) =

∞

t n sm √ √ Hn (ξ)Hm (ξ). n! m! n,m=0

Integrating over R with respect to N1 , we have ∞

t n sm √ √ F (t, ξ)F (s, ξ)N1 (dξ) = Hn (ξ)Hm (ξ)N1 (dξ), n! m! R R n,m=0 and also ∞ ∞ n n

1 t s ets 2 F (t, ξ)F (s, ξ)N1 (dξ) = √ e− 2 (ξ−t−s) dξ = ets = . n! 2π −∞ R n=0 Therefore,

R

Hn (ξ)Hm (ξ)N1 (dξ) = δnm .

To show completeness, let f ∈ L2 (R, N1 ) be orthogonal to Hn for all 2 n ∈ N0 . Then g(ξ) := f (ξ)e−ξ /4 is in L2 (R), and for all t ∈ R, ∞ 1 2 1 2 1 f (ξ)F (t, ξ)N1 (dξ) = √ g(ξ)e− 2 t +tξ− 4 ξ dξ. 0= 2π −∞ R Since − 12 t2 + tξ − 14 ξ 2 = − 14 (2t − ξ)2 + 12 t2 , this implies that the convolution 2 = 0, g ∗ ϕ = 0 for ϕ(ξ) := e−ξ /4 . Taking the Fourier transform, we have gϕ

338


and since ϕ is non-zero everywhere, g = 0 in L2 (R). This implies g = 0 almost everywhere, and therefore f = 0 almost everywhere. Using Theorem 2.12 below and Proposition 2.2, we construct an orthonormal basis of L2 (H, µ). Define the index set of finitely supported sequences in N, F := {ν ∈ NN 0 ; # supp ν < ∞},

(2.13)

where supp ν := {m ∈ N ; νm = 0},

ν ∈ NN 0.

For all ν ∈ F, we define the tensor product Hermite polynomial ' Hνm , Hν :=

(2.14)

(2.15)

m∈N

i.e., for all y ∈ R∞ , since H0 (ξ) = 1, ) Hν (y) = Hνm (ym ) =

)

Hνm (ym ).

(2.16)

m∈supp ν

m∈N

The degree of the polynomial Hν for ν ∈ F is given by

νm = νm . |ν| := m∈N

(2.17)

m∈supp ν

We use the pullback Y ∗ from (2.5) to define Hν on H. For all x ∈ H and ν ∈ F, ) Hν (x) := (Y ∗ Hν )(x) = Hν (Y (x)) = Hνm (Wem (x)). (2.18) m∈N

As in (2.16), the product in (2.18) is finite, since all but finitely many factors are one by definition of F and H0 (ξ) = 1. Theorem 2.5. (Hν )ν∈F is an orthonormal basis of L2 (H, µ). Proof. By Proposition 2.4 and Theorem 2.12 below, (Hν )ν∈F from (2.15) is an orthonormal basis of L2 (R∞ , B(R∞ ), γ). Since the pullback Y ∗ is an isometric isomorphism from L2 (R∞ , B(R∞ ), γ) to L2 (H, µ) by Proposition 2.2, (Hν )ν∈F from (2.18) is an orthonormal basis of L2 (H, µ). We call (Hν )ν∈F the Hermite chaos basis of L2 (H, µ). Wiener–Itˆ o decomposition For all n ∈ N0 , we define the Wiener chaos of order n as the closed subspace L2n (H, µ) := span {Hn (Wf (x)) ; f ∈ H, f H = 1} ⊂ L2 (H, µ),

(2.19)

339


where the white noise map Wf is defined by (C.52) and continuous extension to H. We recall the identity 1 2 eWf (x) µ(dx) = e 2 f H ∀f ∈ H (2.20) H

from Proposition C.36. Lemma 2.6.

For all f, g ∈ H with f H = g H = 1 and all n, m ∈ N0 , H

Hn (Wf (x))Hm (Wg (x))µ(dx) = δnm f, gnH .

(2.21)

As in the proof of Proposition 2.4, for t, s ∈ R, ∞

t n sm √ √ F (t, Wf )F (s, Wg ) dµ = Hn (Wf )Hm (Wg ) dµ, n! m! H H n,m=0

Proof.

1

2

2

and also, using (2.20) and F (t, Wf )F (s, Wg ) = e− 2 (t +s )+tWf +sWg , 1 2 1 2 1 2 2 2 F (t, Wf )F (s, Wg ) dµ = e− 2 (t +s ) eWtf +sg dµ = e− 2 (t +s ) e 2 tf +sg H H

H

= ets f,gH =

∞ n n

t s n=0

n!

f, gnH .

Theorem 2.7. (Wiener–Itˆ o decomposition) L2 (H, µ) = L2n (H, µ).

(2.22)

n∈N0

Proof. Orthogonality of the spaces L2n (H, µ) follows from Lemma 2.6. It remains to be shown that these spaces span L2 (H, µ). Let g ∈ L2 (H, µ) be orthogonal to Hn (Wf ) for all n ∈ N0 and all f ∈ H with f H = 1. Then, for all t ∈ R and any f ∈ H with f H = 1, − 21 t2 0= F (t, Wf )g dµ = e etWf g dµ. H

H

Consequently, the entire function

etWf g dµ

ϕ(t) := H

vanishes on R, and thus is equal to 0 on C. Let ϑ be the signed measure dϑ = g dµ. An arbitrary element h ∈ H is of the form h = tQ−1/2 f for an f ∈ Q1/2 (H) with f H = 1 and some t ∈ R. The Fourier transform of ϑ evaluated at h is i x,tQ−1/2 f H ϑ(h) = e g(x)µ(dx) = eitWf g dµ = ϕ(it) = 0. H

H

340


Therefore, ϑ = 0 and it follows that g = 0 almost everywhere. The following proposition describes the connection between the Wiener– Itô decomposition (2.22) and the Hermite chaos basis (Hν )ν∈F of L2 (H, µ) from Theorem 2.5. For all n ∈ N0 ,

Proposition 2.8.

Proof.

L2n (H, µ) = span {Hν (x) ; ν ∈ F, |ν| = n}.

(2.23)

It suffices to show that for n ∈ N0 and ν ∈ F with |ν| = n, Hν (x)Hn (Wf (x))µ(dx) = 0 ∀f ∈ H : f H = 1.

(2.24)

H

Then the inclusions ‘⊂’ and ‘⊃’ follow from Theorem 2.5 and Theorem 2.7, respectively. Let f ∈ H with f H = 1. Since supp ν is finite for ν ∈ F, there is an N ∈ N0 with νi = 0 for all i ≥ N + 1. In particular, Hν (x) = Hν1 (We1 (x))Hν2 (We2 (x)) · · · HνN (WeN (x)),

x ∈ H.

For t1 , . . . , tN +1 ∈ R, we compute I := F (t1 , We1 ) · · · F (tN , WeN )F (tN +1 , Wf ) dµ H

twice. Using (2.6), linearity of W and (2.20), we have − 12 (t21 +···+t2N +1 ) eWt1 e1 +···+tN eN +tN +1 f dµ I=e H

=e

− 12 (t21 +···+t2N +1 )

e

1 t e +···+tN eN +tN +1 f 2H 2 1 1

.

Abbreviating fi := f, ei H , i ∈ N, since f12 + f22 + · · · = f 2H = 1,

t1 e1 + · · · + tN eN + tN +1 f 2H 2 2 = (t1 + tN +1 f1 )2 + · · · + (tN + tN +1 fN )2 + t2N +1 (fN +1 + fN +2 + · · · )

= t21 + · · · + t2N + 2tN +1 (t1 f1 + · · · + tN fN ) + t2N +1 . Since the quadratic terms cancel, we are left with I=e

tN +1 (t1 f1 +···+tN fN )

=

∞ n

tN +1 (t1 f1 + · · · + tN fN )n n=0

n!

.

Also, (2.7) implies I=

∞

k1 ,...,kN +1

k

+1 tk1 · · · tNN+1 (1 k1 ! · · · kN +1 ! =0

H

Hk1 (We1 ) · · · HkN (WeN )HkN +1 (Wf ) dµ.

Comparing the last two equations leads to (2.24).

341


2.2. Generalized polynomial chaos The construction of an orthonormal basis in Section 2.1 is not specific to Gaussian measures or Hermite polynomials. The important ingredient is the countable product structure of the probability space (H, µ), which we illustrated by the measure-preserving map Y into the product measure space (R∞ , B(R∞ ), γ). We generalize the construction of the chaos basis to countable products of arbitrary probability spaces. Again, all of the following also holds for finite products with the obvious modifications; to simplify notation, we consider only the countable case. We refer to Gittelson (2011a) for further details and a more general construction. Countable products of probability spaces For all m ∈ N, let Γm be an arbitrary non-empty set, endowed with a σalgebra Σm . Let (Ym )m∈N be independent random variables on a probability space (Ω, Σ, P), such that Ym maps into (Γm , Σm ). This sequence constitutes a map ) Γm , ω → (Ym (ω))m∈N , (2.25) Y : Ω → Γ := m∈N

* which is measurable with respect to the product σ-algebra Σ := m∈N Σm on Γ. By the independence of (Ym )m∈N , the distribution of Y is the countable product probability measure ' µ := µm (2.26) m∈N

on (Γ, Σ), where µm = (Ym )# (P) is the distribution of Ym on (Γm , Σm ). Countable product bases For all m ∈ N, let (ϕm,i )i∈N0 be an orthonormal basis of L2 (Γm , Σm , µm ) such that ϕm,0 = 1; the constant 1 is normalized in L2 (Γm , Σm , µm ) since µm is a probability measure. As in Section 2.1, we define the index set F := {ν ∈ NN 0 ; # supp ν < ∞}.

(2.27)

If L2 (Γm , Σm , µm ) is finite-dimensional for some m, then of course its orthonormal basis (ϕm,i )N i=0 is finite, and we restrict νm to the values 0, 1, . . . , N in the definition of F. For all ν ∈ F, define the tensor product ' ϕν := ϕm,νm , (2.28) m∈N

i.e., for all y = (ym )m∈N ∈ Γ, since ϕm,0 = 1 for all m ∈ N, ) ) ϕν (y) = ϕm,νm (ym ) = ϕm,νm (ym ). m∈N

m∈supp ν

(2.29)

342


Let F(N) denote the set of all finite subsets of N. For I ∈ F(N), define the finite product σ-algebra ' ΣI := Σm = σ(ym ; m ∈ I) ⊂ Σ. (2.30) m∈I

A function is ΣI -measurable if it is Σ-measurable and only depends on (ym )m∈I . Also, let FI := {ν ∈ F ; supp ν ⊂ I}. Lemma 2.9. For all I ∈ F(N), the set (ϕν )ν∈FI is an orthonormal basis of L2 (Γ, ΣI , µ). Proof.

Since ϕm,0 = 1 for all m ∈ N, if supp ν ⊂ I, then ) ) ϕν (y) = ϕm,νm (ym ) = ϕm,νm (ym ), y ∈ Γ. m∈I

m∈N

Due to the assumption that I is finite, ' L2 (Γ, ΣI , µ) ∼ L2 (Γm , Σm , µm ). = m∈I

As (ϕm,i )i∈N is an orthonormal basis of L2 (Γm , Σm , µm ) for each m ∈ I by assumption, the claim follows since finite tensor products of orthonormal bases form an orthonormal basis in the product space. The monotone class theorem implies that any function in L2 (Γ, Σ, µ) can be approximated by ΣI -measurable functions with I ∈ F(N). We recall that a set M of real-valued functions on Γ is multiplicative if vw ∈ M whenever v, w ∈ M. A monotone vector space over Γ is a real vector space H of bounded, real-valued functions on Γ such that all constants are in H, and if (vn )n∈N is a sequence in H with 0 ≤ vn ≤ vn+1 for all n ∈ N and v := supn vn is a bounded function on Γ, then v ∈ H. Theorem 2.10. (monotone class theorem) Let M be a multiplicative class of bounded, real-valued functions on Γ and let H be a monotone vector space containing M. Then H contains all bounded σ(M)-measurable functions. We refer to Protter (2005, Theorem I.8) for a proof of Theorem 2.10. 2 2 Proposition 2.11. I∈F (N) L (Γ, ΣI , µ) is dense in L (Γ, Σ, µ). Proof. Let V := I∈F (N) L2 (Γ, ΣI , µ) ⊂ L2 (Γ, Σ, µ) and define H := V ∩ L∞ (Γ, Σ, µ) as the vector space of bounded functions in V. Let M be the set of indicator functions in L2 (Γ, ΣI , µ) for any I ∈ F(N). Then M ⊂ H, 1 ∈ H, and M is closed under multiplication. Let 0 ≤ v1 ≤ v2 ≤ · · · be a pointwise monotonic sequence in H and v := supn vn its pointwise supremum. If v ∈ L∞ (Γ, Σ, µ) ⊂ L2 (Γ, Σ, µ), then (vn )n converges to v

343


in L2 (Γ, Σ, µ) by dominated convergence. Since V is closed in L2 (Γ, Σ, µ), it follows that v ∈ V and therefore v ∈ H. Thus H is a monotone vector space and, using that Σ = σ(M) is the σ-algebra generated by M, the monotone class theorem implies H = L∞ (Γ, Σ, µ). If v ∈ L2 (Γ, Σ, µ), then for any N ∈ N, v1{|v|≤N } ∈ L∞ (Γ, Σ, µ) = H ⊂ V and v ∈ V by dominated convergence. (ϕν )ν∈F is an orthonormal basis of L2 (Γ, Σ, µ).

Theorem 2.12. Proof.

Orthonormality follows from Lemma 2.9 since, for any ν, ν ∈ F, I := supp ν ∪ supp ν ∈ F(N).

Density follows from Proposition 2.11 since (ϕν )ν∈FI spans L2 (Γ, ΣI , µ) for all I ∈ F(N). Let (ϕm,i )i∈N0 be a graded basis of L2 (Γm , Σm , µm ) for each m ∈ N, i.e., there is a map m : N0 → N0 assigning to each index i ∈ N0 a level (i). This might be the degree of a polynomial, or the level of a wavelet, depending on (ϕm,i )i∈N0 . We assume that m (0) = 0 for all m ∈ N. This allows us to define a grading function for the orthonormal basis (ϕν )ν∈F of L2 (Γ, Σ, µ) by

m (νm ) = m (νm ), ν ∈ F. (2.31) (ν) := m∈suppν

m∈N

This function induces a decomposition of L2 (Γ, Σ, µ) into the closed subspaces L2n (Γ, Σ, µ) := span {ϕν ; ν ∈ F, (ν) = n} ⊂ L2 (Γ, Σ, µ), Corollary 2.13. L2 (Γ, Σ, µ) =

n ∈ N0 . (2.32)

L2n (Γ, Σ, µ).

(2.33)

n∈N0

Proof.

By Theorem 2.12 and (2.32), L2 (Γ, Σ, µ) = span ϕν = span ϕν = L2n (Γ, Σ, µ). ν∈F

n∈N0 (ν)=n

n∈N0

We note that other choices of are possible. For example, the dimensions m ∈ N can be weighted differently, leading to an anisotropic decomposition. Also, the 1 -norm in (2.32) can be generalized to an arbitrary p -quasi-norm for any p > 0. Rounding the final value ensures that maps into N0 . Orthogonal polynomials We assume that Γm is a Borel subset of R and that µm has finite moments ξ n µm (dξ), n ∈ N0 . (2.34) Mn := Γm

344


Orthonormal polynomials with respect to µm can be constructed by the well-known three-term recursion βn+1 Pn+1 (ξ) = (ξ − αn )Pn (ξ) − βn Pn−1 (ξ),

n ∈ N0 ,

(2.35)

with the initialization P−1 (ξ) = 0 and P0 (ξ) = 1. The coefficients are cn−1 ξPn (ξ)2 µm (dξ) and βn := , (2.36) αn := cn Γm where cn is the leading coefficient of Pn , and β0 := 1. The values of (αn )n∈N0 and (βn )n∈N0 are tabulated for many common distributions µm (Gautschi 2004). Formula (2.35) can be derived by Gram–Schmidt orthogonalization of the monomials (ξ n )n∈N0 . Note that βn+1 depends on Pn+1 , and can be computed by normalizing the right-hand side of (2.35) in L2 (Γm , Σm , µm ). Lemma 2.14. For all n ∈ N0 , Pn is a polynomial of degree n if n < N := dim L2 (Γm , Σm , µm ) and zero otherwise. The sequence (Pn )n∈N0 (resp. −1 2 (Pn )N n=0 if N is finite) is orthonormal in L (Γm , Σm , µm ). Proof. By the Gram–Schmidt orthogonalization process applied to the monomials (ξ n )n∈N0 , µm -orthonormal polynomials (Pn )n∈N exist. If N is −1 2 finite, then (ξ n )N n=0 is a basis of L (Γm , Σm , µm ), and thus Pn = 0 for all n ≥ N . We show that the orthonormal polynomials constructed by Gram– Schmidt orthogonalization satisfy (2.35). Note that βn+1 Pn+1 (ξ) − ξPn (ξ) is a polynomial of degree at most n. Therefore, and since (Pk )n+1 k=0 are orthonormal, βn+1 Pn+1 (ξ) − ξPn (ξ) = γn Pn (ξ) + γn−1 Pn−1 (ξ) + · · · + γ0 with

(βn+1 Pn+1 (ξ) − ξPn (ξ))Pk (ξ)µm (dξ) = −

γk = Γm

ξPn (ξ)Pk (ξ)µm (dξ) Γm

for k = 0, 1, . . . , n. In particular, γn = −αn , and γk = 0 for k ≤ n − 2 since ξPk (ξ) is a polynomial of degree at most n − 1. We note that ξPn−1 (ξ) = βn Pn (ξ) + q(ξ) for a polynomial q of degree at most n − 1. This implies γn−1 = −βn . If N = dim L2 (Γm , Σm , µm ) is finite, then it follows from Lemma 2.14 −1 2 that (Pn )N n=0 is an orthonormal basis of L (Γm , Σm , µm ). In general, this requires an additional assumption. We consider the case N = ∞ in the following. The measure µm is called determinate if it is uniquely characterized by its moments (Mn )n∈N0 ⊂ R. We note that µm is always determinate if Γm ⊂ R is bounded (Gautschi 2004, Theorem 1.41). The following result was shown by F. Riesz in 1923 (see, e.g., Szeg˝o (1975) for a proof).

345


Proposition 2.15. If µm is determinate, then (Pn )n∈N0 is an orthonormal basis of L2 (Γm , Σm , µm ). Examples of generalized polynomial chaos bases We combine Theorem 2.12 with Proposition 2.15 to construct a countable tensor product basis of L2 (Γ, Σ, µ). Again, we assume for simplicity that L2 (Γm , Σm , µm ) is infinite-dimensional for all m ∈ N. Analogous results hold in the general setting. For all m ∈ N, let (Pnm )n∈N0 be the orthonormal polynomial basis of 2 L (Γm , Σm , µm ) from Proposition 2.15. Then, by Theorem 2.12, the tensor product polynomials ' Pν := Pνmm , ν ∈ F, (2.37) m∈N

form an orthonormal basis of L2 (Γ, Σ, µ), which we call the generalized polynomial chaos basis. If µm = N1 for all m ∈ N, then (Pnm )n∈N0 are the Hermite polynomials (2.8) and (Pν )ν∈F is the Hermite chaos basis, interpreted as a basis of L2 (R∞ , B(R∞ ), γ) instead of L2 (H, µ). In this case, Corollary 2.13 reduces to the Wiener–Itô decomposition, Theorem 2.7, due to Proposition 2.8. We consider as another example the case when µm is the uniform distribution on Γm := [−1, 1] for all m ∈ N, i.e., µm (dξ) = 12 dξ. The corresponding orthonormal polynomials are the Legendre polynomials, which are defined by the three-term recursion √

n+1 n √ √ Ln+1 (ξ) = ξLn (ξ) − √ Ln−1 (ξ), 2n + 3 2n + 1 2n + 1 2n − 1

n ∈ N0 ,

(2.38) with L−1 (ξ) = 0 and L0 (ξ) = 1. The Legendre polynomials satisfy Rodrigues’ formula √ 2n + 1 dn 2 (ξ − 1)n , n ∈ N0 . (2.39) Ln (ξ) = 2n n! dξ n The first few Legendre polynomials are √ L1 (ξ) = 3 ξ,

√

5 2 (2.40) (3ξ − 1). 2 The tensor product Legendre polynomials Lν are defined as in (2.37) for ν ∈ F. In this case, the measure space (Γ, Σ, µ) is a countable product of identical factors ([−1, 1], B([−1, 1]), 12 dξ), ' Γ = [−1, 1]∞ , Σ = B([−1, 1])∞ = B([−1, 1]∞ ), µ = µm , (2.41) L0 (ξ) = 1,

L2 (ξ) =

m∈N

346


with µm (dξ) = 12 dξ for all m ∈ N. Note that even though each µm is absolutely continuous with respect to the Lebesgue measure, i.e., it has a density, the product of these densities is zero. Also, the countable product of the Lebesgue measure on [−1, 1] is not well-defined since the factors are not normalized. Thus µ cannot be defined via a density function. Corollary 2.16. The tensor product Legendre polynomials (Lν )ν∈F form an orthonormal basis of L2 ([−1, 1]∞ , B([−1, 1]∞ ), µ). Proof.

The claim follows from Theorem 2.12 and Proposition 2.15.

We shall refer to (Lν )ν∈F as the Legendre chaos basis. 2.3. PDEs with uniform stochastic parameters Parametric and stochastic operators Let V be a separable real Hilbert space with dual V , and let ·, · denote the (V , V )-duality pairing. We consider operator equations of the form Au = f,

(2.42)

with f ∈ V and A ∈ L(V, V ) a bounded linear operator from V to V . If A is boundedly invertible, then (2.42) has the unique solution u = A−1 f . Let Γ be a topological space. A parametric operator from V to V is given by a continuous map A : Γ → L(V, V ). (2.43) We assume that A(y) is boundedly invertible for all y ∈ Γ, and consider the parametric operator equation A(y)u(y) = f (y)

∀y ∈ Γ

(2.44)

for a map f : Γ → V . Proposition 2.17. Equation (2.44) has a unique solution u : Γ → V . It is continuous if and only if f : Γ → V is continuous. Proof. Since A(y) is boundedly invertible for all y ∈ Γ, (2.44) has the unique solution u(y) = A(y)−1 f (y). If u is continuous in y, then f (y) = A(y)u(y) is also continuous in y, since A is continuous by definition and application of an operator to a vector is continuous on L(V, V ) × V . Furthermore, the map y → A(y)−1 is continuous as a consequence of the abstract property (Kadison and Ringrose 1997, Proposition 3.1.6) of Banach algebras, so continuity of u follows from continuity of f by the same argument as above. We derive a weak formulation of (2.44) in the parameter y under the additional assumptions that A(y) is symmetric positive definite for all y ∈ Γ,


347

and there exist constants c and cˇ such that

A(y)−1 V →V ≤ cˇ ∀y ∈ Γ,

c,

A(y) V →V ≤

(2.45)

i.e., the bilinear form V A(y)·, ·V is a scalar product on V that induces a norm equivalent to · V . The estimates (2.45) always hold if Γ is compact. Let µ be a probability measure on the Borel-measurable space (Γ, B(Γ)). Then the operator A(y) becomes stochastic in the sense that it depends on a parameter y in a probability space (Γ, B(Γ), µ). Similarly, f is a random variable on Γ with values in V . We assume f ∈ L2 (Γ, µ; V ).

(2.46)

Multiplying (2.44) by a test function v : Γ → V and integrating over Γ, we formally derive the linear variational problem A(y)u(y), v(y)µ(dy) = f (y), v(y)µ(dy) (2.47) Γ

Γ

as the weak formulation of (2.44). By (2.45) and (2.46), both integrals are well-defined if v ∈ L2 (Γ, µ; V ). Theorem 2.18. Under the conditions (2.45) and (2.46), the solution u of (2.44) is the unique element of L2 (Γ, µ; V ) satisfying (2.47) for all v ∈ L2 (Γ, µ; V ). Furthermore,

u L2 (Γ,µ;V ) ≤ cˇ f L2 (Γ,µ;V ) .

(2.48)

Proof. We first show that there is a unique u ∈ L2 (Γ, µ; V ) such that A(y) u(y), v(y)µ(dy) = f (y), v(y)µ(dy) ∀v ∈ L2 (Γ, µ; V ). (∗) Γ

Γ

By Cauchy–Schwarz and (2.45), for all v, w ∈ L2 (Γ, µ; V ), A(y)w(y), v(y)µ(dy) ≤ c w(y) V v(y) V µ(dy) Γ

Γ

≤ c v L2 (Γ,µ;V ) w L2 (Γ,µ;V ) . Let R : V → V denote the Riesz isomorphism. By positivity of A(y), there is a unique positive √ S(y) ∈ L(V ) such that A(y) = RS(y)S(y) for all c. Furthermore, S(y) is invertible for all y ∈ Γ and y ∈ Γ, and S(y)

≤ √

S(y)−1 ≤ cˇ. Consequently, for all v ∈ L2 (Γ, µ; V ), A(y)v(y), v(y)µ(dy) = S(y)v(y) 2V µ(dy) ≥ cˇ−1 v 2L2 (Γ,µ;V ) . Γ

Γ

Similarly, by Cauchy–Schwarz, f (y), v(y)µ(dy) ≤ f L2 (Γ,µ;V ) v L2 (Γ,µ;V ) Γ

348


for all v ∈ L2 (Γ, µ; V ). The Lax–Milgram lemma implies existence and uniqueness of the solution u of (∗), and (2.48) for u . By (∗) with v(y) = v0 1E (y) for v0 ∈ V and E ∈ B(Γ), A(y) u(y) − f (y), v0 µ(dy) = 0. E

Since this holds for all measurable sets E, the integrand is 0 a.e. in Γ for satisfies (2.44) for µ-a.e. y ∈ Γ. This implies any v0 ∈ V , and therefore u u = u in L2 (Γ, µ; V ). Remark 2.19. (tensor product structure) For any separable Hilbert space X, the Lebesgue–Bochner space L2 (Γ, µ; X) is isometrically isomorphic to the Hilbert tensor product L2 (Γ, µ) ⊗ X. In particular, the solution u of (2.45) can be interpreted as an element of L2 (Γ, µ) ⊗ V , and f can be seen as an element of L2 (Γ, µ) ⊗ V . Theorem 2.18 implies that the stochastic operator A induces an isomorphism between L2 (Γ, µ) ⊗ V and L2 (Γ, µ) ⊗ V , whose inverse maps f onto u. The diffusion equation with a stochastic diffusion coefficient Let D be a bounded Lipschitz domain in Rd , and (Ω, Σ, P) a probability space. We consider as a model problem the isotropic diffusion equation on D with a stochastic diffusion coefficient and, for simplicity, homogeneous Dirichlet boundary conditions, −∇ · a(ω, x)∇U (ω, x) = f (x), x ∈ D, ω ∈ Ω, (2.49) U (ω, x) = 0, x ∈ ∂D, ω ∈ Ω. The differential operators in (2.49) are understood with respect to the physical variable x ∈ D. We assume there are constants a− and a+ such that 0 < a− ≤ a(ω, x) ≤ a+ < ∞ ∀x ∈ D, ∀ω ∈ Ω.

(2.50)

Furthermore, we select some deterministic approximation a ¯ ∈ L∞ (D) to the stochastic diffusion coefficient a(·, ·). For example, a ¯ could be the mean field, a(ω, x) P(dω),

a ¯(x) :=

x ∈ D,

(2.51)

Ω

√ ¯ := a+ a− or a ¯ := 1. or simply a constant such as a ¯ := (a+ + a− )/2, a We consider a series expansion of the difference a(ω, x) − a ¯(x). Let ∞ 2 (ϕm )m∈N ⊂ L (D) be a biorthogonal basis of L (D) with associated dual basis (ϕ m )m∈N ⊂ L2 (D), i.e., n L2 (D) = δmn ϕm , ϕ

∀m, n ∈ N,

(2.52)

349


and v=

∞

v, ϕ m L2 (D) ϕm

∀v ∈ L2 (D),

(2.53)

m=1

with unconditional convergence in L2 (D). For a positive sequence (αm )m∈N , to be determined below, we define the random variables 1 (a(ω, x) − a ¯(x))ϕ m (x) dx, m ∈ N. (2.54) Ym (ω) := αm D By (2.53), for all ω ∈ Ω, a(ω, x) = a ¯(x) +

∞

Ym (ω)αm ϕm (x),

(2.55)

m=1

with unconditional convergence in L2 (D). Lemma 2.20. There is a positive sequence (αm )m∈N such that Ym (ω) ∈ [−1, 1] for all ω ∈ Ω and all m ∈ N. Proof.

By Hölder’s inequality, (a(ω, x) − a ¯(x))ϕ m (x) dx ≤ a(ω, ·) − a m L1 (D) . ¯ L∞ (D) ϕ D

Due to (2.50), the first term is bounded independently of ω, and we can choose αm := sup a(ω, ·) − a ¯ L∞ (D) ϕ m L1 (D) . ω∈Ω

Motivated by Lemma 2.20, we define as a parameter domain the compact topological space ∞ ) [−1, 1]. (2.56) Γ := [−1, 1]∞ = m=1

Let (αm )m∈N be a sequence as in Lemma 2.20. We assume that the series ∞

αm |ϕm (x)|

(2.57)

m=1

converges in L∞ (D), i.e., lim ess sup

M →∞

x∈D

∞

αm |ϕm (x)| = 0.

(2.58)

m=M

Then aϕ (y, x) := a ¯(x) +

∞

m=1

ym αm ϕm (x),

y = (ym )m∈N ∈ Γ,

x ∈ D, (2.59)

350


converges uniformly in L∞ (D), and the stochastic diffusion coefficient satisfies a(ω, x) = aϕ (Y (ω), x)

∀x ∈ D, ∀ω ∈ Ω,

where Y (ω) := (Ym (ω))m∈N ∈ Γ. ¯ Am : H 1 (D) → H −1 (D) by We define the operators A(y), A, 0 aϕ (y, x)∇v(x) · ∇w(x) dx, y ∈ Γ, H −1 A(y)v, wH01 := D ¯ a ¯(x)∇v(x) · ∇w(x) dx, H −1 Av, wH01 := D αm ϕm (x)∇v(x) · ∇w(x) dx, m ∈ N, H −1 Am v, wH01 :=

(2.60)

(2.61) (2.62) (2.63)

D

for all v, w ∈ H01 (D). By (2.60), A(y) is the operator associated to (2.49) for all ω ∈ Ω. Therefore, U (ω) = u(Y (ω))

∀ω ∈ Ω,

(2.64)

for the solution u of (2.44) for (2.61), provided it exists. Lemma 2.21.

Under condition (2.58), A(y) = A¯ +

∞

ym Am ,

y ∈ Γ,

(2.65)

m=1

with convergence in L(H01 (D), H −1 (D)) uniformly in y. Furthermore, A(y) depends continuously on y ∈ Γ. Proof. Let y ∈ Γ and v, w ∈ H01 (D). By (2.59) and Fubini’s theorem, using (2.58), ¯ w + A(y)v, w = Av,

∞

ym Am v, w.

m=1

Similarly, for all M ∈ N, using |ym | ≤ 1 for all m ∈ N, ! ∞ " ∞

y A v, w y α ϕ (x) ∇v(x) · ∇w(x) dx = m m m m m D m=M m=M ! ∞ "

αm |ϕm (x)| v H01 w H01 . ≤ ess sup x∈D

m=M

Convergence of the series in L(H01 (D), H −1 (D)) follows with (2.58).

351


n → y A sequence (y n )n∈N ⊂ Γ converges to y ∈ Γ if ym m for all m ∈ N. n 1 −1 In this case, A(y ) → A(y) in L(H0 (D), H (D)) since, as above, using n − y | ≤ 2, |ym m ∞ ! ∞ "

n (ym − ym )Am ≤ 2 ess sup αm |ϕm (x)| . 1 x∈D −1 m=M

H0 (D)→H

m=M

(D)

The right-hand side is independent of n, and can be made smaller than

for sufficiently large M ∈ N. Then

A(y ) − A(y) H01 (D)→H −1 (D) ≤ + n

M −1

n |ym − ym | Am H01 (D)→H −1 (D) ,

m=1

which is less than 2 for sufficiently large n ∈ N. We assume that the bilinear form associated to the operator A¯ is coercive on H01 (D), or equivalently, that ∃¯ a− : Proposition 2.22.

ess inf a ¯(x) ≥ a ¯− > 0. x∈D

(2.66)

If ∞

γ :=

1 ess sup αm |ϕm (x)| < 1, a ¯− x∈D

(2.67)

m=1

then A(y) :

H01 (D)

→

H −1 (D)

is boundedly invertible for all y ∈ Γ, and

sup A(y)−1 H −1 (D)→H01 (D) ≤ y∈Γ

a ¯−1 − . 1−γ

(2.68)

Furthermore, A(y) is bounded with a L∞ (D) (1 + γ). sup A(y) H01 (D)→H −1 (D) ≤ ¯

(2.69)

y∈Γ

Proof. The operator A¯ : H01 (D) → H −1 (D) is invertible due to (2.66) and the Lax–Milgram lemma. The norm of its inverse is bounded by 1/¯ a− . By (2.67), as in the proof of Lemma 2.21, ∞

1

A¯−1 (A¯ − A(y)) H01 (D)→H01 (D) ≤ ess sup αm |ϕm (x)| = γ < 1. a ¯− x∈D m=1

Therefore, I − A¯−1 (A¯ − A(y)) = A¯−1 A(y) is invertible by a Neumann series and has norm less than (1 − γ)−1 . The ¯ claim follows by multiplying from the left by A.

352


Discretization by the Legendre chaos basis For a separable Hilbert space V , we consider a parametric operator in L(V, V ) of the form A(y) = A¯ +

∞

ym Am ,

y ∈ Γ = [−1, 1]∞ ,

(2.70)

m=1

¯ Am ∈ L(V, V ) and convergence in L(V, V ) uniformly in y ∈ Γ. As with A, in Section 2.3, we assume that A(y) is positive and boundedly invertible for all y, depends continuously on y ∈ Γ, and satisfies (2.45). By Proposition 2.22, this holds for (2.61) under the assumptions of Section 2.3. We make the additional assumption ∞

Am V →V < ∞,

(2.71)

m=1

which is stronger than (2.58) in the setting of Section 2.3. Let the measure µ on (Γ, B(Γ)) be the countable product of uniform measures on [−1, 1] as in (2.41). Then, by Corollary 2.16, the tensor product Legendre polynomials (Lν )ν∈F form an orthonormal basis of L2 (Γ, µ), called the Legendre chaos basis. We use it to discretize the parameter domain Γ, i.e., to reformulate (2.44) and (2.47) as an equation on a space of sequences in V . By Remark 2.19, the parametric operator A(y) induces a boundedly invertible operator between the Hilbert tensor product spaces L2 (Γ, µ) ⊗ V and L2 (Γ, µ) ⊗ V . The structure of (2.70) carries over to this operator. For all m ∈ N, we define the multiplication operator Mym : L2 (Γ, µ) → L2 (Γ, µ),

g(y) → ym g(y).

(2.72)

It follows from ym ∈ [−1, 1] that Mym is self-adjoint and

Mym L2 (Γ,µ)→L2 (Γ,µ) = 1,

m ∈ N.

(2.73)

Proposition 2.23. The operator in L(L2 (Γ, µ)⊗V, L2 (Γ, µ)⊗V ) induced by A(y) via (2.47), as in Remark 2.19, is A = I ⊗ A¯ +

∞

Mym ⊗ Am ,

(2.74)

m=1

L2 (Γ, µ).

The sum in (2.74) converges uncondiwhere I is the identity on tionally in L(L2 (Γ, µ) ⊗ V, L2 (Γ, µ) ⊗ V ). Proof.

The operator A is well-defined by (2.74) since, by (2.73), N N N

Mym ⊗ Am ≤

Mym

Am =

Am , m=M

m=M

m=M

353


which can be made arbitrarily small for sufficiently large M by (2.71). The convergence is unconditional since the convergence of (2.71) is unconditional. Let g ∈ L2 (Γ, µ) and v ∈ V . Then using (2.72), ¯ + A(g ⊗ v)(y) = g(y)Av

∞

ym g(y)Am v = A(y)(g(y)v),

y ∈ Γ.

m=1

Therefore, A is the operator induced by A(y). Since the tensor product Legendre polynomials (Lν )ν∈F form an orthonormal basis of L2 (Γ, µ), the map

c ν Lν , (2.75) TL : 2 (F) → L2 (Γ, µ), (cν )ν∈F → ν∈F

is a unitary isomorphism by Parseval’s identity. Tensorizing with the identity IV on V , we get the isometric isomorphism TL ⊗ IV : 2 (F) ⊗ V → L2 (Γ, µ) ⊗ V

(2.76)

with adjoint (TL ⊗ IV ) = TL ⊗ IV : L2 (Γ, µ) ⊗ V → 2 (F) ⊗ V .

(2.77)

We define the semidiscrete operator A := (TL ⊗ IV ) A(TL ⊗ IV ) : 2 (F) ⊗ V → 2 (F) ⊗ V .

(2.78)

Similarly, interpreting f ∈ L2 (Γ, µ; V ) as an element of L2 (Γ, µ) ⊗ V , we define f (y)Lν (y)µ(dy) , (2.79) f := (TL ⊗ IV ) f = Γ

ν∈F

which is simply the sequence of Legendre coefficients of f . This leads to the semidiscrete operator equation Au = f. Theorem 2.24.

(2.80)

The operator A from (2.78) has the form A = I ⊗ A¯ +

∞

K m ⊗ Am ,

(2.81)

m=1

with convergence in L(2 (F) ⊗ V, 2 (F) ⊗ V ), where I is the identity on 2 (F) and K m := TL Mym TL . Furthermore, A is boundedly invertible, and the solutions of (2.44) and (2.80) are related by u = (TL ⊗ IV )u.

(2.82)

354


Proof. Equation (2.81) follows from (2.74) since TL = TL−1 . The operator A is boundedly invertible since A is boundedly invertible by Remark 2.19 and (TL ⊗ IV ) and (TL ⊗ IV ) are isomorphisms by definition. Applying the inverse of (TL ⊗ IV ) to (2.80) and inserting (2.78) and (2.79), it follows that A(TL ⊗ IV )u = f, which characterizes u by Theorem 2.18. Lemma 2.25.

For all m ∈ N, the operator K m := TL Mym TL : 2 (F) → 2 (F)

(2.83)

K m (cν )ν∈F = (βνm +1 cν+ m + βνm cν− m )ν∈F ,

(2.84)

has the form

where β0 := 0, and βn := √

1 n √ =√ ∈ 2n + 1 2n − 1 4 − n−2

+ 1 1 ,√ , 2 3

n ∈ N.

(2.85)

Here, m is the Kronecker sequence ( m )n := δmn , and if νm = 0, the term cν− m is irrelevant in (2.84) since it is multiplied by β0 = 0. Furthermore, K m is self-adjoint and

K m 2 (F)→2 (F) = 1, Proof.

m ∈ N.

By definition, since TL−1 = TL , TL K m (cν )ν∈F = Mym TL (cν )ν∈F =

(2.86)

cν ym Lν (y).

ν∈F

Therefore, (2.84) is equivalent to ym Lν (y) = βνm +1 Lν+ m (y) + βνm Lν− m (y). By (2.38), ξLn (ξ) = βn+1 Ln+1 (ξ) + βn Ln−1 (ξ),

ξ ∈ [−1, 1],

n ∈ N0 .

Then the √ claim follows from (2.29). Note that (βn )n∈N is decreasing in n, β1 = 1/ 3, and βn → 1/2. Corollary 2.26.

The solution u of (2.44) is

u(y) = uν Lν (y) ∈ V, y ∈ Γ,

(2.87)

ν∈F

with convergence in L2 (Γ, µ; V ), where the coefficients (uν )ν∈F ∈ V are determined uniquely by the equations ¯ ν+ Au

∞

m=1

Am (βνm +1 uν+ m + βνm uν− m ) = fν ,

ν ∈ F,

(2.88)


for (βn )n∈N0 and ( m )m∈N as in Lemma 2.25, and f (y)Lν (y) µ(dy) ∈ V , ν ∈ F. fν :=

355

(2.89)

Γ

Proof. The claim follows from Theorem 2.24 using the definitions (2.75), (2.76), (2.78), (2.79), Lemma 2.25, and the identification of Lebesgue– Bochner spaces with Hilbert tensor product spaces as in Remark 2.19. Finite element approximation The discretization from Section 2.3 does not include any approximations. The infinite system of equations in Corollary 2.26 determines the Legendre coefficients (uν )ν∈F ∈ V of the exact solution u of (2.44). However, this system of equations lends itself to discretization by standard finite elements. For all ν ∈ F, let VN,ν ⊂ V be a finite-dimensional space. We assume that VN,ν = {0} for all but finitely many ν ∈ F and define the finite-dimensional space VN := {v ∈ L2 (Γ; V ) ; vν ∈ VN,ν ∀ν ∈ F}, (2.90) where vν ∈ V is the νth coefficient in the expansion of v ∈ L2 (Γ; V ) with respect to the tensor product Legendre polynomials (Lν )ν∈F . This space can be interpreted as a subspace of L2 (Γ; V ), as in (2.90), or as the space of sequences (vν )ν∈F in V with vν ∈ VN,ν for all ν ∈ F, which is a subspace of 2 (F; V ). By Parseval’s identity, the norms induced by these two spaces coincide. Accordingly, the Galerkin projection of u onto VN can be characterized in two equivalent ways. We define the Galerkin approximation uN of u on VN as the unique element of VN satisfying A(y)uN (y), vN (y)µ(dy) = f (y), vN (y)µ(dy) ∀vN ∈ VN . (2.91) Γ

Γ

As in Corollary 2.26, the Legendre coefficients of uN satisfy a system of equations ¯ N,ν , vN + Au

∞

Am (βνm +1 uN,ν+ m +βνm uN,ν− m ), vN = fν , vN (2.92)

m=1

for all vN ∈ VN,ν and all ν ∈ F. Since VN,ν = {0} for all but finitely many ν ∈ F, there are only finitely many non-trivial equations (2.92). Also, for the same reason, the sum in each equation is finite. Therefore, without any explicit truncation, the infinite system of equations (2.88) becomes a finite system when considered on a finite-dimensional space. Proposition 2.27. The Galerkin projection uN of u onto VN is welldefined by (2.91), and satisfies

uN L2 (Γ,µ;V ) ≤ cˇ f L2 (Γ,µ;V ) .

(2.93)

356


Its Legendre coefficients (uN,µ )µ∈F are uniquely characterized by (2.92) for ν ∈ F. Furthermore, √

u − uN L2 (Γ,µ;V ) ≤ ccˇ inf u − vN L2 (Γ,µ;V ) . (2.94) vN ∈VN

Proof. As shown in the proof of Theorem 2.18, the bilinear form in (2.91) is continuous with constant c and coercive with constant cˇ−1 . Existence and uniqueness of uN as well as (2.93) follow from the Lax–Milgram lemma applied to the space VN . The equivalence of (2.91) and (2.92) is a consequence of Theorem 2.24, using Lemma 2.25. The quasi-optimality property (2.94) holds since the bilinear form in (2.91) is a scalar product, and the 2 norm induced √ by it√on L (Γ, µ; V ) is equivalent to the standard norm with c and cˇ. constants Given a space VN , the Galerkin projection uN of u onto VN can be computed iteratively, for example by a conjugate gradient iteration; see Gittelson (2011b). The inverse of the deterministic operator A¯ can be used as a preconditioner. The sparse tensor product construction of VN , which amounts to a problem-adapted selection of finite element spaces VN,ν , is discussed in Section 4.1. Approximation results are presented in Section 3.1. 2.4. PDEs with Gaussian parameters The log-normal diffusion equation We consider again the diffusion equation (2.49) with a stochastic diffusion coefficient a(·, ·) on a bounded Lipschitz domain D ⊂ Rd . However, instead of expanding a(·, ·) in a series, we expand its logarithm. More precisely, we take a series expansion of log(a − a∗ ), where a∗ is a bounded function on D with a∗ (x) ≥ 0 for all x ∈ D. Then, instead of (2.59), we have a diffusion coefficient of the form ! ∞ "

a(y, x) = a∗ (x) + a0 (x) exp ym am (x) , x ∈ D, (2.95) m=1

R∞ .

We assume that the coefficients (ym )m∈N are for y = (ym )m∈N ∈ independent standard Gaussian random variables. This is the case, for instance, if log(a − a∗ ) is Gaussian and we expand it in its Karhunen–Loève series, or more generally if (am )m∈N are orthonormal in the Cameron–Martin space of the distribution of log(a−a∗ ): see Section 2.1 and Gittelson (2010b). The diffusion equation with the stochastic coefficient (2.95) and, for simplicity, homogeneous Dirichlet boundary conditions, is −∇ · a(y, x)∇u(y, x) = f (y, x), x ∈ D, (2.96) u(y, x) = 0, x ∈ ∂D.

357


By the above assumptions, the parameter y is in the probability space (R∞ , B(R∞ ), γ), where γ=

∞ '

N1 ,

(2.97)

m=1

as in (2.4). For the sake of generality, we allow the forcing term f in (2.96) to depend on y ∈ R∞ . The series in (2.95) may not converge for all y ∈ R∞ . We assume that am ∈ L∞ (D) for all m ∈ N0 , a0 (x) ≥ a ˇ0 > 0 for all x ∈ D, and ∞

am L∞ (D) < ∞,

(2.98)

m=1

i.e., the sequence αm := am L∞ (D) , m ∈ N, is in 1 (N). Then the series in (2.95) converges in L∞ (D), at least for all y in the set , ∞

∞ Γ := y ∈ R ; αm |ym | < ∞ . (2.99) m=1

Lemma 2.28. Proof.

Γ ∈ B(R∞ ) and γ(Γ) = 1.

Borel-measurability of Γ follows from Γ=

∞ / ∞ , .

∞

y∈R

;

M

αm |ym | ≤ N .

m=1

N =1 M =1

By the monotone convergence theorem, using 0 2 ∞ 2 2 ξ dξ = , |ym |γ(dy) = √ ξ exp − 2 π 2π 0 R∞ it follows that ∞ ∞

αm |ym |γ(dy) = αm R∞ m=1

m=1

0 R∞

|ym |γ(dy) =

∞ 2 αm < ∞, π m=1

which implies that the sum converges γ-a.e. on R∞ , and thus γ(Γ) = 1 by (2.99). Lemma 2.29. and satisfies

For all y ∈ Γ, the diffusion coefficient (2.95) is well-defined

0 1 for all j ≥ 1, i.e., such that the polydisc U is strictly larger than U in every variable. These increasing δ-admissible sequences will next be exploited to obtain bounds on Taylor and Legendre coefficients by shifting paths of integration in the complex domain into the polydiscs U . Estimates of Taylor coefficients Estimates for the Taylor coefficients tν for ν ∈ F are given by the following result. Lemma 3.5. If UEAC(r, R) holds for some 0 < r ≤ R < ∞ and if = (j )j≥1 is a δ-admissible sequence for some 0 < δ < r, then for any ν ∈ F we have the estimate

f V ) −νj

f V −ν j = (3.27)

tν V ≤ , δ δ j≥1

where we use the convention that t−0 = 1 for any t ≥ 0. To prove Lemma 3.5, let ν = (νj )j≥1 ∈ F and J = max {j ∈ N ; νj = 0}. For J and for z ∈ U , we define the set EJ = {1, . . . , J} and the parameter vector zEJ obtained from z by setting to 0 all entries zj for j > J. We then have ∂ |ν| uJ ∂ ν u(0) = ν1 (0, . . . , 0). ∂z1 · · · ∂zJνJ From the assumption that is δ-admissible, we have that

f V , δ for all (z1 , . . . , zJ ) in the J-dimensional polydisc ) U,J := {zj ∈ C ; |zj | ≤ j }.

uJ (z1 , . . . , zJ ) V ≤

1≤j≤J

(3.28)

(3.29)

375


Introducing the sequence defined by j = j + ε if j ≤ J,

j = j if j > J,

ε :=

2

δ , |ψ j | L∞ (D) j≤J

it is easily checked that is 2δ -admissible and therefore U ⊂ Aδ/2 . We infer from Lemma 3.4 that for each z ∈ U, u is holomorphic in each variable zj . function with Therefore uJ is strongly holomorphic as a V -valued function 6 respect to each of the variables z1 , . . . , zJ on the polydisc 1≤j≤J {|zj | < j }. This polydisc is an open neighbourhood of U,J . In this disc, we apply a suitable version of Cauchy’s integral formula (e.g., Theorem 2.1.2 of Hervé (1989)) with respect to each zj , and write z1 , . . . , zJ ) uJ ( = (2πi)−J

|z1 |=1

···

|zJ |=J

uJ (z1 , . . . , zJ ) dz1 · · · dzJ . ( z1 − z1 ) · · · ( zJ − z J )

Differentiating this expression with respect to zj , we find ∂ |ν| uJ (0, . . . , 0) ∂z1ν1 · · · ∂zJνJ −J ··· = ν!(2πi) |z1 |=1

|zJ |=J

uJ (z1 , . . . , zJ ) dz1 · · · dzJ , z1ν1 · · · zJνJ

and therefore, using (3.28), we obtain the estimate |ν| ) −νj ν∂ uJ ν (0, . . . , 0) ≤ ν! f V j , ∂z 1 · · · ∂z J δ 1 V J j≤J

which is equivalent to (3.27).

Proof of Theorem 3.2 With the analyticity of the mapping z → u(z) on the domains Aδ in hand, the proof of Theorem 3.2 under the uniform ellipticity assumption UEAC(r, R) involves two steps: (a) a particular choice of r/2-admissible sequences , and (b) establishing p (F)-summability of the Taylor coefficient sequence. With δ = r/2, (3.27) of Lemma 3.5 reads

tν V ≤

2 f V ) −νj 2 f V −ν . j = r r

(3.30)

j≥1

There are many sequences that are δ-admissible. We now indicate one such choice from Cohen et al. (2011), which is ν-dependent, in order to

376


yield possibly sharp coefficient bounds. We begin our choice by selecting J0 ∈ N so large that

r (3.31)

ψj L∞ (D) ≤ , 12 j>J0

Such a J0 exists under the assumptions of Theorem 3.2 because ( ψj L∞ (D) )j≥1 ∈ p (N) ⊂ 1 (N). Without loss of generality, the basis elements ψj of the sequence are assumed to be enumerated in such a way that ( ψj L∞ )j≥1 is non-increasing. To construct a δ = r/2 admissible vector of weights, we partition N into two sets E := {1 ≤ j ≤ J0 } and F := N\E. Next we choose κ > 1 such that

r (3.32)

ψj L∞ (D) ≤ . (κ − 1) 4 j≤J0

For each multi-index ν ∈ F we select = (ν) by , rνj j := κ, j ∈ E; j := max 1, , 4|νF | ψj L∞ (D)

j ∈ F.

(3.33)

Here, νE denotes the restriction of ν to a set E and |νF | := j>J0 νj . We ν also make the convention that |νFj | = 0 when |νF | = 0. It can be verified that the sequence defined in (3.33) is 2r -admissible (see Cohen et al. (2011) for details). The general bound (3.30) is in particular valid for this sequence , for which it takes the form (with the convention that a factor equals 1 if νj = 0) ! "! " ) |νF |dj νj 2 f V ) νj , (3.34)

tν V ≤ η r νj j∈E

where η :=

1 κ

j∈F

< 1 and dj :=

4 ψj L∞ . r

We note that from (3.31)

d 1 =

j>J0

1 dj ≤ . 3

(3.35)

To prove p (F) summability of the Taylor coefficients tν , we observe that the estimate (3.34) has the general form

tν V ≤ Cr α(νE ) β(νF ).

(3.36)

377


We let FE (respectively FF ) be the collection of ν ∈ F supported on E (respectively on F ). Then, for any 0 < p < ∞, we have

tν pV ≤ Crp α(νE )p β(νF )p = Crp AE AF , (3.37) ν∈F

ν∈F

where

!

AE :=

" p

! ,

α(ν)

AF :=

ν∈FE

" p

β(ν)

The first factor AE is estimated as follows: ! "

) ) α(ν)p = η pνj = η np = AE = ν∈FE j∈E

ν∈FE

.

ν∈FF

j∈E

n≥0

1 1 − ηp

J0

< ∞. (3.38)

To show that AF is finite, we observe ) |νF |dj νj |νF ||νF | νF ≤6 β(ν) := νj d , νj j∈F νj where dνF =

ν ∈ FF ,

(3.39)

j∈F

6

ν

j∈F

dj j and 00 := 1. By Stirling estimates, n!en n!en √ ≤ nn ≤ √ √ , e n 2π n

(3.40)

which hold for all n ≥ 1, we obtain |νF ||νF | ≤ |νF |!e|νF | . On the other hand, using the left inequality in (3.40), we obtain )

ν νj j ≥ 6 j∈F

j∈F

νF !e|νF | . √ max{1, e νj }

With these estimates we obtain from (3.39) that |νF |! ¯νF |νF |! νF ) √ max{1, e νj } ≤ β(ν) ≤ d d , νF ! νF !

(3.41)

j∈F

¯ 1 = e d 1 ≤ where d¯j := edj , j ∈ F . We conclude by noticing that d

e p ¯ 3 < 1. Since d is (N) summable, we may apply Theorem 3.1 to conclude the p (F) summability of tν . With the p (F) summability of tν , the best N -term convergence rate estimate (3.18) follows from (3.13). . Convergence rates of Legendre expansions The analyticity result Theorem 3.2 contains, as a special case, the rate of convergence of best N -term truncations of Taylor expansions in the

378


stochastic coordinates yj . This result and the proof, however, also allow us to obtain corresponding results for Legendre expansions, as we shall show next. We shall obtain bounds in L2 and pointwise bounds in the parameter vector y. To this end, it will be convenient to introduce two types of Legendre expansions with different normalization of the Legendre basis: the Legendre basis (Pn )n≥0 with L∞ normalization

Pn L∞ ([−1,1]) = Pn (1) = 1 (3.42) √ and the L2 normalized sequence Ln (t) = 2n + 1Pn (t), which satisfies 1 dt |Ln (t)|2 = 1. 2 −1 We recall that L0 = P0 = 1 and, for ν ∈ F, ) ) Pνj (yj ) and Lν (y) := Lνj (yj ). Pν (y) := j≥1

(3.43)

j≥1

Note that (Lν )ν∈F is an orthonormal basis of L2 (Γ, µ) where dµ denotes the dy tensor product of the (probability) measures 2 j on [−1, 1] and is therefore a probability measure on Γ = [−1, 1]N . Since u ∈ L∞ (Γ, µ; V ) ⊂ L2 (Γ, µ; V ), it admits unique expansions

u(y) = uν Pν (y) = vν Lν (y), (3.44) ν∈F

ν∈F

that converge in L2 (Γ, µ; V ), where the coefficients uν , vν ∈ V are defined by ! "1/2 ) u(y)Lν (y)µ(dy) and uν := (1 + 2νj ) vν . (3.45) vν := Γ

j≥1

Once again, the key step in establishing sharp rates of convergence of best N -term Legendre GPC approximations of the solution of the parametric, deterministic problem are sharp a priori bounds on the Legendre coefficients. In order to prove their p (F) summability, we estimate the quantities uν V and vν V . By (3.45), ! "1 2 ) (1 + 2νj ) vν V , ν ∈ F. (3.46)

uν V = j≥1

Therefore vν V ≤ uν V and it will be sufficient to prove the p summability of ( uν V )ν∈F . We have the following analogue to Lemma 3.5 from Cohen et al. (2011).

379


Lemma 3.6. Assume that UEAC(r, R) holds for some 0 < r ≤ R < ∞. Let = (j )j≥1 be a δ-admissible sequence for some 0 < δ < r that satisfies j > 1 for all j such that νj = 0. Then, for any ν ∈ F we have the estimate

f V ) −ν ϕ(j )(2νj + 1)j j , (3.47)

uν V ≤ δ j≥1,νj =0

where ϕ(t) :=

πt 2(t−1)

for t > 1.

Based on Lemma 3.6 and a judicious choice of δ-admissible sequence, the following theorem was shown in Cohen et al. (2011). It is the analogue to Theorem 3.2 for Legendre GPC expansions. Theorem 3.7. If a(x, z) satisfies UEAC(r, R) for some 0 < r ≤ R < ∞ and if ( ψj L∞ )j≥1 ∈ p (N) for some p < 1, then the sequences ( uν V )ν∈F and ( vν V )ν∈F belong to p (F) for the same value of p. The Legendre expansions (3.44) converge in L∞ (Γ, µ; V ) in the following sense. If (ΛN )N ≥1 is any sequence of finite sets which exhausts F, then the partial sums SΛN u(y) := ν∈ΛN uν (x)Pν (y) = ν∈ΛN vν (x)Lν (y) satisfy lim sup u(y) − SΛN u(y) V = 0.

(3.48)

N →+∞ y∈Γ

If ΛN is a set of ν ∈ F corresponding to indices of N maximal uν V , sup u(y) − SΛN u(y) V ≤ ( uν V ) p (F) N −s ,

s :=

y∈Γ

1 − 1. p

(3.49)

If ΛN is a set of ν ∈ F corresponding to indices of N maximal vν V ,

u − SΛN u L2 (Γ,µ;V ) ≤ ( vν V ) p (F) N −s ,

s :=

1 1 − . p 2

(3.50)

Spatial regularity and finite element discretization So far, we have considered approximations of u(y) with respect to the parameter vector y ∈ Γ under the assumption that the coefficients tν , vν and uν could be obtained exactly. In practice, however, such coefficients must be approximated by a discretization scheme such as the finite element method. An additional discretization error arises in doing so which can be analysed using standard convergence results for finite element approximations. It is interesting to note that the regularity required of the solution u(y) must then involve both smoothness in the stochastic parameter vector y and in the spatial domain D. We will now present some convergence results of this type. We assume that D is a bounded Lipschitz polyhedron D and that in D we are given a one-parameter, affine family of continuous, piecewise linear finite element spaces (Vh )h>0 on a shape-regular family of simplicial triangulations of mesh width h > 0 in the sense of Ciarlet (1978).

380


To obtain regularity of the parametric solution u(y) in D, additional regularity on f is required: we shall assume f ∈ L2 (D) ⊂ V . Then

f V ≤ CP f L2 (D) ,

(3.51) √

where CP is the Poincaré constant of D (i.e., CP = 1/ λ1 with λ1 being the smallest eigenvalue of the Dirichlet Laplacian in D). Then the smoothness space W ⊂ V is the space of all solutions to the Dirichlet problem −∆u = f

u|∂D = 0,

(3.52)

W = {v ∈ V ; ∆v ∈ L2 (D)}.

(3.53)

in D,

with f ∈ L2 (D) We define the W -seminorm and the W -norm by |v|W = ∆v L2 (D) ,

v W := v V + |v|W .

(3.54)

It is well known that W = H 2 (D) ∩ V for convex D ⊂ Rd . Then any w ∈ W may be approximated in V with convergence rate O(h) by continuous, piecewise linear finite element approximations on regular quasi-uniform simplicial partitions of D of mesh width h (see, e.g., Ciarlet (1978), Braess (2007), Brenner and Scott (2002)). Therefore, denoting by M = dim(Vh ) ∼ h−d the dimension of the finite element space, we have for all w ∈ W the convergence rate, as M = dim(Vh ) → ∞, of inf w − vh V ≤ Ct M −t |w|W ,

vh ∈Vh

(3.55)

with some 0 < t ≤ 1/d (with t = 1/d if W ⊂ H 2 (D)). Spatial regularity of the parametric solution u(y) now takes the form of p-summability of W -norms of the tν , uν and vν . In addition to the requirement f ∈ L2 (D), we add a fourth assumption to conditions (C1)–(C3) on the coefficient a(z, x). (C4) The gradients of the functions a ¯ and ψj , for j ≥ 1, are defined for ∞ every x ∈ D and belong to L (D). Then the following regularity holds (see Cohen et al. (2011)). Theorem 3.8. Let f ∈ L2 (D) and let a(z, x) satisfy UEAC(r, R) for some 0 < r ≤ R < ∞. If ( ψj L∞ (D) )j≥1 ∈ p (N) and ( ∇ψj L∞ (D) )j≥1 ∈ p (N) for some 0 < p < 1, then ( tν W )ν∈F , ( uν W )ν∈F and ( vν W )ν∈F belong to p (F). We now obtain bounds on the convergence rates of the fully discrete approximation of u by linear combinations

u ν Pν (y), or vν Lν (y), tν y ν , ν∈Λ

ν∈Λ

ν∈Λ


381

where Λ ⊂ F is finite and when the coefficients tν , u ν and vν are finite element approximations of tν , uν and vν , respectively, from finite element spaces (Vν )ν∈Λ . For efficiency of approximation as well as in the specific sparse tensor approximation schemes, it will be crucial that for given ν ∈ Λ ⊂ F, the approximation space Vν may depend on ν. To this end, we introduce the vector M = (Mν )ν∈Λ of the dimensions Mν = dimVν , ν ∈ Λ, of the finite element approximation spaces Vν used for approximating the tν . Without loss of generality, we may assume that the error bound (3.55) holds for all such M up to increasing Ct , and express the approximation rate in terms of Ndof , i.e., the total number of degrees of freedom involved :

Mν . (3.56) Ndof := ν∈Λ

Then we may estimate

supu(y) −

tν − tν V +

tν V . tν y ν ≤ y∈Γ ν∈Λ

V

(3.57)

ν ∈Λ /

ν∈Λ

The first term on the right-hand side of (3.57) corresponds to the error occurring from the finite element discretization of the tν ; the second term on the right-hand side corresponds to the error incurred by truncating the Taylor series. By taking Λ := ΛN , the set of indices corresponding to N maximal tν W , it is bounded by

tν W ≤ CV N −s ,

ν ∈Λ /

s :=

1 − 1. p

The global error can then be bounded by

ν Mν−t |tν |W + CV N −s . tν y ≤ C t supu(y) − y∈Γ ν∈ΛN

W

(3.58)

(3.59)

ν∈ΛN

We now have an optimization problem: minimize the degrees of freedom Mν such that Ndof is minimized for a fixed contribution Ct ν∈ΛN Mν−t |tν |W to the error, i.e., we consider the minimization

(3.60) min Mν ; Mν−t |tν |W ≤ N −s . ν∈ΛN

ν∈ΛN

In Cohen et al. (2011), this minimization problem is solved and the following result is obtained. To state it, we let tν , u ν and vν denote the V -projection of tν , uν and vν , respectively, onto Vν .

382


Theorem 3.9. Assume that the finite element spaces have the approximation property (3.55). Then, under the same assumptions as in Theorem 3.8, the following hold. (a) Let ΛN be a set of indices corresponding to N maximal tν W . Then there exists a choice of finite element spaces Vν of dimension Mν , ν ∈ ΛN , such that

1 − min{s,t} ν supu(y) − , s := − 1, tν y ≤ CNdof p y∈Γ ν∈ΛN V where Ndof = ν∈ΛN Mν , C = (C¯t + ( tν V ) p (F) ) ( tν W ) p (F) . (b) Let ΛN be a set of indices corresponding to N maximal uν W . Then there exists a choice of finite element spaces Vν of dimension Mν , ν ∈ ΛN , such that

1 − min{s,t} supu(y) − u ν Pν (y) ≤ CNdof , s := − 1, p y∈Γ ν∈ΛN V where Ndof = ν∈ΛN Mν , C = (C¯t + ( uν V ) p (F) ) ( uν W ) p (F) . (c) Let ΛN be a set of indices corresponding to N maximal vν W . Then there exists a choice of finite element spaces Vν of dimension Mν , ν ∈ ΛN , such that

1 1 − min{s,t} vν Lν ≤ CNdof , s := − , u − 2 p 2 ν∈ΛN

where Ndof =

L (Γ,µ;V )

ν∈ΛN Mν ,

1 C = (C¯t2 + ( vν V ) 2p (F) ) 2 ( vν W ) p (F) .

3.2. Parabolic problems A class of random parabolic problems For 0 < T < ∞, we consider in the bounded time interval I = (0, T ) linear, parabolic initial boundary value problems with random coefficients where, for ease of exposition, we assume that these coefficients are independent of t. We still denote by D ⊂ Rd a bounded Lipschitz domain and we denote the associated space–time cylinder by QT = I × D. In QT , we consider the random parabolic initial boundary value problem ∂u − ∇ · (a(x, ω)∇u) = g(t, x), u|∂D×I = 0, ∂t As before, we make the following assumption. Assumption 3.10.

u|t=0 = h(x).

(3.61)

There exist constants 0 < a− ≤ a+ < ∞ such that

∀x ∈ D, ∀ω ∈ Ω :

0 < a− ≤ a(x, ω) ≤ a+ .


383

It will be convenient to impose a stronger requirement. Assumption 3.11.

The functions a ¯ and ψj satisfy

κ a ¯− ,

ψj L∞ (D) ≤ 1+κ j≥1

¯(x) > 0 and κ > 0. with a ¯− = minx∈D a Assumption 3.10 is then satisfied by choosing ¯− − a− := a

κ 1 a ¯− = a ¯− . 1+κ 1+κ

(3.62)

We consider a space–time variational formulation of problem (3.61). To state it, we denote by V = H01 (D) and H = L2 (D) and identify H with its dual: H H . Then V ⊂ H H ⊂ V = H −1 (D) is a Gelfand evolution triple. For the variational formulation of (3.61), we introduce the Bochner spaces X = L2 (I; V ) ∩ H 1 (I; V )

and

Y = L2 (I; V ) × H.

(3.63)

We equip X and Y with norms · X and · Y , respectively, which are for u ∈ X and v = (v1 , v2 ) ∈ Y given by

u X = ( u 2L2 (I;V ) + u 2H 1 (I;V ) )1/2 and v Y = ( v1 2L2 (I;V ) + v2 2H )1/2 . Given a coefficient realization a(ω, ·) with ω ∈ Ω, a weak solution of problem (3.61) is a function u(·, ·, ω) ∈ X such that du , v1 dt + a(x, ω)∇u(t, x, ω) · ∇v1 (t, x) dx dt + u(0, ·, ω), v2 H I dt I D H = g(t, ·), v1 dt + h, v2 H , ∀v ∈ Y. (3.64) I

The following proposition from Schwab and Stevenson (2009) guarantees its well-posedness for all ω ∈ Ω, under Assumption 3.10. Proposition 3.12. Assume that g ∈ L2 (I; V ), h ∈ L2 (D) and that Assumption 3.10 holds. Then, for every ω ∈ Ω, the parabolic operator B ∈ L(X , Y ) induced by (3.61) in the weak form (3.64) is an isomorphism: for given (g, h) ∈ Y and every ω ∈ Ω, problem (3.64) has a unique solution u(·, ·, ω), which satisfies the a priori estimate

u X ≤ C g L2 (I;V ) + h L2 (D) , (3.65) where the constant C is bounded uniformly for all realizations.

384


As before, we assume that the coefficient a in (3.61) is characterized by a sequence of random variables (yj )j≥1 , i.e., that

yj (ω)ψj (x). (3.66) a(x, ω) = a ¯(x) + j≥1

We again assume that the ψj are scaled in L∞ (D) such that yj : Ω → R, j = 1, 2, . . . are distributed identically and uniformly, and that the ψj are scaled in L∞ (D) such that the range of the Yj is [−1, 1] (see Lemma 2.20). Parametric deterministic parabolic problems As before, with (3.61) we associate the following parametric family of deterministic parabolic problems: given a source term g(t, x) and initial data h(x), for y ∈ Γ, find u(t, x, y) such that ∂u (t, x, y) − ∇x · [a(x, y)∇x u(t, x, y)] = g(t, x), ∂t u(t, x, y)|∂D×I = 0, u|t=0 = h(x),

in QT ,

(3.67)

where, for every y = (y1 , y2 , . . .) ∈ Γ, a(x, y) = a ¯(x) +

∞

yj ψj (x)

j=1

in L∞ (D). For the weak formulation of (3.67), we follow (3.64) and define for y ∈ Γ the parametric family of bilinear forms Γ y → b(y; w, (v1 , v2 )) : X × Y → R by dw b(y; w, (v1 , v2 )) = (3.68) , v1 (t, ·) dt dt I H a(x, y)∇w(t, x) · ∇v1 (t, x) dx dt + w(0), v2 H . + D

I

We also define the linear form f (v) = g(t), v1 (t)H dt + h, v2 H , v = (v1 , v2 ) ∈ Y.

(3.69)

I

The variational form for the parametric, deterministic parabolic problems (3.67) reads: given f ∈ Y , find u(y) : Γ y → X such that b(y; u, v) = f (v)

∀v = (v1 , v2 ) ∈ Y, y ∈ Γ.

(3.70)

Proposition 3.13. For each y ∈ Γ, the operator B(y) ∈ L(X , Y ) defined by (B(y)w)(v) = b(y, w, v) is boundedly invertible. The norms of B(y) and B(y)−1 can be bounded uniformly by constants which only depend on a− , a+ , T and the spaces X and Y. In particular, the solution u of problem (3.70) is uniformly bounded in X for all y ∈ Γ. Moreover, the map u(·, ·, y) : Γ → X is measurable as a Bochner function.


385

The proof of this theorem can be found in Appendix A of Schwab and Stevenson (2009), or in Hoang and Schwab (2010b). The variational formulation (3.70) is pointwise in the parameter and is the basis for sampling methods, such as the Monte Carlo method. For stochastic Galerkin approximation, (3.70) needs to be extended to the parameter space. To this end, we introduce Bochner spaces X = L2 (Γ, µ; X )

and

Y = L2 (Γ, µ; Y)

and note X L2 (Γ, µ) ⊗ X ,

Y L2 (Γ, µ) ⊗ Y.

With the bilinear form B(·, ·) : X ×Y → R and the linear form F (·) : Y → R defined by b(y, u, v)µ(dy) and F (v) = f (v)µ(dy). (3.71) B(u, v) = Γ

Γ

we consider the variational problem: find u∈X

such that B(u, v) = F (v)

for all v ∈ Y.

(3.72)

The Galerkin formulation is well-posed, as we see in the next result (see Hoang and Schwab (2010b)). Proposition 3.14. Under Assumption 3.10, for every f as in (3.69) with g ∈ L2 (I; V ) and h ∈ L2 (D), the parametric deterministic variational problem (3.72) admits a unique solution u ∈ X . Since the family {Lν }ν∈F of tensor product polynomials forms a complete orthonormal system of L2 (Γ, µ), each u ∈ X can be represented as

u= uν Lν , (3.73) ν∈F

where the coefficients uν ∈ X are defined by u(·, ·, y)Lν (y)µ(dy) ∈ X , uν = Γ

the integral being understood as a Bochner integral of X -valued functions over Γ. With this result, we recover from the parametric, deterministic solution the random solution u(t, x, ω) by inserting the random variables Ym (ω) for the coordinate vector y ∈ Γ. Theorem 3.15. Under Assumptions 3.10, 3.11, for given g ∈ L2 (I; V ) and h ∈ H, the following variational problem admits a unique solution.

386


Find u ∈ L2 (Ω; X ) such that, for every v(t, x, ω) = (v1 (t, x, ω), v2 (x, ω)) ∈ L2 (Ω; Y), , du (t, ·, ·), v1 (t, ·, ·) dt E I dt H , a(x, ω)∇u(t, x, ω) · ∇v1 (t, x, ω) dx dt +E ,I D +E u(0, x, ω)v2 (x, ω) dx D , , =E g(t, x)v1 (t, x, ω) dx dt + E h(x)v2 (x, ω) dx . (3.74) I

D

D

This unique solution satisfies the a priori estimate

u L2 (Ω;X ) ≤ C(a) g L2 (I;V ) + h H .

(3.75)

Galerkin approximation As in the elliptic case, we obtain GPC approximations by Galerkin projections onto suitable spaces of polynomials in y ∈ Γ. For every finite subset Λ ⊂ F of cardinality not exceeding N , we define spaces of X - and Y-valued polynomial expansions ,

X Λ = uΛ (t, x, y) = uν (t, x)Lν (y) ; uν ∈ X ⊂ X , ν∈Λ

and

, YΛ =

vΛ (t, x, y) =

vν (t, x)Lν (y) ; vν ∈ X

⊂ Y.

ν∈Λ

In the Legendre basis (Lν )ν∈F , we write

v1ν (t, x)Lν (y) and v1Λ (t, x, y) =

v2Λ (x, y) =

ν∈Λ

v2ν (x)Lν (y),

ν∈Λ

respectively, where vν = (v1ν , v2ν ) ∈ Y for all ν ∈ F. We consider the (semidiscrete) Galerkin approximation: find uΛ ∈ X Λ

such that B(uΛ , vΛ ) = F (vΛ )

∀ vΛ ∈ Y Λ .

(3.76)

Theorem 3.16. For any finite subset Λ ⊂ F of cardinality exactly equal to N , the problem (3.76) corresponds to a coupled system of N = #Λ linear parabolic equations. Under Assumptions 3.10, 3.11, this coupled system of parabolic equations is stable uniformly with respect to Λ ⊂ F: for any Λ ⊂ F, problem (3.76) admits a unique solution uΛ ∈ X Λ which satisfies

387


the a priori error bound !

u − uΛ X ≤ c

"1/2

uν 2X

.

ν ∈Λ /

Here, uν ∈ X are the Legendre coefficients of the solution of the parametric problem in (3.73) and the constant c is independent of Λ. Proof. To prove the uniform well-posedness of the coupled parabolic system resulting from the Galerkin discretization in Γ, we prove that the following inf-sup condition holds: there exist α, β > 0 such that for any Λ ⊂ F, sup

uΛ ∈X Λ ,vΛ ∈Y Λ

sup

inf

0=uΛ ∈X Λ 0=vΛ ∈Y Λ

∀0 = vΛ ∈ Y Λ :

|B(uΛ , vΛ )| ≤ α < ∞,

uΛ X vΛ Y

(3.77)

|B(uΛ , vΛ )| ≥ β > 0,

uΛ X vΛ Y

(3.78)

|B(uΛ , vΛ ))| > 0,

(3.79)

sup

0=uΛ ∈X Λ

where the constants α, β are in particular independent of the choice of Λ ⊂ F (a proof can be found in the Appendix of Hoang and Schwab (2010b)). The projected parametric deterministic parabolic problem (3.76) has a unique solution, and, in virtue of the independence of α, β from Λ, is wellposed and stable with stability bounds which are independent of the choice of Λ ⊂ F. Hence, the error incurred by this projection is quasi-optimal:

u − uΛ X ≤ (1 + β −1 ( g L2 (I;V ) + h L2 (D) )) inf u − vΛ X v ∈X Λ

≤ cu − uν Lν = c uν Lν . ν∈Λ

X

ν ∈Λ /

X

By the normalization of the tensorized Legendre polynomials Lν and by Parseval’s equality, 2

uν Lν =

uν 2X . ν ∈Λ /

X

ν ∈Λ /

The conclusion then follows with c = 1 + β −1 ( g L2 (I;V ) + h L2 (D) ). Best N -term GPC approximations As in the elliptic case, Theorem 3.16 again suggests choosing Λ ⊂ F to be the set of the largest N coefficients uν X . Once (sharp and computable) a priori bounds for uν in X are known, one algorithmic strategy could be to optimize the sets Λ ⊂ F according to these a priori bounds (one such strategy is outlined in Section 4.1 for the elliptic case). Alternatively,

388


an optimal, adaptive Galerkin method will yield iteratively quasi-optimal sequences ΛN of active indices. We now determine such a priori bounds. A best N -term convergence rate estimate in terms of N will result from these bounds once more using (3.13). Therefore, the convergence rate of spectral approximations such as (3.76) of the parabolic problem on the infinite-dimensional parameter space Γ is determined by the summability of the Legendre coefficient sequence ( uν X )ν∈F . We shall now prove that summability of this sequence is determined by that of the sequence (ψj (x))j∈N in the input’s fluctuation expansion (3.66). Throughout, Assumptions 3.10 and 3.11 will be required to hold. In addition, we shall make the following requirement. Assumption 3.17.

There exists 0 < p < 1 such that ∞

ψj pL∞ (D) < ∞.

(3.80)

j=1

Based on this assumption, in Hoang and Schwab (2010b) the following result was proved, with a proof along similar lines to that of the elliptic result Theorem 3.7. Theorem 3.18. If Assumptions 3.10, 3.11 and 3.17 hold for some 0 < p < 1, ν∈F uν pX is finite. Moreover, there is a sequence (ΛN )N ∈N ⊂ F of index sets with cardinality not exceeding N such that the solutions uΛN of the Galerkin semidiscretized problems (3.76) satisfy

u − uΛN X ≤ CN −σ ,

σ=

1 1 − . p 2

This establishes a rate of convergence for best N -term GPC approximations in the stochastic Galerkin semidiscretization (3.76), which is analogous to the estimate (3.50) for the parametric elliptic problem. Note that Theorem 3.18 holds in the semidiscrete setting, i.e., under the assumption that the Galerkin projections (3.76) can be computed exactly. To obtain actually computable realizations of such approximations, however, the coefficients need to be approximated in a hierarchical family of finite element spaces in the domain D. In order to obtain convergence rates analogous to Theorem 3.9 in the elliptic setting, regularity results for the parametric, parabolic problems (3.67) are required. Such results on best N -term approximation of expansions whose coefficients are measured in scales of spaces with additional smoothness in x and t are obtained in Hoang and Schwab (2010b).

389


3.3. Second-order hyperbolic problems Analogous results may also be proved for linear, second-order hyperbolic problems with random coefficients. Such equations arise, for example, in the mathematical description of wave propagation in media with uncertain material properties. Below we recapitulate recent results from Hoang and Schwab (2010a) which are analogous to those for the elliptic and parabolic cases. One important distinction to the parabolic case, however, is that a space–time variational principle is not available so that some of the proofs in Hoang and Schwab (2010a) are significantly different from the elliptic case. A class of wave equations with random coefficients For 0 < T < ∞, we consider in I = (0, T ) the following class of linear, second-order hyperbolic equations with random coefficients: let D be a bounded Lipschitz domain in Rd . We define the space–time cylinder QT = I × D. In QT , we consider the stochastic wave equation ∂2u − ∇ · (a(x, ω)∇u) = g(t, x), ∂t2

u|∂D×I = 0,

u|t=0 = g1 ,

ut |t=0 = g2 .

(3.81) As before, we assume the coefficient a(ω, x) to be a random field on a probability space (Ω, Σ, P) over L∞ (D). The forcing g and initial data g1 and g2 are assumed to be deterministic. To ensure well-posedness of (3.81), we once more require Assumptions 3.10 and 3.11. To state the weak form of the initial boundary value problem (3.81), we let V = H01 (D) and H = L2 (D), and require g ∈ L2 (I; H),

g1 ∈ V,

g2 ∈ H.

(3.82)

For the variational formulation of (3.81), we introduce the Bochner spaces X = L2 (I; V ) ∩ H 1 (I; H) ∩ H 2 (I; V ),

Y = L2 (I; V ) × V × H.

(3.83)

A weak solution of the hyperbolic initial boundary value problem (3.81) is any function u ∈ X such that, for every v = (v0 , v1 , v2 ) ∈ Y, 2 d u (t, ·), v0 (t, ·) dt + u(0), v1 V + ut (0), v2 H 2 I dt H + a(x, ω)∇u(t, x, ω) · ∇v0 (t, x) dx dt I D g(t, x)v0 (t, x) dx dt + g1 , v1 V + g2 , v2 H . (3.84) = I

D

We have the following result, from Hoang and Schwab (2010a). Proposition 3.19. Under Assumption 3.10 and under condition (3.82), for every ω ∈ Ω, the initial boundary value problem (3.84) admits a unique

390


weak solution u ∈ X . The following estimate holds:

u X ≤ C( g L2 (I;H) + g1 V + g2 H ),

(3.85)

where the constant C depends only on T and on a− and a+ in Assumption 3.10. We next present results on the stochastic Galerkin approximation of the initial boundary value problem (3.84). Parametric deterministic wave equations Given a forcing function g(t, x) and initial data g1 (x) and g2 (x) satisfying (3.82), for each y ∈ Γ we consider the parametric, deterministic initial boundary value problem ∂ 2 u(t, x, y) − ∇x · (a(x, y)∇x u(t, x, y)) = g(t, x) in QT , ∂t2 u(t, x, y)|∂D×I = 0, u|t=0 = g1 , ut |t=0 = g2 ,

(3.86)

where the parametric coefficient a(x, y) is defined as in the elliptic and parabolic cases in (3.66). Again, for each y ∈ Γ, we define the bilinear map b : X × Y → R by 2 d w b(y; w, (v0 , v1 , v2 )) = (t, ·), v0 (t, ·) dt (3.87) dt2 I H a(x, y)∇w(t, x) · ∇v0 (t, x) dx dt + I

D

+ u(0), v1 V + ut (0), v2 H . We also define the linear form on Y, f (v) = g(t, x)v0 (t, x) dx dt + g1 , v1 V + g2 , v2 H . I

D

The pointwise parametric (with respect to y ∈ U ) variational formulation of the parametric, deterministic problem (3.86) then reads: find u(y) ∈ X :

b(y; u, v) = f (v)

∀v = (v0 , v1 , v2 ) ∈ Y.

(3.88)

Note that here y-dependent data g, g1 and g2 would be equally admissible. This pointwise in y parametric variational formulation is well-posed uniformly with respect to the parameter vector y. More precisely, we have the following result (see Hoang and Schwab (2010a)). Proposition 3.20. Under Assumption 3.10 and under conditions (3.82), for every y ∈ Γ, the problem (3.88) admits a unique weak solution u(y) ∈ X . The parametric weak solutions {u(y) ; y ∈ Γ} ⊂ X satisfy the a priori estimates ∀y ∈ Γ :

u(·, ·, y) X ≤ C( g L2 (I;H) + g1 V + g2 H ),

(3.89)

391


where the constant C is independent of y. The map u : Γ → X is strongly measurable as an X -valued function. An analogous result also holds for the Galerkin formulation of the parametric deterministic problem. To state this, we introduce the Bochner spaces X = L2 (Γ, µ; X ) and Y = L2 (Γ, µ; Y) and define on these spaces the bilinear form B(·, ·) : X × Y → R and the linear form F (·) : Y → R as B(u, v) = b(y; u, v)µ(dy), F (v) = f (v)µ(dy). Γ

Γ

We may then consider the variational problem: find u∈X

such that B(u, v) = F (v)

∀v ∈ Y.

(3.90)

Proposition 3.21. Under Assumptions 3.10 and 3.11, problem (3.90) admits a unique solution u ∈ X , i.e., the parametric solution map belongs to to the Bochner space L2 (Γ, µ; X ). Moreover, in terms of the orthonormal basis of L2 (Γ, µ) given by the tensorized Legendre polynomials (Lν )ν∈F , each function u ∈ X can be written as an (unconditionally convergent in X ) expansion in tensorized Legendre polynomials:

uν Lν , uν ∈ X . (3.91) u= ν∈F

Semidiscrete Galerkin approximation Spectral approximations of the parametric, deterministic wave equation (3.86) are once again obtained by projection onto finite linear combinations of tensorized Legendre polynomials. We briefly present the corresponding results from Hoang and Schwab (2010a). For any set Λ ⊂ F of finite cardinality, we define polynomial subspaces of X and Y: ,

uν (t, x)Lν (y) ; uν ∈ X ⊂ X , X Λ = uΛ (t, x, y) = ν∈Λ

and

, YΛ =

vΛ (t, x, y) =

vν (t, x)Lν (y) ; vν ∈ Y

⊂ Y.

ν∈Λ

Denoting vν = (v0ν , v1ν , v2ν ), we may write the test functions v ∈ Y Λ componentwise as Legendre GPC expansions:

v0Λ (t, x, y) = v0ν (t, x)Lν (y), viΛ (x, y) = viν (x)Lν (y) i = 1, 2. ν∈Λ

ν∈Λ

We consider the following semidiscrete Galerkin projection of u onto X Λ .

392


Find uΛ ∈ X Λ

such that B(uΛ , vΛ ) = F (vΛ )

∀vΛ ∈ Y Λ .

(3.92)

Theorem 3.22. Under Assumptions 3.10 and 3.11, for every subset Λ ⊂ F of finite cardinality, there exists a unique solution uΛ ∈ X Λ to the Galerkin equations (3.92). It is to be expected that the semidiscrete Galerkin approximations are quasi-optimal, i.e., their error is controlled by the best approximation error. This is indeed once more the case. However, due to the lack of a space– time variational formulation, additional regularity of solutions, in particular point values with respect to t, is required. To this end, we introduce the space Z := H 1 (I; V ) ∩ H 2 (I; H) ⊂ C 0 (I; V ) ∩ C 1 (I; H).

(3.93)

Note that Z ⊂ X . The following error estimate for semidiscrete approximations of parametric solutions in Z holds. Proposition 3.23. Assume that u ∈ L2 (Γ, µ; Z). Then, for all ν ∈ F the coefficient uν in (3.91) belongs to Z. Assume further that for a subset Λ ⊂ F, uΛ ∈ L2 (Γ, µ; Z). Then we have the error bound "1/2 !

u − uΛ L2 (Γ,µ;X ) ≤ c uν Lν =c

uν 2Z . (3.94) 2 ν∈F\Λ

L (Γ,µ;Z)

ν∈F\Λ

Here, the constant c > 0 depends only on the coefficient bounds a− and a+ in Assumption 3.10. For a proof, we refer to Hoang and Schwab (2010a). Proposition 3.23 once more implies quasi-optimality of the L2 (Γ, µ; X ) projection uΛ ∈ X Λ defined in (3.92). We note, however, that in its proof, the extra regularity u ∈ L2 (Γ, µ; Z) was required. It is therefore of interest to establish a regularity result for uΛ which ensures u ∈ L2 (Γ, µ; Z) and, hence, implies the semidiscrete error bound (3.94). To this end, we recall the smoothness space W ⊂ V defined in (3.52) and (3.53). On W , we define the W -seminorm and the W -norm as in (3.54). The role of the space W for the regularity of the solution of the parametric wave equation becomes clear from the following result. Proposition 3.24.

If Assumption 3.11 holds and if, moreover, g ∈ H 1 (I; H),

g1 ∈ W,

g2 ∈ V,

(3.95)

then for every y ∈ Γ we have u(·, ·, y) ∈ Z, and its Z-norm is bounded uniformly for all y ∈ Γ.


393

The Galerkin projections require the corresponding space–time solution space W = L2 (I; W ) ∩ H 1 (I; V ) ∩ H 2 (I; H),

(3.96)

where W is defined in (3.53). Note that W ⊂ Z for Z defined in (3.93). We have the following regularity result for the Galerkin-projected GPC approximations of the parametric, deterministic wave equation (3.86). Proposition 3.25. Under Assumptions 3.10 and 3.11, and if, in addition, a(·, ·) ∈ L∞ (Γ; W 1,∞ (D)), g ∈ H 1 (I; H), g1 ∈ W and g2 ∈ V , then, for every subset Λ ⊂ F of finite cardinality, uΛ ∈ L2 (Γ, µ; W) ⊂ L2 (Γ, µ; Z). For the parametric wave equations, the extra regularity uΛ ∈ L2 (Γ, µ; W) is therefore already required to ensure quasi-optimality of the Galerkin projections. It is also necessary in order to prove best N -term convergence rates analogous to those for the parametric elliptic and the parabolic problems. To state these results, we therefore require the following. Assumption 3.26. We assume in (3.66) that a ¯ ∈ W 1,∞ (D) and ψj ∈ W 1,∞ (D) are such that ∞

ψj W 1,∞ (D) < ∞.

j=1

Moreover, we assume that, for some 0 < p < 1, ∞

( ψj pL∞ (D) + ∇ψj pL∞ (D) ) < ∞.

j=1

Note that Assumption 3.26 implies Assumption 3.11. Under these assumptions we then have the following regularity and best N -term convergence result (see Hoang and Schwab (2010a) for a proof). Proposition 3.27. Under Assumptions 3.10, 3.11 and 3.26, ν∈F uν pZ is finite for the same value of 0 < p < 1 as in these assumptions. If, moreover, the compatibility condition (3.95) holds, there exists a sequence (ΛN ) ⊂ F of index sets with cardinality not exceeding N such that the solutions uΛN of the Galerkin semidiscretized problem (3.92) satisfy, as N → ∞, the error bound

u − uΛN X ≤ CN −σ , Here, the constant C depends only on

σ=

ν∈F

1 1 − . p 2

uν pZ .

394


4. Sparse tensor discretizations 4.1. Sparse tensor stochastic Galerkin discretizations We consider the diffusion equation with a stochastic diffusion coefficient, as in Section 2.3, discretized by the Legendre chaos basis; see Section 2.3. Foundations for subsequent Galerkin approximation were laid in Section 2.3. In Section 3.1, we proved convergence rates of optimal approximations by non-constructive methods. In this section, we consider Galerkin approximations in problem-adapted subspaces of L2 (Γ, µ; V ) with a sparse tensor product structure. As in Bieri, Andreev and Schwab (2009), we show that these computable approximations achieve the optimal convergence rates up to logarithmic factors. We refer to Gittelson (2011c, 2011d) for an analysis of adaptive solvers with similar convergence properties. For a bounded Lipschitz domain D ⊂ Rd , we set V = H01 (D). The dual space of V is V = H −1 (D). We denote the parameter domain by Γ := [−1, 1]∞ ; µ is the product of uniform measures on Γ, as in Section 2.2. Wavelet finite element discretization on the physical domain D Finite element wavelets provide a stable, hierarchical multi-level basis of V = H01 (D). We give a brief overview of the construction of such bases and refer to Cohen (2003) and Nguyen and Stevenson (2009) for details. Let D ⊂ Rd be a bounded Lipschitz polyhedron with plane faces, and let T0 be a regular simplicial mesh of D. For all ∈ N, let T be the mesh of D constructed by regular refinements of the initial triangulation T0 . We denote by I the interior nodes of T and by N := I \ I−1 the new nodes on discretization level ∈ N. On the coarsest level, we have N0 = I0 . Also, denote by E the set of edges of T . Let VD be the space of continuous piecewise linear functions on the T . We recall the standard approximation result

v − PD v H01 (D) ≤ CτD 2−τ v H 1+τ (D)

∀v ∈ H 1+τ (D),

0 ≤ τ ≤ 1, (4.1)

where PD is the orthogonal projection in H01 (D) onto VD and CτD > 0 is independent of . The dimension of VD satisfies dim VD 2d

(4.2)

as → ∞. The standard, so-called ‘one-scale’ basis of VD consists of the piecewise linear, nodal basis functions (λn )n∈I determined by λn (m) = δnm

∀m ∈ I .

(4.3)

Following Nguyen and Stevenson (2009), we construct an alternative, hierarchical basis of VD .


395

We define auxiliary functions (ηn )n∈I−1 in VD satisfying −1 ηn , λ−1 m L2 (D) δnm ηn L2 (D) λm L2 (D)

for all n, m ∈ I−1 and all ∈ N.   3 ηn (m) := − 1/2   0

(4.4)

For d = 1, m = n, [m, n] ∈ E , all other m ∈ I ,

n ∈ I−1 ,

(4.5)

and for d = 2,

  m = n,  14 ηn (m) := − 1 [m, n] ∈ E ,   0 all other m ∈ I ,

n ∈ I−1 .

(4.6)

0 := λ0 for m ∈ I and Wavelets are given by ψm 0 m := λm − ψm

λm , λ−1 n L2 (D) n∈I−1

ηn , λ−1 n L2 (D)

ηn ,

m ∈ N .

(4.7)

We define the detail spaces ; m ∈ N }, WD := span {ψm

∈ N0 .

(4.8)

Then W0D = V0D , and VD is the direct sum of WiD over all i ≤ . The wavelets whose construction was outlined above satisfy properties (W1)– (W5). Hierarchical polynomial spaces By Corollary 2.26, the solution u of (2.44) can be expanded in the Legendre chaos basis as

u(y) = uν Lν (y) ∈ V, y ∈ Γ. (4.9) ν∈F

For any subset Λ ⊂ F, we consider the truncated series

SΛ u(y) := uν Lν (y) ∈ V, y ∈ Γ.

(4.10)

ν∈Λ

Since (Lν )ν∈F is an orthonormal basis of L2 (Γ, µ; V ) (see Corollary 2.16), SΛ u is the orthogonal projection of u onto the span of (Lν )ν∈Λ in L2 (Γ, µ; V ). We denote this space by ,

2 vν Lν (y) ; (vν )ν∈Λ ∈ (Λ; V ) . (4.11) Λ; V := v(y) = ν∈Λ

396


For any Λ ⊂ F, Λ; V is a closed subspace of L2 (Γ, µ; V ) since it is the kernel of SF\Λ , and thus it is a Hilbert space with the same norm. We abbreviate Λ for the subspace Λ; R of L2 (Γ, µ). Let 0 ≤ τ ≤ 1 and γ > 0. For all k ∈ N0 , we define Λγk as the set of the first 2γk indices ν ∈ F in a decreasing rearrangement of ( uν H 1+τ (D) )ν∈F , i.e., Λγk consists of the 2γk indices ν for which uν H 1+τ (D) is largest. We assume that the same decreasing rearrangement is used for all k ∈ N0 , such that Λγk ⊂ Λγk+1 . The approximation spaces in L2 (Γ, µ) induced by the sets Λγk are VkΓ := Λγk ,

k ∈ N0 ,

(4.12)

and the detail spaces are given by W0Γ := V0Γ and Γ WkΓ := VkΓ Vk−1 = Λγk \ Λγk−1 ,

k ∈ N.

(4.13)

Then VkΓ ⊗ V and WkΓ ⊗ V serve as approximation and detail spaces in L2 (Γ, µ; V ). Remark 4.1. The sets Λγk are not computationally accessible. We present a problem-adapted construction for an alternative sequence of index sets in Section 4.2. Sparse tensor product spaces The approximation spaces VkΓ ⊂ L2 (Γ, µ) and VD ⊂ V = H01 (D) can be combined to define finite-dimensional subspaces of L2 (Γ, µ; V ). We recall that the Lebesgue–Bochner space L2 (Γ, µ; V ) is isometrically isomorphic to the Hilbert tensor product L2 (Γ, µ) ⊗ V . For any L ∈ N0 , the tensor product of VLΓ and VLD can be expanded as VLΓ ⊗ VLD =

L

WkΓ ⊗ VLD =

k=0

WkΓ ⊗ WD .

(4.14)

0≤,k≤L

The sparse tensor product space of level L ∈ N0 is defined by restricting the last sum in (4.14) to only the most important component spaces, LD := VLΓ ⊗V

WkΓ ⊗ WD =

0≤+k≤L

L

D WkΓ ⊗ VL−k .

(4.15)

k=0

D LD is equal to VN from (2.90), with VN,ν equal to VL−k if ν ∈ Thus VLΓ ⊗V γ γ γ Γ D L is of the form Λk \ Λk−1 , where Λ−1 := ∅. Any element v of VL ⊗V

v(y, x) =

L

k=0

ν∈Λγk \Λγk−1

L−k

=0 n∈N

cν,n ψn (x)Lν (y),

x ∈ D,

y ∈ Γ. (4.16)

397


LD by u We denote the Galerkin projection of u onto VLΓ ⊗V L . It is determined Γ D L . By Proposition 2.27, u by (2.91) for VN = VL ⊗V L is a quasi-optimal LD . approximation of u in VLΓ ⊗V Convergence estimate LD , the Due to the quasi-optimality of the Galerkin solution u L in VLΓ ⊗V convergence of u L to u is equivalent to that of the best approximation. Let LD . By (4.15), PL denote the orthogonal projection in L2 (Γ, µ; V ) onto VLΓ ⊗V it has the form PL v =

L

k=0


D (PL−k vν )Lν ,

v=

vν Lν ∈ L2 (Γ, µ; V ),

(4.17)

ν∈F

where Λγ−1 := ∅. Theorem 4.2. Let 0 < τ ≤ 1, γ > 0, 0 < p ≤ 2 and s := 1/p − 1/2. Furthermore, let a+ and a− be the bounds from (2.50). Then "1/p 0 ! a + p

u − u L L2 (Γ,µ;V ) ≤ CtD λsγ,τ (L)

uν H 1+τ (D) 2− min(sγ,τ )L a− ν∈F

(4.18) with

( 2 + 22 min(sγ,τ ) L λsγ,τ (L) = ( 2 + |2−2sγ − 2−2τ |−1

Proof.

(4.19)

By (4.17) and Parseval’s identity in L2 (Γ, µ), using (4.1),

u − PL u 2L2 (Γ,µ;V ) = ! ≤

always, if sγ = τ .

(CτD )2

L

k=0

L

k=0


−2(L−k)τ

2

D

uν − PL−k uν 2V +

uν 2V

ν∈F\ΛγL

uν 2H 1+τ (D)


+

uν 2H 1+τ (D)

" .

ν∈F\ΛγL

Applying Stechkin’s lemma (3.13), we have for any 0 < p ≤ 2 and s := 1/p − 1/2, "1/2 ! "1/p !

γ p 2 −s

uν H 1+τ (D) ≤ (#Λk−1 + 1)

uν H 1+τ (D) ν∈F\Λγk−1

ν∈F

for k = 1, . . . , L + 1. Since also "1/2 ! "1/p !

p 2

uν H 1+τ (D) ≤

uν H 1+τ (D) , ν∈F

ν∈F

398


using #Λγk−1 + 1 ≥ 2γ(k−1) , it follows that !

1/2 D

u − PL u L2 (Γ,µ;V ) ≤ Cτ Σ

uν p

"1/p

H 1+τ (D)

L

ν∈F

with ΣL = 2−2Lτ + 2−2sγL +

L

2−2(L−k)τ 2−2sγ(k−1) .

k=1

If sγ = τ , then ΣL simplifies to ΣL = (22sγ L + 2)2−2sγL = (22τ L + 2)2−2τ L . More generally, if sγ = τ , estimating the sum by L times the maximal summand, ΣL ≤ 2−2Lτ + 2−2sγL + 2−2 min(sγ,τ )(L−1) L. Alternatively, if sγ = τ , summing the geometric series leads to ΣL = 2−2Lτ + 2−2sγL + If sγ < τ , we have ΣL ≤ 2

−2sγL

and similarly, if sγ > τ , ΣL ≤ 2

−2τ L

2−2sγL − 2−2τ L . 2−2sγ − 2−2τ

1 2 + −2sγ , 2 − 2−2τ

2+

2−2τ

1 . − 2−2sγ

Then the claim follows using the quasi-optimality property from Proposition 2.27 with c = a+ and cˇ = a−1 − . We express the convergence estimates in Theorem 4.2 with respect to the total number of degrees of freedom used to approximate u. Lemma 4.3. as L := N Proof.

LD scales The dimension of the sparse tensor product VLΓ ⊗V

LD dim VLΓ ⊗V

(L + 1)2dL 2max(γ,d)L

if γ = d, if γ = d,

∀L ∈ N0 .

Using the last expression in (4.15), L = dim VLΓ ⊗V LD = N

L

k=0

D #(Λγk \ Λγk−1 ) dim VL−k .

(4.20)

399


D 2d(L−k) . By definition, #(Λγk \Λγk−1 ) ≤ #Λγk ≤ 2γk , and by (4.2), dim VL−k Therefore, L L

γk d(L−k) dL NL 2 2 =2 2(γ−d)k . k=0

k=0

If γ = d, then all the summands are one, and we arrive at L (L + 1)2dL . N Otherwise, we sum the geometric series to find −1 2max(γ,d)L 2γ−d − 1

(γ−d)L

L 2dL 2 N

with constants independent of L. Corollary 4.4.

In the setting of Theorem 4.2, if γ = d,

u − u L L2 (Γ,µ;V ) 0 ≤

CτD λsγ,τ (L)

!

a+

uν pH 1+τ (D) a−

"1/p

(4.21)

− min(s,sγ/d,τ /γ,τ /d) . N L

ν∈F

If γ = d, then

u − u L L2 (Γ,µ;V ) 0 ≤ CτD λsd,τ (L)

!

a+

uν pH 1+τ (D) a−

"1/p

(4.22)

− min(s,τ /d) . (L + 1)min(s,τ /d) N L

ν∈F

Proof. The claim follows from Theorem 4.2 using Lemma 4.3 to estimate L . 2L from below by a power of N Remark 4.5. Up to logarithmic factors, the convergence rates in CorolL reaches lary 4.2 with respect to the total number of degrees of freedom N the optimum min(s, τ /d) from Theorem 3.9 for two choices of γ. Therefore, assuming a sparse tensor product structure of the Galerkin subspace of L2 (Γ, µ; V ) does not significantly deteriorate the convergence behaviour. If γ = τ /s = d, then sγ/d = τ /d and τ /γ = s, so (4.21) becomes "1/p 0 ! ( a + − min(s,τ /d) .

u − u L L2 (Γ,µ;V ) ≤ CτD 2 + 22τ L N

uν pH 1+τ (D) L a− ν∈F

(4.23) As already stated in the corollary, γ = d also leads to this convergence rate, L . Nevertheless, this albeit with an additional factor that is logarithmic in N choice has the advantage of being independent of the regularity parameters s and τ .

400


Remark 4.6. The above derivation of Theorem 4.2 and Corollary 4.4 corrects some mathematical inaccuracies in the proof of Proposition 3.5 of Bieri et al. (2009). In particular, it is apparent from the above result that the sets Λγk for k = 0, . . . , L − 1 should be defined to contain the indices ν ∈ F for which uν H 1+τ (D) is maximal, and not uν V as claimed in Bieri et al. (2009). For ΛγL , it is also reasonable to use uν V ; doing so sacrifices the sparse tensor product structure, but may reduce the total number of degrees of freedom. Remark 4.7. Corollary 4.4 should be compared with the convergence of full tensor product discretizations. In this case, the dimension of the Galerkin subspace of L2 (Γ, µ; V ) scales as 2(γ+d)L , and the error satisfies Theorem 4.2 without the logarithmic terms. Therefore, the convergence rate of the error with respect to the dimension is min(sγ, τ )/(γ + d). For example, if γ = d, this is min(s, τ /d)/2, which is just half of the rate from (4.22). If γ = τ /s, then the convergence rate (1/s + d/τ )−1 is half of the harmonic mean of s and τ /d, which is only a minor improvement. Remark 4.8. The convergence rates in Corollary 4.4 and Remark 4.5 are limited by the order of convergence τ /d ≤ 1/d of linear finite elements on D. This can be overcome by using higher-order finite elements. For example, if u ∈ p (F; H 1+τ (D)) with s = 1/p − 1/2 and τ = sd, using piecewise polynomial finite elements of degree τ leads to a convergence rate of s L of degrees of freedom. with respect to the total number N 4.2. Algorithmic aspects of polynomial chaos Hierarchical index sets The index sets Λγk from Section 4.1 are not computationally accessible. We consider a different family of index sets that can be constructed explicitly. ∞ 0 For any sequence η = (ηm )∞ m=1 ∈ R , since ηm = 1 for all m ∈ N, η ν :=

∞ )

νm ηm =

)

νm ηm ,

ν ∈ F.

(4.24)

m∈supp ν

m=1

R∞

with ηm ∈ (0, 1), ηm → 0, up to a constant We assume that for some η ∈ factor, (4.24) is an estimate for uν H 1+t (D) for all ν ∈ F; see Section 4.2. Then, for any > 0, Λ (η) := {ν ∈ F ; η ν ≥ }.

(4.25)

Due to the assumptions 0 < ηm < 1 and ηm → 0, Λ (η) is a finite subset of F for all > 0. The construction of the sets (4.25) does not require exact knowledge of uν H 1+t (D) , only estimates with a certain structure. Also, the sets Λ (η) differ from Λγk in that they are defined by a thresholding tolerance instead of a prescribed cardinality.

401


It is clear from the definition that the sets Λ (η) are monotonic in both parameters and η. If ≤ ¯, then Λ (η) ⊇ Λ ¯(η). Similarly, if η ∈ R∞ with ηm ≤ η¯m < 1 for all m ∈ N, then Λ (η) ⊆ Λ (¯ η ). σ ∞ σ For a σ > 0, let η ∈ R be defined by ηm := (m + 1)−σ , m ∈ N. For η σ , sharp asymptotics on the cardinality of the index sets Λ (η σ ) follow from results on integer factorization (Bieri et al. 2009, Proposition 4.5). Proposition 4.9.

For any σ > 1/2, as → 0, √ −1 −1 e2 σ log σ −1/σ #Λ (η )

. √ 2 π(σ −1 log −1 )3/4

(4.26)

In particular, {(η σ )ν }ν∈F ∈ p (F) for any p > 1/σ. Proof. We observe that for all > 0 and σ > 0, Λ (η σ ) = Λ 1/σ (η 1 ). Let f (n) denote the number of multiplicative partitions of n ∈ N, disregarding the order of the factors. For example, f (12) = 4 since 12, 2 · 6, 2 · 2 · 3 and 3 · 4 are the only factorizations of 12. Sharp asymptotics for FΣ (x) :=

x

f (n)

n=1

as x → ∞ were obtained in Canfield, Erd˝os and Pomerance (1983), based on earlier work (Oppenheim 1927, Szekeres and Turán 1933). The result required here is √

FΣ (x) = F (x)(1 + O(1/ log n))

x e2 log x with F (x) = √ . 2 π(log x)3/4

Let > 0 and ν ∈ Λ (η 1 ). By definition, n :=

∞ ) m=1

1 (m + 1)νm ≤ .

Since n is an integer, the index ν represents a multiplicative partition of an integer n ≤ 1/ . Conversely, for 6any integer n ≤ 1/ , any multiplicative partition of n is of the form n = m (m + 1)νm for a ν ∈ F, and therefore ν ∈ Λ (η 1 ). Consequently, #Λ (η σ ) = #Λ 1/σ (η 1 ) = FΣ ( −1/σ ) F ( −1/σ ) as → 0, which implies (4.26). ( Note that −1/σ is the dominating term in (4.26). Let z = σ −1 log −1 . Then (4.26) is bounded by a constant times 2

2

ez e2z e = e(z+1) .

402


√ √ κ z for all z ≥ ( κ − 1)−1 , and thus √ = −κ/σ ∀z ≥ ( κ − 1)−1 .

For any κ > 1, we have z + 1 ≤ 2

e(z−1) ≤ eκz

2

Consequently, by (4.26), for all κ > 1, #Λ (η σ ) = #{ν ∈ F ; (η σ )ν ≥ } −κ/σ . This is a characterizing property of the weak Lebesgue sequence space qw (F) with q = κ/σ (DeVore 1998). Since qw (F) ⊂ p (F) for all p > q, and since κ > 1 is arbitrary, it follows that {(η σ )ν }ν∈F ∈ p (F) for all p > 1/σ. As demonstrated in Bieri et al. (2009, Figure 4.1), (4.26) provides a good approximation for thresholds as large as 10−2 . Proposition 4.9 gives bounds on the size of index sets Λ (η). In order to control the complexity of the numerical construction of these index sets, it is also important to bound the length # supp ν of indices ν ∈ Λ (η). To this end, we define the maximal dimension reached by Λ (η), M (η) := max {m ∈ N ; ηm ≥ } = max {m ∈ N ; m ∈ Λ (η)},

(4.27)

where m ∈ F denotes the Kronecker sequence ( m )n = δmn . Proposition 4.10. and

Let η = (ηm )∞ m=1 satisfy 0 < ηm+1 ≤ ηm < 1, ηm → 0, c1 m−σ1 ≤ ηm ≤ c2 m−σ2

∀m ∈ N,

(4.28)

with c1 , c2 > 0 and σ1 ≥ σ2 > 0. Then there is a constant C > 0 such that # supp ν ≤ C log+ M (η) ≤ C log #Λ (η)

∀ν ∈ Λ (η)

(4.29)

for all 0 < ≤ 1, i.e., whenever Λ (η) = ∅. Proof.

By (4.28) and (4.27), due to the definition of Λ (η),

> ηM (η)+1 ≥ c1 (M (η) + 1)−σ1 ,

and therefore M (η) + 1 ≤ ( /c1 )−1/σ1 . Since ηm ≤ η1 < 1 for all m ∈ N, we can assume without loss of generality that c2 = 1, possibly at the cost of decreasing σ2 . Then, for all ν ∈ F, ) ) νm ην = ηm ≤ m−νm σ2 ≤ [(# supp ν)!]−σ2 . m∈supp ν

m∈supp ν

We note that, by Stirling’s approximation, (x + 1)τ = o(τ log x√ !) for any τ > 0 as x → ∞. Indeed, abbreviating n := τ log x , since n! ≥ 2πnn e−n , (x + 1)τ 1 ≤ √ eτ log x+n−n log n → 0, τ log x ! 2π Consequently, also (τ log x !)−1 = o((x + 1)−τ ).

x → ∞.


403

Suppose that ν ∈ F with # supp ν ≥ (1 + σ1 /σ2 ) log M (η). Then, by the above estimates with x = M (η) and τ = 1 + σ1 /σ2 , ; < −σ2 σ1 ν η ≤ 1+ log M (η) ! σ2 = o (M (µ) + 1)−σ1 −σ2 = o( (σ1 +σ2 )/σ1 ) = o( ). The o(·) is with respect to M (η) → ∞, which by (4.27) is equivalent to

→ 0. Therefore, there is an 0 > 0 such that, for all 0 < ≤ 0 , # supp ν ≥ (1 + σ1 /σ2 ) log M (η) implies η ν < . Equivalently, σ1 log M (η) ∀ν ∈ Λ (η). # supp ν ≤ 1 + σ2 As there are only finitely many distinct sets Λ (η) with > 0 , this estimate holds for all > 0, with a larger constant and log+ (n) := max(log(n), 0) in place of log(n) to accommodate M (η) = 0. The second part of (4.29) follows using the observation that M (η) ≤ #Λ (η) since the Kronecker sequences ( m )n = δmn are in Λ (η) for m = 1, . . . , M (η). Numerical construction By Proposition 4.10, any ν ∈ Λ (η) for η = (ηm )∞ m=1 satisfying (4.28) can be stored in O(log #Λ (η)) memory in the form {(m, νm ) ; m ∈ supp ν},

(4.30)

assuming that an arbitrary integer can be stored in a single storage location. Therefore, the total memory required to store the full set Λ (η) is of the order #Λ (η) log #Λ (η). We show how Λ (η) can be constructed in O(#Λ (η) log #Λ (η)) time. For any sequence c = (cm )∞ m=1 , we define the translation τ1 c := (cm+1 )∞ m=1 .

(4.31)

Furthermore, we define the concatenation of integers with subsets of F. For any n ∈ N0 and Λ ⊂ F, [n, Λ] := {ν ∈ F ; ν1 = n, τ1 ν ∈ Λ} = {[n, ν] ; ν ∈ Λ},

(4.32)

where [n, ν] := (n, ν1 , ν2 , . . .). We observe that =

N (η)

Λ (η) =

[n, Λ η−n (τ1 η)] 1

n=0

with

(4.33)

>

N (η) := max {m ∈ N0 ;

η1n

? log

≥ } = . log η1

(4.34)

404


This suggests a recursive construction of Λ (η). The precise algorithm is given in Construct(η, ). We store indices ν ∈ F in the sparse form (4.30). We append to each ν ∈ Λ (η) the value η ν , which we compute during the construction of Λ (η). In the concatenation [n, Λδ (τ1 η)], these values are updated by η [n,ν] = η1n (τ1 η)ν , where (τ1

η)ν

ν ∈ Λδ (τ1 η),

(4.35)

is known from the construction of Λδ (τ1 η) for all ν ∈ Λδ (τ1 η).

Construct(η, ) → Λ (η) if η1 < then if > 1 then return ∅ else return {0} end end > ? log

N ←− log η1 for n = 0, 1, . . . , N do Λn ←− Construct(τ1 η, η1−n ) Λn ←− [n, Λn ] end N = Λ ←− Λn n=0

return Λ Lemma 4.11. For any > 0 and any η satisfying the assumptions of Proposition 4.10, Construct(η, ) constructs Λ (η) at a computational cost of O(#Λ (η) log #Λ (η)). Proof. It is clear from (4.33) that Construct(η, ) does construct Λ (η). Due to the sparse storage format (4.30), concatenation [0, ν] of ν ∈ F with zero does not involve any work. Therefore, each index ν ∈ Λ (η) is constructed in # supp ν identical steps. The union operations can be performed in constant time, for example, with a linked list data structure. The claim follows since # supp ν = O(log #Λ (η)) by Proposition 4.10. Remark 4.12. For the construction of a sparse tensor product similar to (4.15), we need not only one index set Λ (η), but a hierarchical sequence of such sets. For a sequence of thresholds

0 ≥ 1 ≥ · · · ≥ L = > 0,

(4.36)


405

we use the index sets Λk := Λ k (η), which satisfy ∅ =: Λ−1 ⊆ Λ0 ⊆ Λ1 ⊆ · · · ⊆ ΛL = Λ (η).

(4.37)

The detail spaces (4.13) induced by (4.37) are WkΓ = Λk \ Λk−1 . Their construction requires the partitioning of Λ (η) into Λ (η) =

L =

Λk \ Λk−1 .

(4.38)

k=0

This can be done after constructing Λ (η) by sorting the indices ν ∈ Λ (η) with respect to the values η ν , which are computed during the construction of the index set using the recursion (4.35). Once sorted, the indices ν are easily assigned to the partitions Λk \ Λk−1 by comparing η ν to k . Alternatively, for a parameter γ > 0 as in Section 4.1, the thresholds 0 , . . . , L−1 can be chosen such that #Λk = 2γk for k = 0, . . . , L − 1. The total work is of order O(#Λ (η) log #Λ (η) + L) as → 0 or L → ∞. Remark 4.13. The stochastic Galerkin matrix has the form (2.81), i.e., the Legendre coefficients of the Galerkin projection u L satisfy a discretized version of the system of equations (2.88), which has direct dependencies between indices ν, ν¯ ∈ Λ (ν) that are identical in all dimensions but one, and differ by one in this dimension. We call ν and ν¯ neighbours if there is an m ¯ ∈ N such that |νm − ν¯m | = δmm ¯ . Fast access to the neighbours in Λ (η) of ν ∈ Λ (η) is crucial to solving the Galerkin system. The neighbourhood data are stored efficiently in a directed graph, in which there is an edge from ν to ν¯ if there is an m ¯ ∈ supp ν such that ν¯ = ν − m ¯ , where m ¯ is the Kronecker sequence ( m ¯ )m = δm m ¯ . Then there are exactly # supp ν edges starting at ν ∈ Λ (η). It is useful to label each edge by the appropriate m ¯ ∈ N. The routine Construct(η, ) can be extended to compute the neighbourhood relations during the construction of Λ (η) using the observation that ν and ν¯ are neighbours as above if and only if, at a call of Construct at recursion depth m, ¯ ν and ν¯ are constructed by appending n and n − 1, respectively, to the same index ν ∈ F, for some n ∈ N, and all subsequent additions are the same. Choice of parameters The preceding sections leave open the choice of η = (ηm )∞ m=1 . Due to Theorem 4.2, η ν should approximate uν H 1+τ (D) for the unknown u and some 0 ≤ τ ≤ 1; see Remark 4.6. A reasonable choice is αm ηm =

ϕm W τ,∞ (D) , m ∈ N, (4.39) a ¯− using the notation from Section 2.3. We refer to Bieri et al. (2009) for numerical experiments with this and other choices of ηm and τ = 0. Better

406


a priori choices of ηm may be obtainable from sharper a priori bounds on

uν H 1+τ (D) . As mentioned in Remark 4.12, the thresholds 0 , . . . , L−1 can be chosen such that #Λk = 2γk for k = 0, . . . , L − 1 with a parameter γ > 0 as in Section 4.1. 4.3. Sparse tensor stochastic collocation Stochastic collocation is an alternative to the stochastic Galerkin method. It is also based on a polynomial approximation in the parameter domain. However, the Galerkin projection is replaced by a suitable interpolation operator. A sparse tensor product construction similar to that presented in Section 4.1 is possible in this setting. We consider the diffusion equation with a stochastic diffusion coefficient, as in Section 2.3. The following discussion is based primarily on Bieri (2009a, 2009b). Stochastic collocation We recall the spaces of polynomials on Γ = [−1, 1]∞ from Section 4.1, ,

2 vν Lν (y) ; (vν )ν∈Λ ∈ (Λ) (4.40) Λ = v(y) = ν∈Λ

for Λ ⊂ F, where Lν is the tensor product Legendre polynomial from Section 2.2. If Λ is finite and monotonic in the sense that if µ ∈ Λ, then Λ also contains all ν ∈ F with νm ≤ µm for all m ∈ N, then ,

Λ = v(y) = vν y ν ; (vν )ν∈Λ ∈ RΛ . (4.41) ν∈Λ

Let sets

VkΓ

:= Λk , k ∈ N0 , for a nested sequence of finite monotonic index Λ0 ⊂ Λ1 ⊂ · · · ⊂ Λk ⊂ Λk+1 ⊂ · · · ⊂ F.

(4.42)

We assume that for each k ∈ N0 , there is a finite set Yk = {yik ; i = 1, . . . , NkΓ } ⊂ Γ and an interpolation operator Γ

Ik : RNk → VkΓ ,

NΓ

NΓ

Γ

k ∀i = 1, . . . , NkΓ , ∀(ai )i=1 ∈ RNk . (4.43) These interpolation operators extend to maps k (Ik (ai )i=1 )(yik ) = ai

Ik : C(Γ) → VkΓ ,

(Ik f )(yik ) = f (yik )

∀i = 1, . . . , NkΓ , ∀f ∈ C(Γ), (4.44)


407

which we assume to be the identity on VkΓ , i.e., Ik is a projection of C(Γ) onto VkΓ . Let p ∈ F and

Example 4.14.

Λp := {ν ∈ F ; νm ≤ pm ∀m ∈ N}.

(4.45)

(m)

m For each m ∈ N, let (yi )pi=0 ⊂ [−1, 1] be an arbitrary set of nodes. The corresponding Lagrange polynomials are 6 j=i (z − yj ) (m) , z ∈ [−1, 1], i = 1, . . . , pm + 1. i (z) = 6 (4.46) j=i (yi − yj )

= 1 if pm = 0. Nodes on Γ = [−1, 1]∞ are given by (m) (4.47) Yp := (yim )∞ m=1 ; 0 ≤ im ≤ pm ∀m ∈ N ⊂ Γ. (m)

In particular, 1

The tensor product Lagrange polynomials are ∞ '

y =

(m)

im ∈ Λp ,

y = (yim )∞ m=1 ∈ Yp .

(4.48)

∀y, z ∈ Yp .

(4.49)

(m)

m=1

By construction, they satisfy y (z) = δyz

Consequently, the Lagrange polynomials are linearly independent, and since #Yp = #Λp =

∞ )

(pm + 1) =

m=1

)

(pm + 1),

(4.50)

m∈supp p

(y )y∈Yp spans Λp . Therefore, the interpolation operator (4.44) has the form

f (y)y (z), f ∈ C(Γ), (4.51) (Ip f )(z) = y∈Yp

and it is a projection of C(Γ) onto Λp . The product grids from Example 4.14 are prohibitive for high-dimensional parameter domains since by (4.50), #Yp grows exponentially in # supp p. For each dimension m ∈ N, we consider a non-decreasing sequence of (m) (m) (m) polynomial degrees (qk )∞ k=0 ⊂ N0 , qk+1 ≥ qk . For every m ∈ N and (m) q

(m)

k k ∈ N0 , let (yk,j )j=0 be a set of nodes in [−1, 1], and define the interpolation operator

6

(m)

qk (m) (Ik f )(ξ)

=

j=0

(m) f (yk,j )

(m) i=j (ξ − yk,i ) , 6 (m) (m) (y − y ) i=j k,j k,i

f ∈ C([−1, 1]).

(4.52)

408


Furthermore, we define the univariate differences (m)

∆0

(m)

:= I0

(m)

,

∆k

(m)

:= Ik

(m)

− Ik−1 ,

k ∈ N.

(4.53)

Then the interpolation operator Ip from (4.51) in Example 4.14 with pm = (m) qkm can be expanded as Ip =

∞ '

(m)

Ikm =

m=1

∞ '

∆(m) nm .

(4.54)

0≤nm ≤km m=1

We approximate Ip by truncating the sum in the last term of (4.54). (m) We illustrate this construction for pm = qk if m ≤ M and pm = 0 if m ≥ M + 1. The parameter M truncates the dimensions of the parameter domain Γ, and k determines the order of interpolation in the dimensions (m) m ≤ M . Note that qk may still depend on m, even though k is fixed. For this p, (4.54) can be written as

(M +1) (M +2) (M ) ∆(1) ⊗I0 ⊗· · · . (4.55) IM,k := Ip = n1 ⊗· · ·⊗∆nM ⊗I0 0≤n1 ,...,nm ≤k

The corresponding Smolyak interpolation operator is

(M +1) (M +2) (M ) ∆(1) ⊗ I0 ⊗ · · · . (4.56) IM,k := n 1 ⊗ · · · ⊗ ∆ n M ⊗ I0 0≤n1 +···+nM ≤k

Inserting (4.53), we arrive at the representation

M − 1 (M +1) (M +2) k−|n| ) I (1) ⊗ · · · ⊗ In(M (−1) ⊗ I0 ⊗ I0 ⊗· · · , IM,k = M k − |n| n1 0≤|n|≤k

(4.57) where |n| = n1 + · · · + nM . Remark 4.15. By definition, the Smolyak interpolation operator maps M,k for into Λ . M,k := Λ {ν ∈ F ; νm ≤ qn(m) , 1 ≤ m ≤ M, νm = 0, m ≥ M }. m 0≤|n|≤k

(4.58) Let ν ∈ ΛM,k . For any n = (n1 , . . . , nM ) with |n| ≥ k + 1, there is an (1) (m) (M ) m ≤ M for which νm ≤ qnm −1 , and consequently ∆n1 ⊗ · · · ⊗ ∆nM y ν = 0 (m)

(m)

νm = I νm νm for all y ∈ [−1, 1]M since Inm ym nm −1 ym = ym . As the difference of IM,k and IM,k is merely a sum of such operators, IM,k coincides with IM,k M,k , and the latter acts as the identity on this space. Therefore, the on Λ M,k , Smolyak interpolation operator IM,k is a projection of C(Γ) onto Λ


409

though it is generally not actually an interpolation operator in the sense of (4.51). As in (2.26), we consider product distributions µ :=

∞ '

µm

(4.59)

m=1

on Γ = [−1, 1]∞ , where each µm is a probability measure on [−1, 1]. For every m ∈ N and any q ∈ N0 , the (q + 1)-point Gaussian quadrature rule for the distribution µm consists of abscissae (ξj )qj=0 in [−1, 1] and weights (wj )qj=0 such that 1 q

f (ξ)µm (dξ) ≈ wj f (ξj ), (4.60) −1

j=0

and (4.60) is exact if f is a polynomial of degree at most 2q + 1. The points (ξj )qj=0 are the roots of the orthonormal polynomial of degree q + 1 with respect to the measure µm . (m) γk For a parameter γ to be specified below, we define (yk,j )j=0 as the abscissae of the (γk + 1)-point Gaussian quadrature rule for the measure (m) µm . In this case, we have qk = γk in the construction of Smolyak interpolation operators Ik,M from (4.56). By Bieri (2009b, Lemma 6.2.2), the operator Ik,M based on these nodes uses γk + 2M Γ (4.61) Nk = 2M collocation points in Γ. Sparse tensorization Let VD ⊂ V = H01 (D) denote the finite element spaces from Section 4.1, and let PD be the orthogonal projection in H01 (D) onto VD . We recall that ND := dim VD 2d , and these spaces satisfy the approximation property (4.1). For parameters M and L, the stochastic collocation solution for the Smolyak interpolation operator IM,L and the finite element space VLD is uM,L := (IM,L ⊗ PLD )u. It can be expanded as uM,L =

D (IM,k − IM,k−1 ) ⊗ (PD − P−1 )u.

(4.62)

(4.63)

0≤k,≤L

This approximation can be computed by solving a linear system in VLD for

410


each collocation point of IM,L . Therefore, a measure for the computational cost of computing uM,L is the product NL := NLΓ NLD of the number of collocation points and the dimension of the finite element space. The sparse tensor approximation can be derived by truncating the sum in (4.63), analogously to the sparse tensor product construction in Section 4.1, as

u M,L := (IM,k − IM,k−1 ) ⊗ (P D − P D )u. (4.64)

−1

0≤k+≤L

A (rough) measure for the computational cost of obtaining u M,L is the total number of degrees of freedom, i.e.,

L := N NkΓ ND , (4.65) 0≤k+≤L

which is significantly smaller than NL due to the geometric growth of NkΓ and ND . We turn next to the error analysis of this sparse collocation (m) approximation. We assume that y0 = 0 for all m ≥ M +1. This is satisfied by Gaussian abscissae if the distributions µm are symmetric. All collocation (M +1) (M +2) points in (4.62) and (4.65) are of the form y = (y , y0 , y0 , . . .) with y = (ym )m ∈ [−1, 1]M , and thus the diffusion coefficient a((y , y0

(M +1)

(M +2)

, y0

, . . .), x) = a ¯(x) +

M

ym am (x) +

m=1

=a ¯(x) +

M

∞

(m)

y0 am (x)

m=M +1

ym am (x),

(4.66)

m=1

am (x) = αm ϕm (x), depends only on y ∈ [−1, 1]M . Now let uM ∈ C(Γ; V ) denote the solution of aM (y, x)∇uM (y, x) · ∇v(x) dx = f (x)v(x) dx D

∀v ∈ V, ∀y ∈ Γ,

D

(4.67)

for the truncated series ¯(x) + aM (y, x) = a

M

ym am (x),

am (x) = αm ϕm (x).

(4.68)

m=1

Since the collocation approximations (4.62) and (4.64) depend only on the first M dimensions of Γ, we can only expect convergence to uM , not to the solution u of (4.67) with the exact diffusion coefficient a(y, x). The following statement is Proposition 6.3.1 from Bieri (2009b).

411


Theorem 4.16.

For any 0 ≤ τ ≤ 1, if γ =

M,L L2 (Γ,µ;V ) ≤

uM − u

τ log 2 rmin −1 ,

then

− min C uM C(Σ(Γ,);H 1+τ (D)) N L

where rmin := min {rm ; m = 1, . . . , M }, ( rm := log m + 1 + 2m , 1 < m
0 and all M ∈ N, 1 (4.88) P E[u] − EM [u] V ≥ η ≤ 2 u 2L2 (Ω;V ) . η M Proof.

By Chebyshev’s inequality, using that (ui )M i=1 are uncorrelated, 1 1 P E[u] − EM [u] V ≥ η ≤ 2 Var(EM [u]) = 2 Var(u). η η M

The claim follows since Var(u) ≤ u 2L2 (Ω;V ) . Equation (4.88) is equivalent to

u L2 (Ω;V ) √ ≥1−

P E[u] − EM [u] V ≤

M

∀ > 0.

(4.89)

In this sense, EM [u] converges to E[u] in V at rate M −1/2 , with probability 1 − , which can be chosen arbitrarily close to one. For all i ∈ N and any ∈ N0 , let ui denote the Galerkin projection of i u onto V . The Monte Carlo finite element (MC–FE) method consists of

414


approximating E[u] by EM [u ] =

M 1 i u M

(4.90)

i=1

for a single discretization level . Theorem 4.19. If f ∈ L2 (Ω; Ys ) and a ∈ L∞ (Ω; W s,∞ (D)) with 0 < s ≤ min(p, s∗ ), then for any > 0, @ + 1 + Cs (a)Ca CI h0 2−s f L2 (Ω;Ys ) ≥ 1 − . P E[u] − EM [u ] V ≤ √

M (4.91) Proof. The statement follows from Proposition 4.18, (4.86) and (4.84) by splitting the error as

E[u] − EM [u ] V ≤ E[u] − EM [u] V + EM [u] − EM [u ] V and using linearity of EM . By the assumption a ∈ L∞ (Ω; W s,∞ (D)), the constant Cs (a) in (4.84) is independent of ω. Remark 4.20. The optimal choice of sample size M versus discretization level is reached when the statistical and discretization errors are equilibrated. This is the case when −1/2 M −1/2 = 2−s , or equivalently, M = 22s −1 , with some rounding strategy. Since N = O(2d ), if the computational cost of computing a sample of ui is estimated as N , then the total cost of reaching a tolerance C2−s with probability 1− by the MC–FE method is O(M N ), which is equal to O(2(2s+d) −1 ). In the multi-level Monte Carlo finite element (MLMC–FE) method, the multi-level splitting of the finite element space VL is used to obtain a hierarchy of discretizations which are sampled with level-dependent MC sample sizes M . Specifically, for any L ∈ N0 , setting u−1 := 0, we write, using the linearity of the expectation operator E[·], L L

E[uL ] = E (u − u−1 ) = E[u − u−1 ]. (4.92) =0

=0

Each of the remaining expectations can be approximated by a different number of samples. This leads to the approximation E L [u] :=

L

EM [u − u−1 ]

=0

of E[u], with EM defined as in (4.87).

(4.93)


415

Theorem 4.21. If f ∈ L2 (Ω; Ys ) and a ∈ L∞ (Ω; W s,∞ (D)) with 0 < s ≤ min(p, s∗ ), then there is a constant C depending only on a, D and s such that, for any > 0 and any L ∈ N0 , " ! 0 L

L + 1 −1/2 P E[u] − E L [u] V ≤ C hsL + M hs f L2 (Ω;Ys ) ≥ 1 − .

=0

Proof.

Adding and subtracting E[uL ], we have

E[u] − E [u] V ≤ E[u] − E[uL ] V + L

L

E[u − u−1 ] − E L [u − u−1 ] V .

=0

Using (4.86), the first term is bounded by

E[u] − E[uL ] V ≤ E[ u − uL V ] ≤ Ca CI hsL E[ u Xs ] ≤ Ca CI hsL u L2 (Ω;Xs ) . We apply (4.89) with > 0 to each of the remaining terms. Thus, with probability 1 − 0 − · · · − L , L

E[u − u−1 ] − E L [u − u−1 ] V ≤

=0

L

=0

√

1

u − u−1 L2 (Ω;V ) .

M

Furthermore, due to (4.86),

u − u−1 L2 (Ω;V ) ≤ 3Ca CI hs u L2 (Ω;Xs ) . The claim follows with the choice := (L + 1)−1 , using (4.84). Remark 4.22. Theorem 4.21 holds for arbitrary choices of M in (4.93). For any > 0, we consider B A M := 22s(L−) (L + 1)3 −1 , = 0, 1, . . . , L. (4.94) Then Theorem 4.21 states that, with probability 1 − ,

E[u] − E L [u] V ≤ 2ChsL f L2 (Ω;Ys ) .

(4.95)

Assuming the availability of an optimal finite element solver such as a full multigrid method or, for d = 1, a direct solver, the total work required to compute E L [u] is L

M N ∼ 22sL (L + 1)3 −1

=0

In terms of NL =

O(2Ld ),

L

2(d−2s) .

(4.96)

=0

the computational cost  3 −1  L  NL (log NL )

NL (log NL )4 −1 M N   2s/d =0 NL (log NL )3 −1

is if 2s < d, if 2s = d, if 2s > d.

(4.97)

416


Remark 4.23. If 2s ≤ d, the cost (4.97) of the MLMC–FE method is equivalent to that of the finite element method for solving a single deterministic problem to the same tolerance, up to a logarithmic factor. For example, if d = 2, the MLMC–FE method exhibits this optimal behaviour for p = 1 and s = 1. If d = 3, linear complexity is retained up to order s = 3/2. In the case d = 1, this is only true up to s = 1/2. Already for p = 1 and s = 1, the cost of MLMC–FE in d = 1 is O(NL2 (log NL )3 −1 ). This still compares favourably to the MC–FE method, which requires O(NL3 −1 ) work to achieve the same accuracy. Generally, if 2s > d, the cost of the MLMC– FE method is equivalent to the total number M of Monte Carlo samples, independently of the dimension of the finite element spaces. Thus the efficiency of the MLMC–FE method is dominated by the weaker of its two constituent methods. The above error analysis of the MLMC–FE method is with respect to convergence in probability. Analogous results also hold when the convergence in L2 (Ω; V ) of the MLMC–FE method is analysed, as in Theorem 1.11. We refer to Barth et al. (2010) for proofs and numerical experiments. Approximation of higher moments We recall some notation from Section 1. For any k ∈ N, we denote the k-fold Hilbert tensor product of the Hilbert space V by V (k) := V · · ⊗ V . ⊗ ·

(4.98)

k times

Let u(k) (ω) denote the k-fold tensor product u(ω)⊗· · ·⊗u(ω) ∈ V (k) . Then, if u ∈ Lk (Ω; V ), (k)

u L1 (Ω;V (k) ) = u(ω) ⊗ · · · ⊗ u(ω) V (k) P(dω) Ω (4.99) k k = u(ω) V P(dω) = u Lk (Ω;V ) . Ω

The kth moment of u is Mk u := E[u(k) ] =

u(ω) ⊗ · · · ⊗ u(ω) P(dω) ∈ V (k) . Ω

(4.100)

k times

We use analogous notation for other Hilbert spaces V . ∗ We assume that f ∈ Lr (Ω; Ys∗ ) for some r∗ ≥ 2 and s∗ > 0 such that (4.84) holds. Also, we assume a ∈ L∞ (Ω; W s,∞ (D)) for all s ∈ [0, s∗ ]. The following regularity property generalizes Theorem 1.6 and is shown in Barth et al. (2010, Theorem 5.3).

417


Theorem 4.24. Under the above assumptions, for all 2 ≤ k ≤ r∗ , all 1 ≤ r ≤ r∗ /k and every 0 ≤ s ≤ s∗ ,

u(k) Lr (Ω;X (k) ) ≤ C f (k) Lr (Ω;Y (k) ) ≤ C f kLrk (Ω;Ys ) . s

s

(4.101)

We recall the sparse tensor product construction from Section 1.2. Let (ψj )(,j)∈∇ be a wavelet basis of V such that VL is the span of all ψj with ≤ L. Then the operator PL on V defined as the restriction to the coordinates with ≤ L is a projection onto VL . Let Q := P − P−1 for ∈ N0 , with P−1 := 0, and let W denote the range of Q . Then W is the span of ψj for all indices j. It complements V−1 to V in that V = V−1 ⊕ W , with a direct, but generally not orthogonal sum. We recall from (1.36) that the k-fold sparse tensor product with level L ∈ N0 is given by

(k) V1 ⊗ · · · ⊗ Vk = W1 ⊗ · · · ⊗ Wk ⊂ V (k) . VL := 0≤1 +···+k ≤L

0≤1 +···+k ≤L

(4.102) (k) k−1 k The dimension of VL is only O(NL (log NL ) ), compared to NL for the (k) full tensor product VL . If a hierarchical basis of the subspaces {V }∞ =0 ⊂ V satisfying (W1)– (W5) is explicitly given, the mappings P can be realized by truncating the corresponding expansions: see (1.38) and (1.39). These truncations provide numerically computable, stable and quasi-optimally accurate projections (k) onto VL . They are defined by

(k) Q1 ⊗ · · · ⊗ Qk . (4.103) PL := 0≤1 +···+k ≤L

This is simply the restriction of an element of V (k) , expanded in the ten···k sor product hierarchical basis (ψj11···j ) (assumed to be V -orthogonal k (i ,ji )∈∇ between different levels), to the indices with 1 + · · · + k ≤ L. (k) By Lemma 1.9, the projection PL is stable on V (k) in the sense that (k) (1.42) holds. Furthermore, the quasi-interpolant PL is quasi-optimal (pro vided that the basis (ψj ) is V -orthogonal between different levels ). Using Remark 1.8, for all 0 ≤ s ≤ min(p, s∗ ), there is a constant C(k, s) > 0 such (k) that, for all L ∈ N0 and every U ∈ Xs , (k) −s/d

U − PL U V (k) ≤ C(k, s)NL (log NL )(k−1)/2 U X (k) . s

(4.104)

By Proposition 1.7, the approximation rate in (4.104) is optimal. In terms

418


of the projection PL , the sparse tensor multi-level Monte Carlo approximation of the kth moment Mk u = E[u(k) ] reads (k)

L [u(k) ] := E

L

(k) (k) (k) (k) EM P u − P−1 u−1 ,

(4.105)

=0

where EM is defined as in (4.87), and u−1 := 0. In the case k = 1, (4.105) reduces to (4.93). The proof of the following statement is analogous to the proof of Theorem 4.21. Theorem 4.25. If f ∈ L2k (Ω; Ys ) with 0 < s ≤ min(p, s∗ ), then there exists a constant C depending only on a, D, s and k such that, for any

> 0 and any L ∈ N0 , with probability 1 − , L [u(k) ] (k)

Mk u − E V ! ≤ C hsL |log hL |(k−1)/2 +

0

"

(4.106)

L + 1 −1/2 s M h |log h |(k−1)/2 f kL2k (Ω;Ys ) .

L

=0

Remark 4.26. We consider the same choice of M as in Remark 4.22, B A (4.107) M := 22s(L−) (L + 1)3 −1 , = 0, 1, . . . , L. Then the error bound (4.106) in Theorem 4.25 becomes L [u(k) ] (k) ≤ 2ChsL |log hL |(k−1)/2 f k 2k

Mk u − E V L (Ω;Ys )

(4.108)

with probability 1− . Assuming the availability of an optimal finite element solver, such as a full multigrid method or, for d = 1, a direct solver, since the (k) dimension of V is on the order of N (log N )k−1 , the total work required L [u(k) ] is to compute E L

=0

M N (log N )k−1 22sL (L + 1)k+2 −1

L

2(d−2s) .

(4.109)

=0

In terms of NL = O(2Ld ), the computational cost is  k+2 −1  if 2s < d, L  NL (log NL )

k−1 k+3 −1 if 2s = d, NL (log NL )

M N (log N )   2s/d k+2 −1 =0 NL (log NL )

if 2s > d.

(4.110)

We refer to Barth et al. (2010) for a proof and further details. They also provide an error analysis of MLMC methods in the L2 (Ω, V (k) )-norm.


419

APPENDIX A. Review of probability A.1. Basic notation Definition A.1. (probability space) A triple (Ω, Σ, P) is called a probability space if Ω is a set, Σ is a σ-algebra on Ω, and P is a positive measure on Σ with P(Ω) = 1. Measurable sets E ∈ Σ are events; P(E) is the probability of the event E. P is called the probability measure. Remark A.2. (i) We shall always have {ω} ∈ Σ for all ω ∈ Ω. This is not implied by Definition A.1. (ii) For all E ∈ Σ, P(E) ∈ [0, 1]. (iii) Instead of ‘P-a.e.’, we write ‘P-a.s.’, which stands for ‘P almost surely’. Consider a finite number n of experiments Ei with random outcomes Ei , i = 1, . . . , n. To describe them, we assume we are given probability spaces (Ωi , Σi , Pi ), i = 1, . . . , n. Imagine next a new experiment E consisting of ‘mutually independent parallel’ experiments E1 , . . . , En . What is a suitable probability space (Ω, Σ, P) for the mathematical description of E? Any realization is of the form E (ω1 , . . . , ωn ) ∈ Ω1 × · · · × Ωn . If Ai ∈ Σi , i = 1, . . . , n is the outcome of Ei , then we consider the outcomes A1 , . . . , An of E1 , . . . , En (in this order). Hence, A := A1 × · · · × An ⊂ Ω1 × · · · × Ωn is the outcome of E. ‘Independence’ suggests that we choose P(A) := P1 (A1 ) · · · Pn (An ). The set of all events {A1 * × · · · × An ; Ai ∈ Σi } n is a generator of the product sigma* algebra Σ := i=1 Σi . On Σ, the n product measure P = P1 ⊗ · · · ⊗ Pn = i=1 Pi is the only measure satisfying the consistency condition P(A1 × · · · × An ) =

n )

Pi (Ai )

∀Ai ∈ Σi .

i=1

Obviously, P is a probability measure on Σ. Hence (Ω, Σ, P) =

n ' (Ωi , Σi , Pi ) i=1

is a probability space for E. A.2. Random variables, distributions, moments Definition A.3. (random variable) Let (Ω, Σ, P) be a probability space, and let (Ω , Σ ) be any measurable space. A (Ω , Σ )-valued random variable is any (Σ, Σ )-measurable map X : Ω → Ω .

420


Remark A.4. (i) If the measurable space (Ω , Σ ) is clear from the context, we omit it. (ii) Images of elementary events ω ∈ Ω, i.e., ω → X(ω) ∈ Ω, are referred to as samples (of X), draws (of X), or realizations (of X). (iii) Images X(A) of ‘complex’ events A ∈ Σ are called ensembles. Example A.5. (1) If (Ω , Σ ) = (R, B 1 , R1 ), we call X a random number resp. a random variable (RV). (2) If (Ω , Σ ) = (Rn , B n ), n > 1, we call X a random vector. We always assume for Ω = Rd that Σ = B n . (3) If I = [0, T ] is an interval in R and Ω = C 0 (I), we call Ω ω → Xt (ω) ∈ C 0 (I) a stochastic process; for given fixed ω ∈ Ω, the realization Xt (ω) : I t → Xt (ω) is called a (continuous) sample path (of X). (4) We shall be interested in random variables mapping into a function space Ω over a domain D ⊂ Rd , for example the Sobolev space Ω = H01 (D). Then a ‘sample’ Ω ω → u(ω) ∈ H01 (D) is a random function or random field. The construction of a σ-algebra Σ on such an Ω will be explained below. Notation A.6. Let X be a (Ω , Σ )-valued random variable. For any A ∈ Σ , {X ∈ A } := X −1 (A ) ∈ Σ, (A.1) P{X ∈ A } := P(X −1 (A )). A }

is called the ‘event that X lies in The set {X ∈ the probability of this event. Note that A → P{X ∈ A },

(A.2) A ’,

and P{X ∈ A } is

A ∈ Σ ,

is the image measure of P under X on (Ω , Σ ). Since P{X ∈ Ω } = P(Ω) = 1, it is a probability measure on (Ω , Σ ). Random variables are measurable maps between measurable spaces. Proposition A.7.

Let (Ω, Σ), (Ω , Σ ) be measurable spaces.

(a) A map T : Ω → Ω is measurable if and only if ∀A ∈ E :

T −1 (A ) ∈ Σ

(A.3)

for some generator E of Σ . (b) If T1 : (Ω1 , Σ1 ) → (Ω2 , Σ2 ) and T2 : (Ω2 , Σ2 ) → (Ω3 , Σ3 ) are measurable, then T2 ◦ T1 : (Ω1 , Σ1 ) → (Ω3 , Σ3 ) is measurable.


421

(c) If T : (Ω, Σ) → (Ω , Σ ) is measurable. Then, for every measure µ on Σ the map A −→ µ(T −1 (A )) =: µ (A )

(A.4)

is a measure µ on (Ω , Σ ). Definition A.8. (image measure) The measure µ in (A.4) is the image measure of µ under T , denoted by µ = T# (µ), i.e., T# (µ)(A ) := µ(T −1 (A ))

∀A ∈ Σ .

(A.5)

Note that (T2 ◦ T1 )# (µ) = T2# (T1# (µ)).

(A.6)

Consider now a family ((Ωi , Σi ))i∈I of measurable spaces and a family (T of maps Ti : Ω → Ωi into Ωi . Then the σ-algebra Σ generated by i )i∈I −1 T i∈I i (Σi ) in Ω is the smallest σ-algebra such that each Ti is (Σi , Σ)measurable. We write ! " . −1 σ(Ti ; i ∈ I) := σ Ti (Σi ) . (A.7) i∈I

Definition A.9. (distribution, law) Let X be a (Ω , Σ )-valued random variable on a probability space (Ω, Σ, P). Then PX := X# (P) = P ◦ X −1

(A.8)

is called the distribution of X (with respect to P) or the law of X. Hence PX (A ) = P{X ∈ A },

A ∈ Σ .

(A.9)

Definition A.10. (expectation, mean field) Let X ∈ Rn be a random variable on a probability space (Ω, Σ, P). Then E(X) = EP (X) := X P(dω) ∈ Rn (A.10) Ω

is called the expected value or expectation of X. If X ∈ Ω is a random field in a separable Banach space Ω , then (X) = X P(dω) ∈ Ω (A.10 ) E(X) = EP Ω

is called the mean field or ensemble average of X and is sometimes denoted by X when P is clear from the context. Remark A.11. The Bochner integral in (A.10 ) is well-defined if E( X ) < ∞, where · denotes the norm on Ω .

422


Remark A.12. Let (Ω , Σ ) = (Rn , B n ). Then, for any Borel-measurable function f on Rn which is PX -integrable, we have E(f ◦ X) = f dPX (A.11) Rn

or EP (f ◦ X) = EPX (f ).

(A.11 )

In particular, if X is integrable, f (x) := x gives xPX (dx). E(X) =

(A.12)

Rn

Definition A.13. (covariance) Let X be an integrable (Rn , B n )-valued random variable on a probability space (Ω, Σ, P). Then Cov(X) := E (X − E[X])(X − E[X]) ∈ Rn ⊗ Rn = Rn×n (A.13) is called the covariance of X. Note that

Cov(X) := Rn

(x − E[X])(x − E[X]) dx

(A.14)

is finite if and only if X is square-integrable. Proposition A.14. A real-valued random variable X on a probability space (Ω, Σ, P) is square-integrable if and only if X is integrable and Cov(X) = Var(X) < ∞. Then

Var(X) = E (X − E(X))2 = E(X 2 ) − E(X)2 2 2 = x PX (dx) − xPX (dx) . R

(A.15)

R

A.3. Independence Let (Ω, Σ, P) be a probability space and let I be a set of indices. Definition A.15. (independent events) A family (Ai )i∈I of events in Σ is called independent (with respect to P) if, for every non-empty, finite index set {i1 , . . . , in } ⊂ I, P(Ai1 ∩ · · · ∩ Ain ) = P(Ai1 ) · · · P(Ain ).

(A.16)

Example A.16. Let (Ωi , Σi , Pi ), i = 1, . . . , n, be probability spaces, and * (Ω, Σ, P) = ni=1 (Ωi , Σi , Pi ). For each i, let Ai ∈ Σi . Then the events Ai := Ω1 × · · · × Ωi−1 × Ai × Ωi+1 × · · · × Ωn , in Ω are independent.

i = 1, . . . , n,


423

Definition A.17. (independent families of events) Let (Ei )i∈I be a family of sets in Σ. It is called independent if (A.16) holds for every nonempty, finite index set {i1 , . . . , in } ⊂ I and every possible choice of Aiν ∈ Eiν , ν = 1, . . . , n. It is clear from the definition that independence is preserved if each Ei is reduced. Remark A.18. (i) Independence is preserved if each Ei is increased to its Dynkin system δ(Ei ) ⊂ Σ. (ii) If (Ei )i∈I ⊂ Σ is any independent family of ∩-stable subsets Ei of Σ, then the family (σ(Ei ))i∈I is independent. (iii) If (Ei )i∈I ⊂ Σ is as in (ii), and (Ij )j∈J is a partition of I into mutually disjoint Ij ⊂ I, then the system " ! . Σj := σ Ei , j ∈ J , i∈Ij

is independent. A.4. Independent random variables Let (Ω, Σ, P) be a probability space. By Remark A.18(ii), the family (Ai )i∈I is independent if and only if the family (Σi )i∈I of σ-algebras is independent, where Σi = {∅, Ai , Aci , Ω}. Definition A.19. (independent random variables) A family (Xi )i∈I of random variables (with i-dependent ranges) is independent if (σ(Xi ))i∈I is independent. Theorem A.20. Let (Xi )i=1,...,n be (Ωi , Σi )-valued random variables, and let Gi be a ∩-stable generator of Σi , Ωi ∈ Gi , i = 1, . . . , n. Then (Xi )i=1,...,n is independent if and only if, for all Qi ∈ Gi , i = 1, . . . , n, " !n n / ) −1 Xi (Qi ) = P(Xi−1 (Qi )). (A.17) P i=1

Proof.

i=1

Put Ei := {Xi−1 (Qi ) ; Qi ∈ Gi }.

Then Ei is a generator of σ(Xi ), and, by ∩-stability of Gi , Ei is ∩-stable and Ω ∈ Ei . By Remark A.18(ii), we must show that independence of (Ei )i∈I is equivalent to ∀Ei ∈ Ei :

P(E1 ∩ · · · ∩ En ) = P(E1 ) ∩ · · · ∩ P(En ).

424


This is evident. Let Xi be (Ωi , Σi )-valued random variables, i = 1, . . . , n, on a single probability space (Ω, Σ, P), and define the product map Y := X1 ⊗ · · · ⊗ Xn : Ω → Ω1 × · · · × Ωn by Y (ω) := (X1 (ω), . . . , Xn (ω)),

ω ∈ Ω.

(A.18)

Then, for each A1 × · · · × An , Ai ∈ Σi , i = 1, . . . , n, (A.19) Y −1 (A1 × · · · × An ) = X1−1 (A1 ) ∩ · · · ∩ Xn−1 (An ). 6n *n Hence Y is a ( i=1 Ωi , i=1 Σi )-valued random variable on (Ω, Σ, P), and the distributions PXi , i = 1, . . . , n, and PY are well-defined. We call PY = PX1 ⊗ · · · ⊗ PXn the joint distribution of X1 , . . . , Xn . It is a probability measure on the product-measurable space " ! n n n ) ' ) Ωi , Σi = (Ωi , Σi ). i=1

i=1

i=1

Theorem A.21. Finitely many random variables Xi , i = 1, . . . , n, are independent if and only if their joint distribution is the product of their marginal distributions, i.e., if PX1 ⊗···⊗Xn = PX1 ⊗ · · · ⊗ PXn . Proof.

For each i = 1, . . . , n, let Ai ∈ Σi be an event. By (A.19), ! n " ! n " !n " ) ) / −1 −1 =P PY Ai = P Y Ai Xi (Ai ) , i=1

i=1

i=1

and PXi (Ai ) = P(Xi−1 (Ai )),

i = 1, . . . , n.

Hence, PY is the measure of the PXi if and only if, for any Ai ∈ Σi , PY (A1 × · · · × An ) = PX1 (A1 ) · · · PXn (An ). This is equivalent to !n " n / ) −1 P Xi (Ai ) = P(Xi−1 (Ai )) i=1

∀Ai ∈ Σi , i = 1, . . . , n.

i=1

By Theorem A.20, (A.17), X1 , . . . , Xn are independent.

(A.20)


425

A.5. Infinite products of probability spaces We saw that the proper model for the description of n < ∞ independent experiments with random outcome is the product probability space * (Ω, Σ, P) = ni=1 (Ωi , Σi , Pi ). We now consider the case where we have an infinite family E = (En )∞ n=1 of ‘independent’, ‘random’ experiments. Each experiment is described by a probability space (Ωn , Σn , Pn ), n = 1, 2, . . . . We build a probability space to describe E. It should satisfy the following conditions. (1) Each elementary event ω ∈ Ω is a sequence (ωn )∞ n=1 of elementary events ωn ∈ Ωn , i.e., Ω=

∞ )

Ωn = Ω1 × Ω2 × · · · .

n=1

(2) If A1 ∈ Σ1 , . . . , An ∈ Σn are possible outcomes at the first n experiments, we view the set A = A1 × · · · × An × Ωn+1 × Ωn+2 × · · · ,

n = 1, 2, . . .

(A.21)

as that outcome of E which gives A1 , . . . , An in the first n experiments of the (infinite) sequence (Ei )∞ i=1 of experiments. Therefore, we require that A defined in (A.21) satisfies A ∈ Σ and that P(A) = P1 (A1 ) · · · Pn (An ).

(A.22)

The requirements (A.21) and (A.22) and a certain minimum property define the measure P uniquely. Given an index set I = ∅ and a family ((Ωi , Σi , Pi ))i∈I of probability spaces, for each K ⊆ I we define ) ΩK := Ωi , (A.23) i∈K

and we set Ω := ΩI =

)

Ωi .

(A.24)

i∈I

Note that ΩK is the set of all maps . ωK : K → Ωi such that ωK (i) ∈ Ωi

∀i ∈ K.

i∈K

Restricting ωK to J ⊂ K, we get the projection map pK J := ΩK → ΩJ .

(A.25)

K If K = I, we write pJ := pIJ ; if j = {i}, pK i := p{i} . Then K L pL J = p J ◦ pK ,

J ⊂ K ⊂ I.

(A.26)

426


By F = F(I) we denote the set of all finite subsets of I. For each J ∈ F, we have ' ' ΣJ := Σi , PJ := Pi . (A.27) i∈J

i∈J

Definition A.22. (infinite product of σ-algebras) We call the prod* uct Σ of the family {Σi ; i ∈ I} of σ-algebras the smallest σ-algebra i i∈I Σ0 in Ω for which each projection pi is (Σ0 , Σi )-measurable, i.e., ' Σ0 = Σi := σ(pi ; i ∈ I). (A.28) i∈I

For all J ∈ F, pJ is (Σ0 , ΣJ )-measurable, since, by (A.26), we have pi = pJi ◦ pJ for all i ∈ J. This allows us to extend (A.28) to ' Σi = σ(pi ; i ∈ I) = σ(pJ ; J ∈ F(I)). (A.28 ) i∈I

We now wish to find a probability measure P on Σ0 such that ! "" ! ) ) −1 = Ai Pk (Ai ) ∀J ∈ F, ∀Ai ∈ Σi , ∀i ∈ J. P pJ i∈J

i∈J

By 6 definition of the image of a measure under a mapping, each pJ (P) of a set i∈J Ai has the value " ! ) ) (pJ )# (P) Ai = Pi (Ai ). (A.29) i∈J

i∈J

For all J ∈ F, the finite product measure PJ in (A.27) is the unique measure such that (A.29) holds. Does there exist a probability measure P on Σ0 such that its image under any projection pJ , J ∈ F, equals PJ ? * Theorem A.23. There exists a unique measure P on Σ0 = i∈I Σi such that (pJ )# (P) = PJ ∀J ∈ F(I). (A.30) P is a probability measure on (Ω, Σ0 ). We refer to Bauer (1996) for the proof. If |I| < ∞, P = PI by (A.30). Definition A.24. (infinite product measure) The unique probability A.23 is called the product of the measures measure P on Σ0 from Theorem * (Pi )i∈I and is denoted by i∈I Pi . The probability space " ! ' ' ) (Ω, Σ0 , P) = Ωi , Σi , Pi i∈I

i∈I

i∈I


427

is called the product of the probability spaces ((Ωi , Σi , Pi )i∈I ) and is denoted by ' (Ω, Σ0 , P) =: (Ωi , Σi , Pi ). i∈I

Now we can extend Theorem A.21 on independence of random variables X1 , . . . , Xn . Theorem A.25. A family (Xi )i∈I of random variables is independent if and only if their joint distribution is the product of the distributions PXi , i.e., ' PX i . (A.31) P Xi = i∈I

Proof.

i∈I

For every ∅ = J ⊂ I, J ∈ F, let ) ) Ωi → Ωj pJ : i∈I

j∈J

denote the projection, let Y denote the mapping 6 * Ω denote j j∈J j∈J Xj . Then YJ = pJ ◦ Y , whence it follows that

* i∈I

Xi , and let YJ : Ω →

PYJ = (pJ )# (PY ), by transitivity (A.6) of image measures. Independence of (Xi )i∈I is equivalent to independence of (Xj )j∈J for all J ∈ F, i.e., by Theorem A.21, ' PXj ∀J ∈ F. P YJ = j∈J

By Theorem A.23, (A.31) is equivalent to ' PX j (pJ )# (PY ) =

∀J ∈ F.

j∈J

The assertion follows. Corollary A.26. For any family ((Ωi , Σi , Pi ))i∈I of probability spaces, there exists an independent family (Xi )i∈I of (Ωi , Σi )-valued random variables Xi on a suitable probability space (Ω, Σ, P) such that ∀i ∈ I :

Pi = PX i

is the distribution of Xi . Proof.

We choose (Ω, Σ, P) =

' (Ωi , Σi , Pi ) i∈I

(A.32)

428


and Xj = pj :

)

Ωi → Ωj .

i∈I

* Then, by the definition of product measure P = i∈I Pi , Pj is the distribution of Xj , for all j ∈ I. * The independence of (Xi )i∈I6follows from Theorem A.25, since i∈I Xi is the identity map from Ω = i∈I Ωi onto itself, whence ' ' Pi = PX i . Pi∈I Xi = P = i∈I

i∈I

B. Review of Hilbert spaces We review several standard notions and definitions of bases in separable Hilbert spaces to the extent necessary in the present work; among the many references for this material, we mention in particular Christensen (2008, 2010). B.1. Basic properties By H, we denote a real , separable Hilbert space, with norm · H and inner product ·, ·. Definition B.1. (Schauder basis) A sequence (ek )∞ k=1 ⊂ H is called a (Schauder ) basis of H if for any x ∈ H there exists a unique sequence (ck )∞ k=1 such that n

ck ek → 0, n → ∞, (B.1) x − k=1

H

or equivalently, x=

∞

c k ek

(B.1 )

in H.

k=1

Proposition B.2. Every separable Hilbert space over R admits a basis ∞ (ek )∞ k=1 . Given any basis (ek )k=1 of H, there exists a unique sequence ∞ (gk )k=1 ⊂ H such that ∀f ∈ H :

f=

∞

f, gk ek

in H.

(B.2)

k=1

Then (gk )∞ k=1 is also a basis of H, called the dual basis. The sequences (ek )∞ and (gk )∞ k=1 k=1 are biorthogonal , ek , gj = δkj .

(B.3)

429

Sparse tensor discretizations for sPDEs ∞ We also say (gk )∞ k=1 is the biorthogonal system for (ek )k=1 .

(fk )∞ k=1 ⊂ H is a Bessel sequence if

Definition B.3. (Bessel sequence) ∃B > 0

∞

|f, fk |2 ≤ B f 2H .

∀f ∈ H :

k=1

B is called a Bessel bound. Lemma B.4. (fk )∞ k=1 ⊂ H is a Bessel sequence with Bessel bound B if and only if ∞

T : (ck )∞ → c k fk k=1 k=1

is a well-defined, bounded operator from 2 (N) into H and T ≤

√ B.

2 Proof. Assume (fk )k ⊂ H is Bessel ∞ and (ck )k ∈ (N). To show that T (ck )k is well-defined, we show that k=1 ck fk converges in H. Let m, n ∈ N, n > m. Then n % n & n m

c f − c f c f c f , g = = sup k k k k k k k k g =1 k=1

k=1

k=m+1

k=m+1

n

≤ sup

|ck fk , g|

g =1 k=m+1

!

n

≤

"1/2

|ck |2

k=m+1

≤

√

!

n

B

! sup g =1

"1/2 |ck |2

n

"1/2 |fk , g|2

k=m+1

.

k=m+1

n ∞ Since (ck ) ∈ ( n1 |ck |2 )∞ n=1 is Cauchy, and thus ( 1 ck fk )n=1 ⊂ H is convergent. Therefore T : 2 (N) → H is well-defined and bounded with √

T ≤ B. Obviously, T is also linear. Since 2 (N),

∞

|f, fk |2 = T ∗ f 22 ≤ T 2 f 2H

∀f ∈ H,

k=1

the claim follows. ∞ Lemma B.5. Let (ek )∞ k=1 be a basis of H and (gk )k=1 the associated ∞ biorthogonal system. If (ek )k=1 is a Bessel sequence with bound B, then ∞

1 |f, gk |2

f 2H ≤ B k=1

∀f ∈ H,

(B.4)

430


and for all finitely supported sequences (ck )∞ k=1 , 2 ∞ ∞ 1 2 |ck | ≤ c k gk . B k=1 k=1 H ∞ Proof. Let f ∈ H. Since f = k=1 f, gk ek , 2 ∞

f 4H = f, gk ek , f

(B.5)

k=1

≤

∞

|f, gk |2

k=1

∞

|ek , f |2 ≤ B f 2H

k=1

∞

|f, gk |2 .

k=1

This shows (B.4). Let (ck )∞ k=1 be a finitely supported sequence. Then (B.5) follows from ∞ 2 % ∞ 2 &2 ∞ ∞ ∞ ∞

|ck |2 = cj gj , ek = c j g j , ek ≤ B c j gj . k=1

k=1 j=1

k=1

δjk

j=1

j=1

H

B.2. Bases of Hilbert spaces As before, we let H denote a separable Hilbert space. Definition B.6. (orthonormal basis) A sequence (ek )∞ k=1 ⊂ H is an orthonormal system in H if ek , ej = δkj . It is an orthonormal basis of H if, in addition, it is a basis of H, i.e. n

∀x ∈ H : x − x, ei ei → 0, n → ∞, (B.6) i=1

H

or equivalently, ∀x ∈ H :

x=

∞

x, ei ei

in H.

(B.6 )

i=1

Remark B.7. Any orthonormal system (ek )k ⊂ H is a Bessel sequence with Bessel constant B = 1. Proof. Let (ck )k ⊂ 2 (N), m, n ∈ N, n > m. Then ( nk=1 ck ek )n converges in H since 2 m m

c e = |ck |2 → 0, m → ∞, k k k=n+1

H

n+1


and we find

431

2 ∞ ∞

c e = |ck |2 = c 22 (N) . k k k=1

H

k=1

Therefore, for f ∈ H, (ck )k = (f, ek )k ∈ 2 (N) and thus B = 1. Theorem B.8. For an orthonormal system (ek )k ⊂ H, the following conditions are equivalent: (a) (ek )k is an orthonormal basis of H, (b) for all f ∈ H, f = ∞ k=1 f, ek ek , (c) for all f, g ∈ H, f, g = ∞ k=1 f, ek ek , g, ∞ (d) for all f ∈ H, k=1 |f, ek |2 = f 2H (Parseval), (e) span(ek )∞ k=1 is dense in H, (f) f, ek = 0 for all k ∈ N implies f = 0. Corollary B.9. If (ek )k is an orthonormal basis of H, it coincides with its biorthogonal basis, and ∀f ∈ H :

∞

f= f, ek ek . k=1

Theorem B.10. basis (ek )k .

Every separable Hilbert space H has an orthonormal

Proof. Since H is separable, there exists a sequence (fk )∞ k=1 for which span(fk )∞ is dense in H. We assume, without loss of generality, that for k=1 n all n ∈ N, fn+1 ∈ / span(fk )k=1 . Applying Gram–Schmidt orthogonalization to (fk )∞ , we obtain an orthonormal system (ek )∞ k=1 k=1 ⊂ H such that n span(ek )k=1 = span(fk )nk=1 for all n ∈ N. By Theorem B.8, (ek )k is an orthonormal basis of H. Example B.11. thonormal basis.

If H = 2 (N), then (δk ) = (0, 0, . . . , 0, 1, 0, . . .) is an or-

Theorem B.12. Every separable infinite-dimensional Hilbert space H is isometrically isomorphic to 2 (N). Proof. Let (ek )∞ k=1 be an orthonormal basis of H. Then, for all (ck ) ∈ 2 (N), ∞ c e k=1 k k is convergent. Also, ∀f ∈ H :

∞

f, ek ek . f= k=1

432


2 If (δk )∞ k=1 is the orthonormal basis of (N) from Example B.11, we define " !∞ ∞

U : H → 2 (N) : U ck ek := ck δk , (ck )k ∈ 2 (N). k=1

k=1

ThenU : H → 2 (N) is well-defined and bijective. Also, for all f ∈ H, f = k f, ek ek and 2 ∞ ∞

2

U f 2 (N) = f, ek δk = |f, ek |2 = f 2H . 2 k=1

(N)

k=1

Theorem B.13. Let (ek )k be an orthonormal basis of a separable Hilbert space H. Then all orthonormal bases of H are of the form (U ek )∞ k=1 for a unitary map U : H → H. Proof.

Let (fk )k be an orthonormal basis of H. Define " !

c k ek = ck fk , (ck )k ∈ 2 (N). U : H → H, U k

(B.7)

k

Then U : H → H is bounded and bijective. For f, g ∈ H, f = f, e k ek , g = k k g, ek ek . Then using (B.7), Theorem B.8 implies U ∗ U f, g = U f, U g & %

f, ek fk , g, e f = f, ek g, ek = f, g. = k

k

U ∗U

Therefore = I, and since U is surjective by (B.7), U is unitary. Conversely, if a unitary map U is given, then U ek , U ej = U ∗ U ek , ej = ek , ej = δkj , so (U ek )k is an orthonormal system. Since U is surjective, (U ek )k is an orthonormal basis of H. B.3. Tensor products of separable Hilbert spaces Tensor products are useful in the design of sparse approximation schemes. We consider tensor products of separable Hilbert spaces and refer to Light and Cheney (1985), Ryan (2002), Schatten (1943), Grothendieck (1955) and Kalton (2003) for a general theory of tensor products of Banach spaces. We follow the construction in Reed and Simon (1980) for separable Hilbert spaces. Let H1 , H2 be two separable Hilbert spaces. For ϕ1 ∈ H1 , ϕ2 ∈ H2 , we denote by ϕ1 ⊗ ϕ2 the conjugate bilinear form on H1 × H2 defined by (ϕ1 ⊗ ϕ2 )(ϕ1 , ψ2 ) := ψ1 , ϕ1 H1 ψ2 , ϕ2 H2

∀ψi ∈ Hi , i = 1, 2.


433

Let E denote the space of all finite linear combinations of such bilinear forms; on E, we define an inner product by ϕ ⊗ ψ, η ⊗ µ := ϕ, ηH1 ψ, µH2 ,

ϕ, η ∈ H1 ,

ψ, µ ∈ H2 .

(B.8)

·, · from (B.8) is well-defined and positive definite.

Proposition B.14.

Proof. To show that ·, · is well-defined, we check that λ, λ is independent of the linear combination of simple tensors used to represent λ and λ in E. By linearity and symmetry, it suffices to show that if µ is a finite sum in E equal to the zero form, then η, µ = 0 Let η=

N

∀η ∈ E.

ci (ϕi ⊗ ψi ).

i=1

Then η, µ =

%N

& ci (ϕi ⊗ ψi ), µ

i=1

=

N

ci µ(ϕi , ψi ) = 0,

i=1

since µ is the zero form. Hence ·, · is well-defined. Next, we show that ·, · is positive definite. Let λ=

M

dk (ηk ⊗ µk ).

k=1

Then M1 := span {ηk }N k=1 ⊂ H1 ,

M2 := span {µk }M k=1 ⊂ H2

N2 1 are subspaces. Let {ϕj }N j=1 , {ψ }=1 be orthonormal bases of M1 and M2 . Then we can write each ηk uniquely in terms of the ϕj , and each µk uniquely via the ψ to get N2 N1

λ= cj (ϕj ⊗ ψ ). j=1 =1

We compute

%

λ, λ =

j,

=

j,,i,m

cj (ϕj ⊗ ψ ),

& cim (ϕi ⊗ ψm )

i,m

cj cim ϕj , ϕi H1 ψ , ψm H2 =

|cj |2 ≥ 0.

j,

Furthermore, if λ, λ = 0, then cj = 0 for all j, , hence λ = 0. Thus ·, · is positive definite. Therefore, E is a pre-Hilbert space with inner product ·, ·.

434


Definition B.15. (tensor product of separable Hilbert spaces) The tensor product H1 ⊗ H2 of H1 and H2 is the completion of E under ·, ·. Proposition B.16. If (ϕk )k and (ψ ) are orthonormal bases of H1 and H2 , respectively, then the set of all dyadic products (ϕk ⊗ ψ )k, is an orthonormal basis of H1 ⊗ H2 . Proof. We assume that H1 , H2 are infinite-dimensional and separable. Then (ϕk ⊗ ψ )k is an orthonormal set in H1 ⊗ H2 , and hence we must show E ⊂ S := spanH1 ⊗H2 (ϕk ⊗ ψ ). Let ϕ ⊗ ψ ∈ E. Since (ϕk )k , (ψ ) are bases, we have

ϕ= c k ϕk , |ck |2 < ∞, ψ = d ψ , |d |2 < ∞. k

k

Therefore, there exists a vector

µ= ck d ϕk ⊗ ψ ∈ S, k,

and

N2 N1

ck d ϕk ⊗ ψ → 0, ϕ ⊗ ψ −

N1 , N2 → ∞.

k=1 =1

Let (M1 , µ1 ), (M2 , µ2 ) denote two measure spaces. We assume that L2 (M2 , µ1 ) and L2 (M2 , µ2 ) are separable. Further, let (ϕk (x))k , (ϕ (y)) denote orthonormal bases of L2 (M1 , µ1 ) and of L2 (M2 , µ2 ), respectively. We show that (ϕk (x) ψ (y))k is then an orthonormal basis of L2 (M1 × M2 , µ1 ⊗ µ2 ). To see this, we assume f (x, y) ∈ L2 (M1 × M2 , µ1 ⊗ µ2 ) and f (x, y)ϕk (x)ψ (y) (µ1 ⊗ µ2 ) d(x, y) = 0 ∀k, . M1 ×M2

By Fubini’s theorem,

f (x, y)ϕk (x) µ1 (dx) ψ (y)µ2 (dy) = 0.

M2

M1

Since (ψ ) is an orthonormal basis of L2 (M2 , µ2 ), we have f (x, y)ϕk (x) µ1 (dx) = 0, for µ2 -a.e. y.

(B.9)

M1

Let Sk = {y ∈ M2 ; (B.9) = 0}. Then µ2 (Sk ) = 0, and . Sk : f (x, y)ϕk (x) µ1 (dx) = 0 ∀k. ∀y ∈ / k

M1

Therefore, f (x, y) = 0 for all y ∈ / k Sk , µ1 -a.e. x ∈ M1 , and consequently f (x, y) = 0 for µ1 ⊗ µ2 -a.e. (x, y) ∈ M1 × M2 . This implies that (ϕk (x) ψ (y))k is a basis of L2 (M1 × M2 , µ1 ⊗ µ2 ).


435

Let U : ϕk ⊗ ψ → ϕk (x)ψ (y). Then U maps the orthonormal bases (ϕk )k of L2 (M1 , µ1 ) and (ϕ ) of L2 (M2 , µ2 ) onto the orthonormal basis (ϕk ⊗ ψ ) of L2 (M1 × M2 , µ1 ⊗ µ2 ). Thus U extends uniquely to a unitary isomorphism U : L2 (M1 , µ1 ) ⊗ L2 (M2 , µ2 ) → L2 (M1 × M2 , µ1 ⊗ µ2 ). Note that for f ∈ L2 (M1 , µ1 ) and g ∈ L2 (M2 , µ2 ), " ! " !

c k ϕk ⊗ d ψ = U c k d ϕk ⊗ ψ U (f ⊗ g) = U =

k

k,

ck d ϕk (x)ψ (y) = f (x)g(y).

k,

Thus U

L2 (M1 × M2 , µ2 ⊗ µ2 ) ∼ = L2 (M1 , µ1 ) ⊗ L2 (M2 , µ2 ). Also, if (M, µ) is a measure space, and H is a separable Hilbert space with orthonormal basis (ϕk )k , then we have in H: ∀g ∈ L2 (M, µ; H) : Define U:

N

k=1

g(x) = lim

N

N →∞

fk (x) ⊗ ϕk →

k=1

N

(ϕk , g(x))H ϕk .

=:fk (x)∈L2 (M,dµ)

fk (x)ϕk .

k=1

Then U is well-defined on a dense subset of L2 (M, µ) ⊗ H onto a dense set in L2 (M, µ; H), preserving norms. Thus U extends uniquely to a unitary operator U : L2 (M, µ) ⊗ H → L2 (M, µ; H). Theorem B.17. Let (M1 , µ1 ) and (M2 , µ2 ) be measure spaces such that L2 (M1 , dµ1 ) and L2 (M2 , dµ2 ) are separable. (a) There is a unique unitary isomorphism from L2 (M1 , µ1 ) ⊗ L2 (M2 , µ2 ) to L2 (M1 × M2 , µ1 ⊗ µ2 ) such that f ⊗ g → f g. (b) For any separable Hilbert space H, there exists a unique unitary isomorphism from L2 (M1 , µ1 )⊗H to L2 (M1 , µ1 ; H) such that f (x)⊗ϕ → f (x)ϕ. (c) There exists a unique unitary isomorphism from L2 (M1 × M2 , µ1 ⊗ µ2 ) to L2 (M1 , µ1 ; L2 (M2 , µ2 )) satisfying f (x, y) → f (x, ·).

436


B.4. Linear operators on Hilbert spaces Again we denote by H and U real , separable Hilbert spaces, with norms

· H , · U , inner products ·, ·H , ·, ·U and associated Borel sets B(H) and B(U ), respectively. Let L(U, H) denote the Banach space of all bounded linear maps from U into H. For U = H, we write L(H) = L(H, H). For T ∈ L(H), T ≥ 0 denotes a non-negative, self-adjoint operator, for which h, T hH ≥ 0 for all h ∈ H. Let L+ (H) be the set of all such operators, L+ (H) := {T ∈ L(H) ; T x, x ≥ 0 ∧ T x, y = x, T y

∀x, y ∈ H}.

By K(U, H) ⊂ L(U, H) we denote the set of compact linear operators from U to H. In building Gaussian measures on Hilbert spaces, we shall be using two important subsets of L(U, H): the nuclear operators, which are also called trace-class operators, and the Hilbert–Schmidt operators. We recapitulate basic properties as needed here and refer to the Appendix of Peszat and Zabczyk (2007) for a more detailed overview. Definition B.18. By L1 (U, H) ⊂ L(U, H) we denote the subset of nuclear operators from U to H: T ∈ L(U, H) is nuclear if there exist sequences {aj }j∈N ⊂ H, {bj }j∈N ⊂ U such that ∞ j=1 aj H bj U < ∞ and such that ∀f ∈ U :

Tf =

∞

f, bj U aj .

(B.10)

j=1

The set L1 (U, H) is a Banach space with norm ∞ ∞

T L1 (U,H) := inf

aj H bj U ; T f = f, bj U aj , ∀f ∈ U . j=1

j=1

(B.11) Note that L1 (U, H) ⊂ K(U, H) since each T ∈ L1 (U, H) can be approximated in operator norm by a sequence of operators of finite rank. Lemma B.19. Let T ∈ L1 (H) and let {ek }∞ k=1 be an orthonormal basis of H. Then ∞

Tr T = T ek , ek H (B.12) k=1

exists and is independent of the particular choice of orthonormal basis. Definition B.20. We call T ∈ L(U, H) a Hilbert–Schmidt operator (HS operator) if ∞

T ek 2H < ∞ (B.13) k=1


437

for some orthonormal basis (ek )∞ k=1 of U . We denote the subset of L(U, H) of HS operators by L2 (U, H). The linear space L2 (U, H) of HS operators from U into H is a separable Hilbert space: its scalar product is defined in terms of the orthonormal basis (ek )k ⊂ U of U , ∞

S, T HS := Sek , T ek H . (B.14) k=1

We denote by · HS the corresponding Hilbert–Schmidt norm. For S ∈ L2 (U, H) and orthonormal bases (ek )k ⊂ U , (fk )k ⊂ H of U and of H, respectively, we have

Sek 2H = Sek , fj 2H = ek , S ∗ fj 2U =

S ∗ fj 2U . k

kj

j

kj

Therefore we have the following. Proposition B.21. The HS operator norm · HS does not depend on the choice of orthonormal basis for U . Also, S ∈ L2 (U, H) if and only if S ∗ ∈ L2 (H, U ) and S L2 (U,H) = S ∗ L2 (H,U ) . Moreover, if (fk )k ⊂ H and (ek )k ⊂ U are orthonormal bases, then the rank-one operators (fj ⊗ ek )j,k∈N defined by (fj ⊗ ek )(u) := fj ek , uU ,

u∈U

are an orthonormal basis of L2 (U, H). We collect further properties of operators in L1 and L2 . Proposition B.22. (a) For S ∈ L1 (U, H) and T ∈ L(H, V ), T S ∈ L1 (U, V ) and

T S L1 (U,V ) ≤ S L1 (U,H) T L(H,V ) . (b) If S ∈ L(U, H), T ∈ L1 (H, V ) then T S ∈ L1 (U, V ) and

T S L1 (U,V ) ≤ S L(U,H) T L1 (H,V ) . (c) If S ∈ L(U, H) and T ∈ L(H, U ), and if either S or T is of trace class, then T S ∈ L1 (U ) and Tr(T S) = Tr(ST ). (d) if S ∈ L(U ) and if T ∈ L2 (U, H) then T S ∈ L2 (U, H) and

T S L2 (U,H) ≤ T L2 (U,H) S L(U ) . (e) If K(U, H) ⊂ L(U, H) denotes the subset of compact linear operators from U to H, L1 (U, H) ⊂ L2 (U, H) ⊂ K(U, H) ⊂ L(U, H).

438


Proof. We show (e): we can write R ∈ L1 (U, H) as R = k bk ⊗ ak , where (ak )k ⊂ U and (bk )k ⊂ H are bases such that k ak U bk H < ∞ and we recall that b ⊗ a(u) = ba, uU . Let (en )n ⊂ U denote an orthonormal basis of U . Then 2

2

Ren H = bk ak , en U H n n k

≤ |bk , bl | |ak , en U | |al , en U | n

≤

kl

bk H bl H

≤

1/2

n

kl

ak , en 2U

n

2

bk H ak U

1/2 al , en 2U

.

k

Taking the infimum over all bases (ak )k ⊂ U and (bk )k ⊂ H such that k ak U bk H < ∞, we obtain the estimate

R L2 (U,H) ≤ R L1 (U,H) , which proves L1 (U, H) ⊂ L2 (U, H). The next inclusion follows from the fact that the HS norm is a stronger norm than the operator norm, and that K(U, H) is closed in L(U, H) and that each HS operator R ∈ L2 (U, H) is the limit, in the HS norm, of the sequence of operators of finite rank,

Tn = fk ⊗ T ∗ fk , n ∈ N. k≤n

We recall the spectral theorem for compact, self-adjoint operators. Proposition B.23. For Q ∈ K(H) and Q = Q∗ , there exists an orthonormal basis (ek )∞ k=1 of eigenfunctions of H such that Qek = λk ek , and a decreasing sequence (λk )k ⊂ R, λ1 ≥ λ2 ≥ · · · ≥ 0 of real, non-negative eigenvalues which accumulate only at zero. Moreover, ∀x ∈ H :

Qx =

∞

λk x, ek ek ,

k=1

and if Q ∈ L+ 1 (H), ∞ > Tr(Q) =

∞

λk .

k=1

We finally add a result which is useful in the context of Karhunen–Loève expansion of random fields. It should be compared to Theorem B.17.

439


Proposition B.24. Let (Ei , Ei ), i = 1, 2 be two measurable spaces and let µi be σ-finite measures on (Ei , Ei ), i = 1, 2. Further, let E = E1 ×E2 and E := E1 ⊗ E2 denote the product space. Then product measure µ = µ1 ⊗ µ2 is σ-finite on (E, E). Further, let U = L2 (E1 , E1 , µ1 ) and H = L2 (E2 , E2 , µ2 ) be separable. Then an operator R ∈ L(U, H) belongs to L2 (U, H) if and only if there is a kernel K ∈ L2 (E, E, µ) such that ∀ψ ∈ U, ξ ∈ E2 : Rψ(ξ) := K(η, ξ)ψ(η)µ1 (dη). (B.15) E1

Then

R 2L2 (U,H)

|K(η, ξ)|2 µ1 (dη)µ2 (dξ).

= E1

(B.16)

E2

2 ∞ Proof. Let (ek )∞ k=1 be an orthonormal basis of L (E1 , E1 , µ1 ) and let (fk )k=1 2 be an orthonormal basis of = L (E2 , E2 , µ2 ). Further, let the operator R be given by (B.15). Then the Parseval equality (Theorem B.8(d)) implies 2 ∞

Ren 2H = K(η, ξ)en (η)µ1 (dη) µ2 (dξ) n=1

E2

E1

|K(η, ξ)|2 µ(dη)µ2 (dξ).

= E2

E1

Conversely, assume that R ∈ L2 (U, H) and let (en )∞ n=1 be an orthonor2 ∞ mal basis of L (E1 , E1 , µ1 ) and let (fk )k=1 be an orthonormal basis of L2 (E2 , E2 , µ2 ). Define the kernel K(η, ξ) : U × H → R by K(η, ξ) :=

∞ ∞

Ren , fk N en (η)fk (ξ).

n=1 k=1

C. Review of Gaussian measures on Hilbert spaces C.1. Measures on metric spaces For any complete metric space E, denote by B(E) the Borel σ-algebra generated by all closed or, equivalently, open sets of E. A random variable in (Ω, Σ, P) with values in E is a mapping X : Ω → E such that ∀I ∈ B(E) : X −1 (I) ∈ Σ. The law of X is the probability measure X# P on (E, B(E)) defined by X# P(I) := P(X −1 (I)) = P(X ∈ I)

∀I ∈ B(E).

440


Proposition C.1. (change of variables) Let X be a random variable on (Ω, Σ, P) with values in E. Let ϕ : E → R be bounded and B(R)measurable. Then ϕ(X(ω))P(dω) = ϕ(x) X# P(dx). (C.1) Ω

E

We show (C.1) for ϕ = 1I , I ∈ B(E). In this case,

Proof.

ϕ(X(ω)) = 1

X −1 (I)

ω ∈ Ω.

(ω),

∈Σ

Hence,

ϕ(X(ω)) P(dω) = P(X −1 (I)) = X# P(I) =

Ω

ϕ(x) X# P(dx). E

The general case follows by approximating ϕ(x) by simple functions. C.2. Gaussian measures on separable Hilbert spaces We present a concrete construction of Gaussian measures on separable Hilbert spaces as countable products of Gaussian measures on R, generalizing naturally the construction of Gaussian measures in Rd for finite dimensions d < ∞, following Da Prato (2006). Gaussian measure on R For a pair (a, λ) of real numbers with a ∈ R, λ ≥ 0, define a measure Na,λ on (R, B(R)) as follows. If λ = 0, Na,0 := δa , where

δa (B) :=

1 if a ∈ B, 0 else,

and if λ > 0, 1 Na,λ (B) := √ 2πλ

e−

B ∈ B(R),

(x−a)2 2λ

dx,

B ∈ B(R).

B

Note that 1 Na,λ (R) = √ 2πλ

∞

e−

(x−a)2 2λ

dx = 1,

−∞

hence Na,λ is a probability measure. As is well known, Na,λ is absolutely continuous with respect to the Lebesgue measure dx, with explicit density Na,λ (dx) = √

(x−a)2 1 e− 2λ dx. 2πλ


Proposition C.2.

441

For any a ∈ R, λ ≥ 0, the mean of Na,λ is xNa,λ (dx) = a, R

and its variance is

R

(x − a)2 Na,λ (dx) = λ.

For all m ∈ N, the mth moment of the Gaussian measure Na,λ is finite, i.e., xm Na,λ (dx) < ∞, R

and its Fourier transform is given by 1 2 eihx Na,λ (dx) = eiah− 2 λh , Na,λ (h) := R

h ∈ R.

Gaussian measures on finite-dimensional Hilbert spaces We consider a finite-dimensional Hilbert space H, d := dim H < ∞ with scalar product ·, · and norm · H . To this end, we recall that for probability space (Ωi , Σi , Pi ), i = 1, . . . , d, the product probability space is given by d ' (Ω, Σ, P) := (Ωi , Σi , Pi ) i=1

(see Appendix A). Let a ∈ H, Q ∈ L+ (H) (see Appendix B). Then Q can be represented in any orthonormal basis (ei )di=1 of H as a d × d symmetric, positive semidefinite matrix (ej , Qej )di,j=1 . We now choose a particular orthonormal basis (ei )di=1 such that Q becomes diagonal, i.e., such that Qek = λk ek ,

k = 1, . . . , d,

λk ≥ 0.

Let ∀x ∈ H :

xk := x, ek ,

k = 1, . . . , d.

Then a = (a1 , . . . , ad ), ak = a, ek , and H is isomorphic to Rd via the unitary map γ : H → Rd , x → γ(x) := (x1 , . . . , xd ). Define a product measure Na,Q on (Rd , B(Rd )) by Na,Q :=

d ' k=1

If a = 0, we write NQ for Na,Q .

Nak ,λk .

442


Proposition C.3. Let H be a Hilbert space with d := dim H < ∞, and let Q ∈ L+ (H). Then xNa,Q (dx) = a ∈ H, H

for all y, z ∈ H,

y, x − az, x − aNa,Q (dx) = Qy, z, H

and for all h ∈ H, a,Q (h) := N

1

ei h,x Na,Q (dx) = ei a,h− 2 Qh,h . H

If, moreover, det Q > 0, then Na,Q (dx) is absolutely continuous with respect to the Lebesgue measure on Rd , and 1 1 −1 Na,Q (dx) = ( e− 2 Q (x−a),(x−a) dx. d (2π) det Q The vector a ∈ H is the mean and Q ∈ H ⊗ H is called the covariance operator of Na,Q . Proposition C.4. Let H be a finite-dimensional Hilbert space, a ∈ H, Q ∈ L+ (H), and let µ be a finite measure on (H, B(H)) such that 1 ei h,x µ(dx) = ei a,h− 2 Qh,h . ∀h ∈ H : H

Then µ = Na,Q . Measures on separable Hilbert spaces Let H be a separable Hilbert space with dim H = ∞, scalar product ·, · and norm · H . For any n ∈ N and any orthonormal basis (ek )∞ k=1 of H, we define the projection Pn : H → Pn (H) = span{e1 , . . . , en } ⊂ H by Pn x :=

n

x, ek ek ,

x ∈ H.

(C.2)

k=1

Then, for any x ∈ H, x − Pn x H → 0 as n → ∞. Denote by M(H) the set of all bounded measures on (H, B(H)). Proposition C.5. (uniqueness of measures on H) For µ, ν ∈ M(H), if, for all continuous, bounded ϕ : H → R, ϕ(x) µ(dx) = ϕ(x) ν(dx), (C.3) H

then µ = ν.

H


443

Proof. Let C ⊂ H be closed, and let (ϕn )n be a sequence of continuous, bounded functions on H such that ∀x ∈ H : ϕn (x) −→ 1C (x), n→∞

sup |ϕn (x)| ≤ 1,

(C.4)

x∈H

e.g.,

  1 if x ∈ C,    1 1 − n d(x, C) if x ∈ / C ∧ d(x, C ) < , ϕn (x) := n   1  0 if d(x, C) ≥ . n By dominated convergence, ϕn dµ = ϕn dν −→ µ(C) = ν(C). H

n→∞

H

Since the closed subsets of H are ∩-stable and generate B(H), µ = ν. Proposition C.6. Let µ, ν be finite measures on (H, B(H)) such that for all n ∈ N, (Pn )# µ = (Pn )# ν. Then in M(H).

µ=ν

Proof. Let ϕ : H → R be continuous and bounded. By dominated convergence, ϕ(x) µ(dx) = lim ϕ(Pn x) µ(dx). n→∞ H

H

The change of variables formula (C.1) implies ϕ(x) µ(dx) = lim ϕ(Pn x) µ(dx) = lim ϕ(ξ)(Pn )# µ(dξ) n→∞ H n→∞ P (H) H n ϕ(ξ)(Pn )# ν(dξ) = lim ϕ(Pn x) ν(dξ) = lim n→∞ P (H) n→∞ H n ϕ(x) ν(dx). = H

Proposition C.5 implies the assertion µ = ν. Define the Fourier transform µ of µ ∈ M(H) as ∀h ∈ H : µ (h) := ei x,h µ(dx).

(C.5)

H

Proposition C.7. (Fourier characterization) Let µ, ν be finite measures on (H, B(H)). Then ∀h ∈ H :

µ (h) = ν(h)

=⇒

µ=ν

in M(H).

444


By (C.1), for all n ∈ N, i x,Pn h e µ(dx) = µ (Pn h) =

Proof.

H

and similarly

ei ξ,h (Pn )# µ(dξ) = (P n )# µ(h), Pn (H)

ν(Pn h) =

e

i x,Pn h

ei ξ,h (Pn )# ν(dξ) = (P n )# ν(h).

ν(dx) =

H

Pn (H)

By assumption, µ (Pn h) = ν(Pn h) for all n ∈ N, hence (P n )# µ = (Pn )# ν, and thus (Pn )# µ = (Pn )# ν by a generalization of Proposition C.4. Then Proposition C.6 implies µ = ν in M(H). Let P(H) ⊂ M(H) be the set of probability measures on (H, B(H)). Let µ ∈ P(H) satisfy

x H µ(dx) < ∞,

x 2H µ(dx) < ∞. H

H

Define

F : H → R,

x, h µ(dx),

F (h) :=

h ∈ H.

H

Then F ∈ H since F (·) is linear and by the Cauchy–Schwarz inequality,

x H µ(dx) h H ∀h ∈ H, |F (h)| ≤ H

so

|F (h)| = sup ≤ 0=h∈H h H

F H

x H µ(dx). H

By the Riesz representation theorem, there is a unique m ∈ H such that x, h µ(dx), ∀h ∈ H; m, h = H

m is called the mean of µ ∈ P(H); we write mean(µ) := m. Consider next the bilinear form G(·, ·) : H × H → R defined by G(h, k) := h, x − mk, x − m µ(dx), h, k ∈ H. H

Then by the Cauchy–Schwarz inequality,

x − m 2H µ(dx) h H k H |G(h, k)| ≤

∀h, k ∈ H.

H

By the Riesz representation theorem, there exists a unique Q ∈ L(H) such that Qh, k = h, x − mk, x − m µ(dx), ∀h, k ∈ H. H


445

The operator Q ∈ L(H) is called the covariance (operator ) of µ ∈ P(H); we write Cov(µ) := Q ∈ L(H). Proposition C.8. Let µ ∈ P(H) such that m = mean(µ) ∈ H and Q = Cov(µ) ∈ L(H) exist. Then Q ∈ L+ 1 (H), i.e., Q is symmetric, positive and of trace class. Proof. Qh, k = h, Qk for all h, k ∈ H by definition. For any orthonormal basis (ek )k of H, ∀k ∈ N : Qek , ek = |x − m, ek |2 µ(dx). H

By monotone convergence and Parseval’s identity, Q ∈ L(H) implies ∞

2 |x − m, ek | µ(dx) =

x − m 2H µ(dx). ∞ > Tr Q = k=1

H

H

We close with a lemma on kth moments of a measure µ on H. Lemma C.9. Let µ ∈ M(H) be a probability measure on (H, B(H)) and let k ∈ N be such that |h, x|k µ(dx) < ∞. ∀h ∈ H : H

Then there exists a constant C(k, µ) such that, for all h1 , . . . , hk ∈ H, |h1 , x · · · hk , x|µ(dx) ≤ C(k, µ) h1 H · · · hk H . H

In particular, the symmetric k-form

· · ⊗ H (h1 , . . . , hk ) → H ⊗ · k times

h1 , x · · · hk , xµ(dx) H

is continuous. Observing that for H (k) = H · · ⊗ H ⊗ ·

we have (H (k) ) H · · ⊗ H , ⊗ ·

k times

k times

by the Riesz representation theorem there exists a unique Mk µ ∈ (H (k) ) , the kth moment of the measure µ, such that, for all h1 ⊗ · · · ⊗ hk ∈ H (k) , k M µ, h ⊗ · · · ⊗ h = h1 , x · · · hk , xµ(dx). 1 k H (k) (H )(k) H

Gaussian measures on separable Hilbert spaces Definition C.10. (Gaussian measure) Let a ∈ H and Q ∈ L+ 1 (H). The Gaussian measure µ := Na,Q

on (H, B(H))

446


with mean mean(µ) = a and covariance Cov(µ) = Q is the µ ∈ P(H) with Fourier transform 1 µ (h) = Na,Q (h) = exp ia, h − Qh, h , h ∈ H. 2 Theorem C.11. On any separable Hilbert space H, for any a ∈ H and Q ∈ L+ 1 (H), there is a unique Gaussian measure Na,Q . Proof. Since H is separable and Q ∈ L+ 1 (H), the spectral theorem implies that there is an orthonormal basis (ek )k of H and a sequence (λk )k ⊂ R≥0 such that Qek = λk ek

∀k ∈ N.

For x ∈ H, set xk := x, ek , k ∈ N. Then (xk )k ∈ 2 (N). Note that H∼ = 2 (N) via γ : H → 2 defined by x → γ(x) := (xk )k ∈ 2 (N). Define, on H = 2 (N), the measure µ :=

∞ ' k=1

Nak ,λk ,

x = (x1 , x2 , . . .).

(C.6)

µk

Note that formally, µ is defined on R∞ rather than 2 (N). Proposition C.12.

2 (N) ∈ B(R∞ ), and for µ as in (C.6), µ(2 (N)) = 1.

Proof. We leave the first statement as an exercise. By monotone convergence, "2 ! ∞ ∞

2 2

x 2 (N) µ(dx) = |xk | µ(dx) = |xk |2 Nak ,λk (dxk ) R∞

∞

= =

R k=1 ∞

k=1 ∞

R

k=1

R

(xk − ak ) Nak ,λk (dxk ) + 2

a2k

(λk + a2k ) = Tr Q + a 2 (N) < ∞.

k=1

Therefore, µ({x ∈ R∞ ; x 2 = ∞}) = 0, which implies the second assertion. We next characterize the Fourier transform of Gaussian measures on infinite-dimensional, separable Hilbert spaces.

447


Theorem C.13. For any a ∈ H and Q ∈ L+ 1 (H), there exists a unique µ ∈ P(H) such that 1 ∀h ∈ H. (C.7) µ (h) = exp ia, h − Qh, h 2 Moreover, if H is identified with 2 (N) via the eigenbasis of Q, then µ is the restriction to H of the product measure ∞ '

µk =

k=1

∞ '

Nak ,λk .

k=1

Proof. Since the characteristic function µ of µ uniquely determines µ by Proposition C.7, we must only show existence. The sequence µk := Nak ,λk of Gaussian measure on R, k = 1, 2, . . . admits a unique product measure µ=

∞ '

µk

k=1

on R∞ , and µ ∈ P(R∞ ). By Proposition C.12, µ is concentrated on 2 (N), i.e., µ(2 (N)) = 1. We denote the restriction µ|2 (N) again by µ. Define ∀n ∈ N :

νn :=

n '

µk .

k=1

Then using Proposition C.3, i x,h e µ(dx) = lim 2 (N)

n→∞ 2 (N)

e

i Pn h,Pn x

µ(dx) = lim

n→∞ Rn

1

ei Pn h,ξ νn (dξ)

1

= lim ei Pn h,Pn a− 2 QPn h,Pn h = ei h,a− 2 Qh,h . n→∞

Corollary C.14.

H

x 2H Na,Q (dx) = Tr Q + a 2H .

(C.8)

Gaussian random fields For the mathematical formulation of results on existence and regularity of solutions to stochastic PDEs, we require Bochner spaces of random variables taking values in separable Hilbert and Banach spaces. Definition C.15. (Lebesgue–Bochner spaces) Let (Ω, Σ, P) be a probability space, K a separable Hilbert space, and X : Ω → K a random variable. (i) We say that X is a K-valued Gaussian random variable if the distribution of X is a Gaussian measure on K.

448


(ii) For any 0 < p < ∞, denote by Lp (Ω, Σ, P; K) the linear space of all random variables X : Ω → K for which

X(ω) pK P(dω) < ∞, Ω

and by

L∞ (Ω, Σ, P; K)

the space of all X with ess supω X(ω) K < ∞.

(iii) The space Lp (Ω, Σ, P; K) endowed with 1/p p

X Lp (Ω,Σ,P;K) :=

X(ω) K P(dω)

(C.9)

Ω

if p < ∞, and X L∞ (Ω,Σ,P;K) := ess supω X(ω) K , is a quasi-Banach space for 0 < p < 1 and a Banach space if p ≥ 1. (iv) For p = 2, L2 (Ω, Σ, P; K) equipped with the inner product X, Y L2 (Ω,Σ,P;K) := X(ω), Y (ω)K P(dω)

(C.10)

Ω

is a Hilbert space. Example C.16. Let X ∈ L2 (Ω, Σ, P; K). Then X has mean mX ∈ K, y, hK X# P(dy) = X(ω), hK P(dω) ∀h ∈ K, (C.11) mX , hK = K

Ω

and covariance QX ∈ L(K), y − mX , hK y − mX , kK X# P(dy) QX h, kK = K = X(ω) − mX , hK X(ω) − mX , kK P(dω)

(C.12)

Ω

for all h, k ∈ K. We abbreviate mean(X) := mX = mean(X# P) and Cov(X) := Q = Cov(X# P). Proposition C.17. (convergence of Gaussian RV) Let (Xn )n be a sequence of Gaussian random variables in (Ω, Σ, P), taking values in a separable Hilbert space K. Let an := mean Xn ∈ K and Qn := Cov Xn ∈ L+ 1 (K) for all n ∈ N, and assume that Xn → X in L2 (Ω, Σ, P; K) as n → ∞. Then X is a Gaussian random variable with law Na,Q where, for every h, k ∈ K, a, hK = lim an , hK , n→∞

Qh, kK = lim Qn h, kK . n→∞

Proof.

Let a := mean(X) and Q := Cov(X). By dominated convergence, Xn (ω), h P(dω) = Xn (ω), h P(dω) = a, h lim an , h = lim

n→∞

n→∞ Ω

Ω

449


for all h ∈ K, and

lim Qn h, k = lim Xn (ω) − an , hXn (ω) − an , k P(dω) n→∞ Ω = X(ω) − a, hX(ω) − a, k P(ω) = Qh, k

n→∞

Ω

for all h, k ∈ K. To show that X is Gaussian, by Theorem C.13 it suffices to show that the Fourier transform of X# P is the Fourier transform of a Gaussian measure, i.e., 1 ei y,k X# P(dy) = ei a,k− 2 Qk,k ∀k ∈ K. K

This follows from i y,k i X(ω),k e X# P(dy) = e P(dω) = lim ei Xn (ω),k P(dω) K

Ω

= lim e

i an ,k− 12 Qn k,k

n→∞

n→∞ Ω i a,k− 12 Qk,k

=e

.

Hence X is Gaussian, and X# P = Na,Q . It is well known that linear transformations of Gaussian random variables are once again Gaussian. For random fields taking values in function spaces, this covariance takes the following form. Theorem C.18. (affine transformations of Gaussians) Let µ = Na,Q be a Gaussian measure on a separable Hilbert space (H, B(H)). Then (a) For all b ∈ H, T : H → H, T (x) := x + b is Gaussian on (H, B(H)), and T# µ = Na+b,Q . (C.13) (b) If T ∈ L(H, K) for a separable Hilbert space K, T is Gaussian and T# µ = NT a,T QT ∗ . Proof.

By (C.1), for all k ∈ K, ei k,y T# µ(dy) = ei k,T x µ(dx) K H 1 ∗ ∗ ∗ ei T k,x µ(dx) = ei T k,a− 2 T QT k,k . =

H

Then Theorem C.13 implies (C.14), and (C.13) follows similarly. Computation of some Gaussian integrals Let H be a separable Hilbert space. We abbreviate L2 (H, Na,Q ) := L2 (H, B(H), Na,Q )

(C.14)

450


for any Gaussian measure Na,Q on (H, B(H)). Proposition C.19.

x Na,Q (dx) = a,

(C.15)

H

x − a, yx − a, z Na,Q (dx) = Qy, z H

x − H

a 2H

Na,Q (dx) = Tr Q =

∀y, z ∈ H, ∞

(C.16)

λk .

(C.17)

k=1

Proof. Let (ek )k be an orthonormal basis of H, and Pn x := nk=1 ek , xek . Then x Na,Q (dx) = lim Pn x Na,Q (dx) n→∞ H H ! " n n ∞ (x −a )

)

−1/2 − 2λ x k λ e dx ek = ak ek = a. = lim n→∞

k=1

=1 R

k=1

Equations (C.16) and (C.17) are proved analogously. C.3. Elliptic operator equations with Gaussian data Elliptic operator equations Let X, Y be separable Hilbert spaces and A ∈ L(X, Y ), with associated bilinear form a(u, v) = Y Au, vY , u ∈ X, v ∈ Y. (C.18) Theorem C.20. continuous: coercive: injective:

Assume that the bilinear form a(·, ·) in (C.18) is ∀u ∈ X, v ∈ Y : |a(u, v)| ≤ C1 u X v Y , a(u, v) ≥ C2 > 0, inf sup 0=u∈X 0=v∈Y u X v Y ∀v ∈ Y \ {0} :

sup |a(u, v)| > 0.

(C.19a) (C.19b) (C.19c)

u∈X

Then, for every f ∈ Y , the problem Au = f,

(C.20a)

i.e., u∈X:

a(u, v) = f (v) = Y f, vY

∀v ∈ Y m,

(C.20b)

admits a unique solution u ∈ X, and

u X ≤

f Y . C2

(C.21)

Conversely, A ∈ L(X, Y ) is boundedly invertible if and only if (C.19) holds.


Example C.21. the equation

451

Let D ⊂ Rd be a bounded Lipschitz domain. Consider

Ak2 u := −∇ · A(x)∇u − k 2 u = f in D,

u|∂D = 0,

(C.22)

where A ∈ L∞ (D, Rd×d sym ) is symmetric positive definite, i.e., there is an α > 0 such that, for all ξ ∈ Rd , ess inf ξ A(x)ξ ≥ α ξ 22 .

(C.23)

x∈D

Let 0 < µ1 ≤ µ2 < µ3 < · · · , µn → ∞, denote the eigenvalues of the Dirichlet problem, σ = {µ1 , µ2 , . . .} = (µn )∞ n=1 , and denote by (wn )n the corresponding sequence of eigenfunctions, −∇ · A∇wn = µn wn in D,

wn |∂D = 0.

(C.24)

We assume that (wn )n are normalized in L2 (D); then wm , wn L2 (D) = δmn , Claim C.22.

m, n = 1, 2, . . . .

(C.25)

For every value of k in (C.22) such that

/ σ = {µ1 , µ2 , . . .} (no resonance condition), k2 ∈

(C.26)

the bilinear form ak (u, v) := ∇v, A(x)∇uL2 (D) − k 2 u, vL2 (D) of (C.22) satisfies (C.19) with X = Y = H01 (D) and C2 = min

|k 2 − µ | . µ

Then, for every f ∈ V = H −1 (D), (C.22) has a unique solution u ∈ H01 (D) and 1

u H01 (D) ≤ (C.27)

f H −1 (D) . −1 2 min µ |k − µ | Elliptic operator equations with Gaussian data We consider now the special case X = Y =: V , A ∈ L(V, V ) coercive, i.e., there is a C2 > 0 such that a(v, v) ≥ C2 v 2V

∀v ∈ V.

(C.28)

Then we obtain (C.19c), and the following problem admits a unique solution: given f ∈ V , find u ∈ V such that a(u, v) := V Au, vV = V f, vV

∀v ∈ V.

(C.29)

Theorem C.23. Assume that f ∈ L2 (Ω, Σ, P; V ) is a Gaussian random field such that af = mean(f ) ∈ V and Qf = Cov(f ) ∈ L+ 1 (V ) exist. Then

452


the following problem admits a unique solution: find u ∈ L2 (Ω, Σ, P; V ) such that Au = f in L2 (Ω, Σ, P; V ). (C.30) Moreover, au = mean(u) = A−1 af , Qu ∈ Proof.

L+ 1 (V

)

satisfies

and

(C.31)

∗

AQu A = Qf in L(V ).

(C.32)

The weak form of (C.30) is

where

∀v ∈ L2 (Ω, Σ, P; V ),

˜ a ˜(u, v) = (v)

u ∈ L2 (Ω, Σ, P; V ) :

a ˜(u, v) := Ω

(C.33)

V v(ω), Au(ω)V P(dω),

˜ := (v) Ω

V

v(ω), f (ω)V P(dω).

Letting V = L2 (Ω, Σ, P; V ), we infer from (C.28) that ∀v ∈ V :

a ˜(v, v) ≥ C2 u 2V ,

hence (C.33) has a unique solution. Since A−1 ∈ L(V , V) is 1 to 1 and onto, we get that u(ω) = A−1 f (ω)

P-a.s.

By Theorem C.18 with T = A−1 , u is Gaussian on V since u = A−1 f , and its distribution is the Gaussian measure NA−1 af ,A−1 Qf (A−1 )∗ , i.e., it is characterized completely by au = mean(u) = A−1 af Remark C.24.

and

Qu = Cov(u) = A−1 Qf (A−1 )∗ .

For any separable Hilbert space H, by Theorem B.17, L2 (Ω, Σ, P; H) ∼ = L2 (Ω, Σ, P) ⊗H. S

C.4. Covariance kernels and the Karhunen–Loève expansion From Theorem C.23, (C.32), we infer that, given a covariance operator Qf ∈ L+ 1 (V ) on the data space V of the boundedly invertible operator A ∈ L(V, V ), we have that if f is Gaussian, then u = A−1 f is Gaussian on V , with mean au satisfying (C.34) Aau = af , and covariance operator Qu given by AQu A∗ = Qf ∈ L(V, V )

resp. Qu = A−1 Qf (A−1 )∗ ∈ L+ 1 (V ).

(C.35)


453

For the computation of u in terms of f , it suffices, by Theorem C.18, to compute au and Qu in terms of af and Qf as in (C.35). As we shall see, this is done most easily by means of covariance kernel representations of Qf and Qu . Let (H1 , ·, ·H1 ) and (H2 , ·, ·H2 ) be separable Hilbert spaces, and let S denote a ‘stochastic’ space of random variables with finite second moments. For example, we have L2 (Ω, Σ, P; Hi ) ∼ = L2 (Ω, Σ, P) ⊗ Hi = S ⊗ Hi ,

i = 1, 2, . . . ,

(C.36)

for S = L2 (Ω, Σ, P). Let (sm )m∈Λ be an orthonormal basis in S, with a countable index set Λ. Then any f ∈ H1 ⊗ S can be uniquely represented as

f= fm ⊗ sm in S ⊗ H1 , (C.37) m∈Λ

with (fm )m ∈ 2 (Λ; H1 ). Proposition C.25.

The mapping

S ⊗ H1 × S ⊗ H2 (f, g) → Cf g :=

fm ⊗ g m ∈ H1 ⊗ H 2

m∈Λ

is well-defined, bilinear, bounded with norm 1, and independent of the choice of basis (sm )m of the stochastic space S. Definition C.26. (correlation kernel) For f ∈ S ⊗ H1 , g ∈ S ⊗ H2 , we call Cf g ∈ H1 ⊗ H2 defined in Proposition C.25 the correlation kernel of the pair (f, g) in H1 × H2 . If H1 = H2 = H, the set {Cf := Cf g ; f ∈ S ⊗ H} of auto-correlation kernels is in one-to-one correspondence with the class L+ 1 (H) of positive definite trace-class operators. Theorem C.27. If (H, ·, ·H ) and (S, ·, ·S ) are separable Hilbert spaces of equal dimension, and (sm )m∈Λ is an orthonormal basis of S, then the autocorrelation kernels of elements f in S ⊗ H are in one-to-one correspondence with the positive definite trace-class operators on H, via the correspondence

fm ⊗ fm = Cf → Cf : H x → x, fm H fm , (C.38) m∈Λ

m∈Λ

where S⊗H f =

fm ⊗ s m ,

fm ∈ H,

m ∈ Λ.

(C.39)

m∈Λ + the Proof. The operator Cf defined in (C.38) is the L1 (H)-norm limit of ∞ n finite rank operators Cf := m∈Λn x, fm H fm , for some sequence (Λn )n=1

454


of subsets Λn ⊂ Λ such that #Λn = n, since

Tr Cf = Cf em , em H = |fm , en H |2 =

fm 2H = f 2S⊗H < ∞, m

m

n

m

for any orthonormal basis (em )m of H. Non-negative definiteness of Cf is obvious. Since ∀x, y ∈ H : Cf x, yH = Cf , x ⊗ yH⊗H , (C.40) the definition (C.38) of Cf is independent of the choice of bases in S and H. Hence the map (C.38) is well-defined. The mapping (C.38) from covariance kernels to covariance operators, i.e., the correspondence Cf → Cf , is injective. To see that Cf → Cf is also surjective, we let C ∈ L+ 1 (H) be given. Then C is compact and has a countable eigensequence (λm , ϕm )m∈N , such that Cϕm = λm ϕm ,

m ∈ N,

ϕm , ϕn H = δmn .

(C.41)

The eigenvalues λm ∈ R have finite multiplicity, and we assume they are ordered decreasingly, i.e., λ1 ≥ λ2 ≥ · · · , and they accumulate only in 0. Then, since C is of trace class,

λm < ∞. (C.42) The series that

√ m

m

λm ϕm ⊗ sm converges, by (C.42), to some f ∈ S ⊗ H such Cf =

λm ϕm ⊗ ϕm in H ⊗ H.

(C.43)

m

(C.40), (C.41) and (C.43) imply that C has the same spectral decomposition as Cf , hence C = Cf . Corollary C.28. Let (H, ·, ·H ) be a separable Hilbert space and let C ∈ H ⊗H be a correlation kernel. Then, with the spectrum (C.8) of its operator C ∈ L+ 1 (H) defined as in (C.40), the corresponding covariance kernel C can be represented as

C= λm ϕm ⊗ ϕm in H ⊗ H. (C.44) m

Theorem C.29. Let (H, ·, ·H ) and (S, ·, ·S ) be separable Hilbert spaces, and let C ∈ H ⊗ H be a correlation kernel with representation (C.44). Then f ∈ S ⊗ H satisfies Cf = C in H ⊗ H if and only if there exists an S-orthonormal family (Xm )m ⊂ S such that

( λm Xm ⊗ ϕm in S ⊗ H. (C.45) f= m


455

Proof. The ‘if’ part follows as in the proof of Theorem C.27, upon completion of the family (Xm )m ⊂ S to an orthonormal basis of S. Conversely, if Cf = C, then we may write

Ym ⊗ ϕm in S ⊗ H, f= m

with (Ym )m ⊂ S, which implies with Proposition C.25 that

Ym , Ym S ϕm ⊗ ϕm . Cf = m,m

Comparing this with (C.44), we find (since (ϕm )m is an orthonormal basis of H) that Ym , Ym S = λm δmm , −1/2

This is (C.44) with Xm := λm

m, m ∈ N.

Ym .

Definition C.30. The expansion (C.45) of f ∈ S ⊗ H in terms of the spectral decomposition of its covariance operator Cf is called Karhunen– Loève expansion of f . Theorem C.31. Let X, Y be separable Hilbert spaces. Let A ∈ L(X, Y ) be boundedly invertible (see Theorem C.20), and let f ∈ L2 (Ω, Σ, P; Y ) ∼ = L2 (Ω, Σ, P) ⊗ Y be a given Gaussian random field on Y , with mean(f ) = af ∈ Y ,

Qf = Cov(f ) ∈ L+ 1 (Y ).

(C.46)

Then u = A−1 f ∈ L2 (Ω, Σ, P; X) is also Gaussian on X, with mean(u) = au ∈ X,

Qu = Cov(u) ∈ L+ 1 (X),

(C.47)

satisfying Aau = af in Y ,

(C.48)

and AQu A∗ = Qf in L+ 1 (Y ),

Qu = A−1 Qf (A−1 )∗ ∈ L+ 1 (X).

(C.49)

The kernels Cu of Qu , resp. Cf of Qf , satisfy the equation (A ⊗ A)Cu = Cf in Y ⊗ Y ∼ = (Y ⊗ Y ) .

(C.50)

C.5. The white noise map Let H be a separable Hilbert space, dim H = ∞, and µ = NQ a nondegenerate centred Gaussian measure on H (i.e., ker Q = {0} ⊂ H); furthermore, let (ek )k be an orthonormal basis of H such that Qek = λk ek , k ∈ N. For x ∈ H, set xk := x, ek . Then, for all k, Q−1 ek = λ−1 k ek ,

456


hence Qx0 ∈ H ⊥ , so Qx0 ∈ L(H) is not boundedly invertible (since λk → 0 as k → ∞ due to Tr Q < ∞). The range Q(H) of Q does not equal H, Q(H) = H. Lemma C.32. Q(H) is a dense subspace of H. Proof.

Let x0 ∈ Q(H)⊥ ⊂ H. Then, since Q is self-adjoint, ∀x ∈ H :

0 = x0 , QxH = Qx0 , xH ,

hence Qx0 = 0. But ker Q = {0}, so x0 = 0. Define the operator Q1/2 by Q1/2 x :=

∞ (

λk x, ek H ,

x ∈ H.

(C.51)

k=1

Its range Q1/2 (H) is the reproducing kernel Hilbert space or the Cameron– Martin space of the measure µ = NQ in H. It is a dense subspace of H, and H = Q1/2 (H). We introduce an isometry W : H → L2 (H, NQ ), the white noise map. Let Q1/2 (H) f → Wf ∈ L2 (H, NQ ) be given by Wf (x) = Q−1/2 f, xH We have

∀x ∈ H.

(C.52)

Wf (x)Wg (x) NQ (dx) = f, gH

∀f, g ∈ H.

(C.53)

H

This map W : Q1/2 (H) → L2 (H, NQ ) is a densely defined isometry which can be extended to all of H. Lemma C.33. For any f ∈ H, Wf is a real Gaussian random variable with mean zero and variance f 2H . Proof.

Define νf := (Wf )# µ. We must show 1 2 2 iξη ∀η ∈ R νf (η) = e νf (dξ) = eiηWf (x) µ(dx) = e− 2 η f H . R

H

To this end, let (zn )n ⊂ Q1/2 (H) be a sequence such that zn → z ∈ H. By dominated convergence, −1/2 z ,x n eiηWz (x) µ(dx) = lim eiη Q µ(dx) H

= We pick z = f to conclude.

n→∞ H 1 2 2 lim e− 2 η zn H n→∞

1

= e− 2 η

2 z 2 H

.

457


Given z ∈ H \ Q1/2 (H), one could try to define Wz by

Remark C.34.

∀x ∈ Q1/2 (H) :

Wz (x) = Q−1/2 x, zH ,

rather than by (C.52). This is meaningless due to µ(Q1/2 (H)) = 0. Lemma C.35. µ(Q1/2 (H)) = 0. For all n, k ∈ N, set ∞

2 2 , Un := y ∈ H ; λ−1 Y < n

Proof.

Un,k :=

y∈H;

=1

2k

2 2 , λ−1 Y < n

=1

Then Un ↑ as n → ∞, and, for every fixed n ∈ N, Un,k ↓ Un as k → ∞. Hence we are done, if Q1/2 (H),

∀n :

µ(Un ) = lim µ(Un,k ) = 0.

(C.54)

k→∞

−1/2

To see (C.54), we use that for all n, k ∈ N, for z := λ

2k '

µ(Un,k ) =

Un,k =1

Y ,

Nλk (dyk ) =

{z∈R2k ; |z| 0. Hence, for all a ∈ H, NQ ∼ Na,Q and for all x ∈ H, 1

−1

1 dNa,Q e− 2 Q (x−a),x−aH −1/2 a 2 + Q−1/2 a,Q−1/2 x H H (x) = = e− 2 Q . (C.58) 1 −1 x,x −

Q H dNQ e 2 We claim:

(1) if a ∈ Q1/2 (H), then Na,Q ∼ NQ , (2) if a ∈ H ∩ Q1/2 (H)⊥ , then Na,Q ⊥ NQ .


459

In the first case, (C.58) still holds if dim H = ∞, but Q−1/2 a, Q−1/2 xH is replaced by WQ−1/2 a (x). Hellinger integral Let µ and ν be probability measures on (Ω, Σ). Then both µ and ν are absolutely continuous with respect to the probability measure ζ = (µ + ν)/2 on (Ω, Σ). Definition C.38. (Hellinger integral) The Hellinger integral of µ, ν is defined by 2 dµ dν H(µ, ν) := ζ(dω). (C.59) dζ dζ Ω Obviously, 0 ≤ H(µ, ν) ≤ 1. By Hölder’s inequality, we have 1/2 1/2 dµ dν = 1. 0 ≤ H(µ, ν) ≤ dζ dζ Ω dζ Ω dζ Remark C.39. If λ is a probability measure on (Ω, Σ) such that µ ) λ and ν ) λ, then also ζ ) λ and dµ dµ dλ = dζ dλ dζ and we find

dν dν dλ = , dζ dλ dζ

∧ 0

H(µ, ν) = Ω

dµ dν dλ. dλ dλ

Remark C.40. Assume µ ∼ ν. Then dµ dν dµ dν dµ = = = dζ dζ dζ dµ dζ and hence

2 H(µ, ν) = Ω

Example C.41.

dν dµ dζ = dµ dζ

dµ dζ

2

2 Ω

dν dµ

dν dµ. dµ

Let Ω = R, µ = Nλ , ν = Na,λ , a ∈ R, and λ > 0. Then a2 ax dν (x) = e− 2λ + λ , dµ

and hence 2

− a2λ

R

Proposition C.42.

a2

e 2λ Nλ (dx) = e− 8λ . ax

H(µ, ν) = e

x ∈ R,

If H(µ, ν) = 0, the µ and ν are singular.

460

Proof.


Let f=

dµ , dζ

g=

dν , dζ

ζ=

1 (µ + ν). 2

Then f g = 0 ζ-a.e., since

( f g dζ = 0. H(µ, ν) = Ω

Define the sets A = {ω ∈ Ω ; f (ω) = 0}, B = {ω ∈ Ω ; g(ω) = 0}, C = {ω ∈ Ω ; (f g)(ω) = 0}. Then ζ(C) = 1, hence µ(C) = ν(C) = 1. Moreover, µ(A) = A f dζ = 0 and ν(B) = B g dζ = 0. Therefore, µ(B \ A) = 1 and ν(A \ B) = 1, i.e., µ and ν are mutually singular. Kakutani’s theorem If H(µ, ν) = 0, µ and ν are mutually singular. Conversely, if H(µ, ν) > 0, then µ and ν are not necessarily equivalent, in general, unless µ, ν are countable products of equivalent ‘factor measures’. This is Kakutani’s theorem. We prepare its exposition with products of two measures. Lemma C.43. Let µi , νi , i = 1, 2, be probability measures on (Ω, Σ). Then H(µ1 ⊗ µ2 , ν1 ⊗ ν2 ) = H(µ1 , ν1 )H(µ2 , ν2 ). Proof.

Let ζ1 , ζ2 be probability measures on (Ω, Σ) such that µ 1 ) ζ1 ,

ν 1 ) ζ1 ,

µ2 ) ζ2 ,

ν 2 ) ζ2 .

Then, by Fubini’s theorem, µ1 ⊗ µ2 ) ζ1 ⊗ ζ2

∧

ν 1 ⊗ ν 2 ) ζ1 ⊗ ζ2 .

Define fi (ωi ) :=

dµi (ωi ), dζi

gi (ωi ) :=

dνi (ωi ), dζi

i = 1, 2.

Then d(µ1 ⊗ µ2 ) = f1 (ω1 )f2 (ω2 ), d(ζ1 ⊗ ζ2 ) Hence

H(µ1 ⊗ µ2 , ν1 ⊗ ν2 ) =

d(ν1 ⊗ ν2 ) = g1 (ω1 )g2 (ω2 ). d(ζ1 ⊗ ζ2 )

(f1 , g1 )(ω1 )(f2 , g2 )(ω2 )

Ω×Ω

= H(µ1 , ν1 )H(µ2 , ν2 ).

1/2

ζ1 (dω1 )ζ2 (dω2 )


461

Kakutani’s theorem is an infinite-dimensional generalization of the previous result. Theorem C.44. (Kakutani) Let (µk )k , (νk )k be sequences of probability measures on (R, Σ), such that µk ∼ νk for all k ∈ N, and define µ :=

∞ '

µk ,

ν :=

k=1

∞ '

νk .

k=1

If H(µ, ν) > 0, then µ ∼ ν and ) dνk dν (x) = lim (xk ) n→∞ dµ dµk n

in L1 (R∞ , µ).

(C.60)

k=1

If H(µ, ν) = 0, then µ and ν are singular. We refer to Kakutani (1948) and Da Prato (2006) for a proof of Theorem C.44.

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Numerical algebraic geometry and algebraic kinematics Charles W. Wampler∗ General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090-9055, USA E-mail: [email protected] URL: www.nd.edu/˜cwample1

Andrew J. Sommese† Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA E-mail: [email protected] URL: www.nd.edu/˜sommese

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

∗ †

This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and by General Motors Research and Development. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and the Duncan Chair of the University of Notre Dame.

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CONTENTS 1 Introduction 2 Notation

470 472

PART 1: Fundamentals of algebraic kinematics

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3 Some motivating examples 4 Algebraic kinematics 5 History of numerical algebraic geometry and kinematics

473 492 511

PART 2: Numerical algebraic geometry

514

6 Finding isolated roots 7 Computing positive-dimensional sets 8 Software

520 530 541

PART 3: Advanced topics

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9 Non-reduced components 10 Optimal solving

542 545

PART 4: Frontiers

553

11 Real sets 12 Exceptional sets

553 555

PART 5: Conclusions

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Appendix: Study coordinates

559

References

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1. Introduction While systems of polynomial equations arise in fields as diverse as mathematical finance, astrophysics, quantum mechanics, and biology, perhaps no other application field is as intensively focused on such equations as rigidbody kinematics. This article reviews recent progress in numerical algebraic geometry, a set of methods for finding the solutions of systems of polynomial equations that relies primarily on homotopy methods, also known as polynomial continuation. Topics from kinematics are used to illustrate the applicability of the methods. High among these topics are questions concerning robot motion.

Numerical algebraic geometry and algebraic kinematics

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Kinematics can be defined as the study of the geometry of motion, which is the scaffolding on which dynamics builds differential equations relating forces, inertia, and acceleration. While not all problems of interest in kinematics are algebraic, a large majority are. For instance, problems concerning mappings between robot joints and their hand locations fall into the algebraic domain. As our interest here is in numerical algebraic geometry, we naturally restrict our attention to the subset of kinematics questions that can be modelled as polynomial systems, a subfield that we call algebraic kinematics. We give a high-level description of kinematics that encompasses problems arising in robotics and in traditional mechanism design and analysis. In fact, there is no clear line between these, as robots are simply mechanisms with higher-dimensional input and output spaces as compared to traditional mechanisms. We discuss which types of mechanisms fall within the domain of algebraic kinematics. While algebraic geometry and kinematics are venerable topics, numerical algebraic geometry is a modern invention. The term was coined in 1996 (Sommese and Wampler 1996), building on methods of numerical continuation developed in the late 1980s and early 1990s. We will review the basics of numerical algebraic geometry only briefly; the reader may consult Sommese and Wampler (2005) for details. After summarizing the developments up to 2005, the article will concentrate on the progress of the last five years. Before delving into numerical algebraic geometry, we will discuss several motivational examples from kinematics. These should make clear that kinematics can benefit by drawing on algebraic geometry, while a short historical section recalls that developments in algebraic geometry in general, and numerical algebraic geometry in particular, have benefited from the motivation provided by kinematics. Numerical algebraic geometry applied to algebraic kinematics could be called ‘numerical algebraic kinematics’, which term therefore captures much of the work described in this paper. With a foundational understanding of kinematics in place, we then move on to discuss numerical algebraic geometry in earnest, occasionally returning to kinematics for examples. Our presentation of numerical algebraic geometry is divided into several parts. The first part reviews the basics of solving a system of polynomial equations. This begins with a review of the basic techniques for finding all isolated solutions and then extends these techniques to find the numerical irreducible decomposition. This describes the entire solution set of a system, both isolated points and positive-dimensional components, factoring the components into their irreducible pieces. The second part describes more advanced methods for dealing effectively with sets that appear with higher multiplicity and techniques that reduce computational work compared to the basic methods. Finally, we turn our attention to the frontiers of the area, where current research is developing methods for treating the real

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points in a complex algebraic set and for finding exceptional examples in a parametrized family of problems. Like any computational pursuit, the successful application of numerical algebraic geometry requires robust implementation in software. In Section 8, we review some of the packages that are available, including the Bertini package (Bates, Hauenstein, Sommese and Wampler 2008) in which much of our own efforts have been invested.

2. Notation This paper uses the following notation. √ • i = −1, the imaginary unit. • a∗ for a ∈ C is the complex conjugate of a. • A for set A ⊂ CN is the closure of A in the complex topology. • For polynomial system F = {f1 , . . . , fn }, F : CN → Cn , and y ∈ Cn , F −1 (y) = {x ∈ CN | F (x) = y}. We also write V(F ) = V(f1 , . . . , fn ) := F −1 (0). (Our use of V() ignores all multiplicity information.) • ‘DOF’ means ‘degree(s) of freedom’. • We use the following abbreviations for types of joints between two links in a mechanism: – R = 1DOF rotational joint (simple hinge), – P = 1DOF prismatic joint (a slider joint allowing linear motion), and – S = 3DOF spherical joint (a ball and socket). These are discussed in more detail in Section 4.2. • C∗ , pronounced ‘Cee-star’, is C \ 0. • Pn is n-dimensional projective space, the set of lines through the origin of Cn+1 . • Points in Pn may be written as homogeneous coordinates [x0 , . . . , xn ] with (x0 , . . . , xn ) = (0, . . . , 0). The coordinates are interpreted as ratios: [x0 , . . . , xn ] = [λx0 , . . . , λxn ] for any λ ∈ C∗ .


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• A quaternion u is written in terms of the elements 1, i, j, k as u = u0 1 + u1 i + u2 j + u3 k,

u0 , u1 , u2 , u3 ∈ C.

We call u0 the real part of u and u1 i + u2 j + u3 k the vector part. We may also write u = (u0 , u), where u is the vector part of u. • If u and v are quaternions, then u ∗ v is their quaternion product. This may be written in terms of the vector dot-product operator ‘·’ and vector cross-product operator ‘×’ as (u0 , u) ∗ (v0 , v) = (u0 v0 − u · v, u0 v + v0 u + u × v). Note that ∗ does not commute: in general, u ∗ v = v ∗ u. • When u = (u0 , u) is a quaternion, u = (u0 , −u) is its quaternion conjugate, and u ∗ u is its squared magnitude.

PART ONE Fundamentals of algebraic kinematics 3. Some motivating examples Before attempting to organize kinematic problems into a common framework, let us begin by examining some examples. 3.1. Serial-link robots To motivate the discussion, let us see how one of the simplest problems from robot kinematics leads to a system of polynomial equations. It will also serve to introduce some basic terminology for robot kinematics. The generalization of this problem from the planar to the spatial case is one of the landmark problems solved in the 1980s. 3.1.1. Planar 3R robot Consider the planar robot arm of Figure 3.1(a) consisting of three moving links and a base connected in series by rotational joints. (Hash marks indicate that the base is anchored immovably to the ground.) Kinematicians use the shorthand notation ‘R’ for rotational joints and hence this robot is known as a planar 3R robot. The rotation of each joint is driven by a motor, and a sensor on each joint measures the relative rotation angle between successive links. Given the values of the joint angles, the forward kinematics problem is to compute the position of the tip, point R, and the orientation of the last link. Often, we refer to the last link of the robot as its ‘hand’. The converse of the forward kinematics problem is the inverse

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C. W. Wampler and A. J. Sommese R

R

c

R θ3

c

Q b

b

O

(a)

θ1

Q φ2

θ2 a

φ3

P

P

a O

(b)

φ1

(c)

Figure 3.1. Three-link planar robot arm and its kinematic skeleton.

kinematics problem, which is to determine the joint angles that will place the hand in a desired position and orientation. The forward kinematics problem is easy. We assume that the position of point O is known: without loss of generality we may take its coordinates to be (0, 0). The relative rotation angles, (θ1 , θ2 , θ3 ) in Figure 3.1(b), are given, and we know the lengths a, b, c of the three links. We wish to compute the coordinates of the tip, R = (Rx , Ry ), and the orientation of the last link, absolute rotation angle φ3 in Figure 3.1(c). To compute these, one may first convert relative angles to absolute ones as (φ1 , φ2 , φ3 ) = (θ1 , θ1 + θ2 , θ1 + θ2 + θ3 ). Then, the tip is at

Rx a cos φ1 + b cos φ2 + c cos φ3 = . Ry a sin φ1 + b sin φ2 + c sin φ3

(3.1)

(3.2)

In this case, the forward kinematics problem has a unique answer. The inverse kinematic problem is slightly more interesting. It is clear that if we can find the absolute rotation angles, we can invert the relationship (3.1) to find the relative angles. Moreover, we can easily find the position of point Q = (Qx , Qy ) as Q = (Qx , Qy ) = (Rx − c cos φ3 , Ry − c sin φ3 ). However, to find the remaining angles, φ1 , φ2 , we need to solve the trigonometric equations a cos φ1 + b cos φ2 Qx = . (3.3) Qy a sin φ1 + b sin φ2 Of course, an alternative would be to use basic trigonometry to solve for θ2 as the external angle of triangle OP Q having side lengths a, b, and Q,


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which can be done with the law of cosines. But, it is more instructive for our purposes to consider converting equation (3.3) into a system of polynomials. One way to convert the trigonometric equations to polynomials is to introduce for i = 1, 2 variables xi = cos φi , yi = sin φi , i = 1, 2, and the trigonometric identities cos2 φi + sin2 φi = 1. With variable substitutions, the four trigonometric equations become the polynomial equations   ax1 + bx2 − Qx  ay1 + by1 − Qy    2 (3.4)  x1 + y12 − 1  = 0. x22 + y22 − 1 This system of two linears and two quadratics has total degree 1 · 1 · 2 · 2 = 4, which is an upper bound on the number of isolated solutions. However, for general a, b, Qx , Qy it has only two isolated solutions. An alternative method often used by kinematicians for converting trigonometric equations to polynomials is to use the tangent half-angle relations for rationally parametrizing a circle. This involves the substitutions sin φ = 2t/(1 + t2 ) and cos φ = (1 − t2 )/(1 + t2 ), where t = tan φ/2. (Denominators must be cleared after the substitutions.) This approach avoids doubling the number of variables but the degrees of the resulting equations will be higher. Either way, the conversion to polynomials is a straightforward procedure. The pathway of first writing trigonometric equations and then converting them to polynomials is not really necessary. We have presented it here to illustrate the fact that for any robot or mechanism built by connecting rigid links with rotational joints there is a straightforward method of arriving at polynomial equations that describe the basic kinematic relations. A different way of arriving at polynomial equations for this simple example is to first concentrate on finding the location of the intermediate joint P . Clearly P is distance a from O and distance b from Q, both of which are known points. Thus, P can lie at either of the intersection points of two circles: one of radius a centred at O and one of radius b centred at Q. Letting P = (Px , Py ), we immediately have the polynomial equations Px2 + Py2 − a2 = 0. (3.5) (Px − Qx )2 + (Py − Qy )2 − b2 Here, only (Px , Py ) is unknown, and we have two quadratic polynomial equations. Subtracting the first of these from the second, one obtains the system Px2 + Py2 − a2 = 0, (3.6) −2Px Qx + Q2x − 2Py Qy + Q2y − b2 + a2 which consists of one quadratic and one linear polynomial in the unknowns (Px , Py ). Hence, we see that the 3R inverse kinematics problem has at most

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two isolated solutions. In fact, by solving a generic example, one finds that there are exactly two solutions. Of course, this is what one would expect for the intersection of two circles. The situation can be summarized as follows. The robot has a joint space consisting of all possible joint angles θ = (θ1 , θ2 , θ3 ), which in the absence of joint limits is the three-torus, T 3 . The parameter space for the family of 3R planar robots is R3 , consisting of link-length triplets (a, b, c). The workspace of the robot is the set of all possible positions and orientations of a body in the plane. This is known as special Euclidean two-space SE (2) = R2 ×SO(2). Since SO(2) is isomorphic to a circle, one can also write SE (2) = R2 × T 1 . Above, we have represented points in SE (2) by the coordinates (Rx , Ry , φ3 ). Equations (3.1) and (3.2) give the forward kinematics map K3R (θ; q) : T 3 × R3 → SE (2). For any particular such robot, say one given by parameters q ∗ ∈ R3 , we have a forward kinematics map that is the restriction of K3R to that set of parameters. Let us denote that map as K3R,q∗ (θ) := K3R (θ; q ∗ ), so K3R,q∗ : T 3 → SE (2). The forward kinematics map K3R,q is generically two-to-one, that is, for each set of joint angles θ ∈ T 3 , the robot with general link parameters q has a unique hand location H = K3R,q (θ) while for the same robot there are two possible sets of joint angles to reach any general hand location, i.e., −1 K3R,q (H) is two isolated points for general H ∈ SE (2). To be more precise, we should distinguish between real and complex solutions. Let TR3 be the set of real joint angles while TC3 is the set of complex ones. Similarly, we write SE (2, R) and SE (2, C) for the real and complex workspaces. The reachable workspace of the robot having link parameters q is the set Wq = {H ∈ SE (2, R) | ∃θ ∈ TR3 , K3R (θ; q) = H)}.

(3.7)

One can confirm using the triangle inequality that the reachable workspace is limited to locations in which point Q lies in the annulus {Q ∈ R2 | |a − b| ≤ Q ≤ |a + b|}. In the interior of the reachable workspace, the inverse kinematics problem has two distinct real solutions, outside it has none, and on the boundary it has a single double-point solution. An exception is when a = b. In that case, the interior boundary of the annulus for Q shrinks to a point, the origin O. When Q is at the origin, the inverse kinematics problem has a positive-dimensional solution with θ2 = ±π and θ1 = T 1 . We will refer to cases where the inverse kinematics problem has a higher than ordinary dimension as exceptional sets. The forward kinematics equations extend naturally to the complexes, where the situation is somewhat simpler due to the completeness of the complex number field. Over the complexes, for generic a, b, the inverse kinematics problem has two isolated solutions everywhere except at singu-


477

larities where it has a double root. The forward and inverse kinematics maps still make sense, but the reachable workspace is no longer relevant. Since a real robot can only produce real joint angles, the maps over the reals and the corresponding reachable workspace are ultimately of paramount interest, but to get to these, it can be convenient to first treat the problem over the complexes. The roots of the inverse kinematics problem over the complexes are real whenever the hand location is within the reachable workspace. The problem of analysing the reachable workspace for the robot with parameters q comes down to finding the singularity surface Sq = {θ ∈ TR3 | rank dθ K3R (θ; q) < 3},

(3.8)

where dθ K3R denotes the 3×3 Jacobian matrix of partial derivatives of K3R with respect to the joint angles. The boundaries of the reachable workspace are contained in the image of the singularities, K3R,q (Sq ). Again, it is often simplest to first treat this problem over the complexes and then to extract the real points in the complex solution set. We note that the forward and inverse kinematics problems are defined by the analyst according to the intended usage of the device; there may be more than one sensible definition for a given mechanism. These are all related to the same essential kinematics of the device, usually related by simple projection operations. As a case in point, consider the variant of the forward and inverse kinematics problems, and the associated reachable workspace problem, occurring when the robot is applied not to locating the hand in position and orientation but rather simply positioning its tip without regard to the orientation with which this position is attained. : In the position-only case, we have a new forward kinematics map K3R 3 3 2 T ×R → R which may be written in terms the projection π : SE (3) → R2 , = π ◦ K3R . Holding the parameters π : (Rx , Ry , φ3 ) → (Rx , Ry ), as K3R of the robot fixed, the forward kinematics problem for K3R is still unique, but the inverse kinematics problem (finding joint angles to reach a given position) now has a solution curve. The positional 3R planar robot is said to be kinematically redundant, and motion along the solution curve for a fixed position of the end-point is called the robot’s self-motion. The reachable workspace for the positional 3R robot is either a disc or an annulus in the plane, centred on the origin. The singularity condition, similar to equation (3.8), Sq = {θ ∈ TR3 | rank dθ K3R (θ; q) < 2},

(3.9)

(S ) = π(K (S )) that is the union of four circles gives an image K3R 3R q q having radii |a ± b ± c|. These correspond to configurations where points O, P , Q, and R are collinear: as shown in Figure 3.2(a), this means the linkage is fully outstretched or folded into one of three jackknife configurations.

478


(a)

(b)

Figure 3.2. Singularities and positional workspace for 3R planar robot.

Rotation of the first joint does not affect the singularity, so the projection of each singularity curve gives a circle in the workspace. The outer circle is a workspace boundary, while the inner circle may or may not be a boundary depending on the particular link lengths. If any point in the interior of the smallest circle can be reached, then the entire inner disc can be reached. In particular, the origin can be reached if and only if the link lengths satisfy the triangle inequality: max(a, b, c) ≤ a + b + c − max(a, b, c). For the particular case we are illustrating, this is satisfied, as is shown in Figure 3.2(b). The two intermediate circles are never workspace boundaries, but they mark where changes occur in the topology of the solution curve for the inverse kinematics problem. This illustrates the general principle that potential boundaries of real reachable workspaces are algebraic sets given by equality conditions, and the determination of which patches between these is in or out of the workspace can be tested by solving the inverse kinematics problem at a test point inside the patch.


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In short, we see that the 3R planar robot has three associated kinematics problems, as follows. For all these problems, the robot design parameters q = (a, b, c) are taken as given. (1) The forward kinematics problem for a given robot q is to find the hand location from knowledge of the joint angles. This has the unique solution (Rx , Ry , φ3 ) := K3R (θ; q). (2) The inverse kinematics problem for a given robot q is to find the joint angle sets that put the hand in a desired location, H ∈ SE (2). The −1 answer, denoted K3R,q (H), is the solution for θ of the equations H − K3R (θ; q) = 0. For generic cases, the solution set is two isolated points. (3) The reachable workspace problem is to find the boundaries of the real working volume of a given robot q. This can be accomplished by computing the singularity surface Sq of equation (3.8). An additional problem of interest is to identify exceptional cases, especially cases where the dimensionality of the solutions to the inverse kinematic problem is larger than for the general case. More will be discussed about this in Section 12. For this problem, we let the robot parameters q = (a, b, c) vary and seek to solve the following. (4) The exceptional set problem for forward kinematics function K3R is to find D1 = {(θ, q, H) ∈ T 3 × R3 × SE (2) | dim{θ ∈ T 3 | K3R (θ; q) = H} ≥ 1}, (3.10) where dim X means the dimension of set X. As will be described later, it is preferable in general to use local dimension instead of dimension, but it makes no difference in the 3R case. We described one exceptional set in the discussion of the inverse kinematic problem above. The exceptional set problem is significantly more difficult than the other three; the difficulty is only hinted at by the fact that all the symbols (θ1 , θ2 , θ3 , a, b, c, Rx , Ry , φ3 ) ∈ T 3 × R3 × SE (2) are variables in the problem. All four problems exist as well for the positional 3R robot and its forward (θ; q). The generic inverse kinematics solutions become kinematics map K3R curves, and the exceptional set problem would find robots in configurations where the inverse solution has dimension greater than or equal to two. In the problem statements of items (1)–(4) above, we have reverted to writing functions of angles, implying a trigonometric formulation. However, it is clear from the earlier discussion that all of these problems are easily restated in terms of systems of polynomial equations.

480

C. W. Wampler and A. J. Sommese z3

x3 z4

x4

z2 z5 x2 z1

z7

z6

x5

x6 y7

x7

z0

x1

p

y0 x0 Figure 3.3. Schematic six-revolute serial-link robot.

3.1.2. Spatial 6R robot The spatial equivalent of the 3R planar robot is a 6R chain, illustrated in Figure 3.3. In the drawing, the six spool-like shapes indicate the rotational joints, which have axes labelled z1 , . . . , z6 . Vectors x1 , . . . , x6 point along the common normals between successive joint axes. Frame x0 , y0 , z0 is the world reference frame, and a final end-effector frame x7 , y7 , z7 marks where the hand of the robot arm would be mounted. The purpose of the robot is to turn its joints so as to locate the end-effector frame in useful spatial locations with respect to the world frame. For example, these destinations may be where the hand will pick up or drop off objects. The set of all possible positions and orientations of a rigid body in threespace is SE (3) = R3 × SO(3), a six-dimensional space. Accordingly, for a serial-link revolute-joint spatial robot to have a locally one-to-one mapping between joint space and the workspace SE (3), it must have six joints. With fewer joints, the robot’s workspace would be a lower-dimensional subspace of SE (3). With more, the robot is kinematically redundant and has a positive-dimensional set of joint angles to reach any point in the interior of its workspace. With exactly six joints, the robot generically has a finite number of isolated points in joint space that produce the same hand location in SE (3). For a general 6R robot and a general hand location, the 6R inverse kinematics problem has 16 isolated solutions over the complexes, while the number of real solutions may change with the hand location. The singularity


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surfaces divide the reachable workspace into regions within which the number of real isolated solutions is constant. This includes, of course, the boundaries between the reachable workspace and its exterior, where no solutions exist. Robots with special geometries, which are very common in practice, may have a lower number of isolated solutions, and for some hand locations and special robot geometries, positive-dimensional solution sets may arise. Thus, we see that the four types of problems discussed for the 3R robot extend naturally to the 6R robot, although one may expect the answers to be somewhat more complicated and harder to compute. The most common way to formulate the kinematics of a serial-link robot is to use 4 × 4 transformation matrices of the form: C d A= , C ∈ SO(3), d ∈ R3×1 , 03 = [0 0 0]. (3.11) 03 1 The set of all such matrices forms a representation of SE (3). Pre-multiplication by A is equivalent to a rotation C followed by a translation d. When put in this form, a rotation of angle θ around the z axis is written   cos θ − sin θ 0 0  sin θ cos θ 0 0 . (3.12) Rz (θ) =   0 0 1 0 0 0 0 1 We may use the 4 × 4 representation to build up the chain of transformations from one end of the robot to the other. The displacement from the origin of link 1, marked by x1 , z1 , to the origin of link 2, marked by x2 , z2 , is Rz (θ1 )A1 , where A1 is a constant transform determined by the shape of link 1. In this manner, the final location of the hand with respect to the world reference system, which we will denote as H ∈ SE (3), can be written in terms of seven link transforms A0 , . . . , A6 ∈ SE (3) and six joint angles as H = K6R (θ; q) := A0

6

Rz (θj )Aj .

(3.13)

j=1

Here, the set of parameters q is the link transforms, so q ∈ SE (3)7 , while the joint space is the six-torus: θ ∈ T 6 . As in the planar case, to convert from this trigonometric formulation to an algebraic one, we simply adopt unit circle representations of the joint angles. It is clear that the four basic problem types discussed for the 3R robot (forward, inverse, workspace analysis, and exceptional sets) carry over to the 6R case with K6R playing the role of K3R . 3.2. Four-bar linkages The reachable workspaces of 3R and 6R robots have the same dimension as the ambient workspace, SE (2) and SE (3), respectively. This is not true

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C. W. Wampler and A. J. Sommese P

B2 B1

Θ2

Φ A2 Θ1 A1 (a)

(b)

Figure 3.4. Four-bar linkage.

for many mechanisms used in practice. In fact, it is precisely the ability to restrict motion to a desired subspace that makes many mechanisms useful. Consider, for example, the four-bar linkage, illustrated in Figure 3.4. As is customary, we draw the three moving links, two of which are attached to the ground at points A1 and A2 . The ground link counts as the fourth bar; a convention that was not yet adopted in some of the older papers (e.g., Cayley (1876), Clifford (1878) and Roberts (1875)). The middle link, triangle B1 B2 P , is called the coupler triangle, as it couples the motion of the two links to ground, A1 B1 and A2 B2 . Four-bars can be used in various ways, such as • function generation, i.e., to produce a desired functional relationship between angles Θ1 and Θ2 ; • path generation, i.e., to produce a useful path of the coupler point P , as shown in Figure 3.4(b); or • body guidance, in which the coordinated motion in SE (2) of the coupler link is of interest. The path of the coupler point P is called the coupler curve. In the example shown in Figure 3.4(b) the coupler curve has two branches; the mechanism must be disassembled and reassembled again to move from one branch to the other. (Over the complexes, the curve is one irreducible set. Whether the real coupler curve has one or two branches depends on inequality relations between the link sizes.) To specify a configuration of a four-bar, one may list the locations of the three moving links with respect to the ground link. That is, a configuration is a triple from SE (3)3 . We may take these as (A1 , Θ1 ), (A2 , Θ2 ) and (P, Φ). For generic link lengths, the mechanism has a 1DOF motion, a curve in

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SE (3)3 . The whole family of all possible four-bars is parametrized by the shapes of the links and the placement of the ground pivots. These add up to 9 independent parameters, which we may enumerate as follows: • positions of ground pivots A1 and A2 in the plane (4 parameters); • lengths |A1 B1 |, |A2 B2 |, |B1 P |, and |B2 P | (4 parameters); and • angle ∠B1 P B2 (1 parameter). Clearly, the motion of a four-bar is equivalent to the self-motion of a kinematically redundant positional 3R robot whose end-point is at A2 . The workspace regions drawn in Figure 3.2 now become the regions within which we may place A2 without changing topology of the motion of the four-bar. The boundaries delineate where one or more of the angles Θ1 , Θ2 , and Φ change between being able to make full 360◦ turns or just partial turns. As in the case of the 3R robot, we may use several different ways to write down the kinematic relations. One particularly convenient formulation for planar mechanisms, especially those built with rotational joints, is called isotropic coordinates. The formulation takes advantage of the fact that a vector v = ax+by in the real Cartesian plane can be modelled as a complex number v = a + b i. Then, vector addition, u + v, is simple addition of complex numbers, u + v, and rotation of vector v by angle Θ is eiΘ v. Suppose that in Cartesian coordinates P = (x, y), so that the location of the coupler link is represented by the triple (x, y, Φ), which we may view as an element of SE (2). To make the Cartesian representation algebraic, we switch to the unit circle coordinates (x, y, cΦ , sΦ ) ∈ V(c2Φ + s2Φ − 1), which is an isomorphic representation of SE (3). In isotropic coordinates, ¯ ∈ V(φφ¯ − 1), where the representation of SE (3) is (p, p¯, φ, φ) p = x + iy,

p¯ = x − iy,

φ = eiΦ ,

φ¯ = e−iΦ .

(3.14)

When (x, y, Φ) is real, we have the complex conjugate relations p∗ = p¯ and

¯ φ∗ = φ,

(3.15)

where ‘∗ ’ is the conjugation operator. The isotropic coordinates are merely a linear transformation of the unit circle coordinates: p 1 i x φ 1 i cΦ . = and = p¯ 1 −i y φ¯ 1 −i sΦ To model the four-bar linkage using isotropic coordinates, we draw a vector diagram in the complex plane, shown in Figure 3.5. Point O marks the origin, from which vectors a1 and a2 mark the fixed ground pivots and vector p indicates the current position of the coupler point. As the coupler rotates by angle Φ, its sides rotate to φb1 and φb2 , where φ = eiΦ . The links

484

C. W. Wampler and A. J. Sommese φb2 φb1 2

p

Φ

Θ2

Θ1

a2

1

a1

O

Figure 3.5. Vector diagram of a four-bar linkage.

to ground have lengths 1 and 2 . Letting θj = eiΘj , j = 1, 2, we may write two vector loop equations: 1 θ1 = p + φb1 − a1 , 2 θ2 = p + φb2 − a2 .

(3.16a) (3.16b)

Since a1 , a2 , b1 , b2 are complex vectors, their isotropic representations include the conjugate quantities a ¯1 , a ¯2 , ¯b1 , ¯b2 , with a ¯1 = a∗1 , etc., for a real linkage. Accordingly, the vector loop equations imply that the following conjugate vector equations must also hold: 1 θ¯1 = p¯ + φ¯¯b1 − a ¯1 , ¯2 . 2 θ¯2 = p¯ + φ¯¯b2 − a

(3.17a) (3.17b)

Finally, we have unit length conditions for the rotations: θ1 θ¯1 = 1,

θ2 θ¯2 = 1,

and

φφ¯ = 1.

(3.18)

¯ 1 , a2 , a ¯2 , b1 , ¯b1 , b2 , ¯b2 , Altogether, given a four-bar with the parameters a1 , a ¯ ¯ 1 , 2 , we have 7 equations for the 8 variables p, p¯, φ, φ, θ1 , θ1 , θ2 , θ¯2 , so we expect in the generic case that the mechanism has a 1DOF motion. While equations (3.16)–(3.18) describe the complete motion of the fourbar, we might only be interested in the path it generates. In general, in such cases we might just invoke a projection operation, ¯ θ1 , θ¯1 , θ2 , θ¯2 ) → (p, p¯), K : (p, p¯, φ, φ, and keep working with all the variables in the problem. However, in the case


485

of the four-bar, we can easily perform the elimination operation implied by the projection. First, the rotations associated to Θ1 and Θ2 are eliminated as follows: 21 = (p + φb1 − a1 )(¯ p + φ¯¯b1 − a ¯1 ),

(3.19a)

p + φ¯¯b2 − a ¯2 ). 22 = (p + φb2 − a2 )(¯

(3.19b)

After expanding the right-hand sides of these and using φφ¯ = 1, one finds ¯ Consequently, one may solve these that the equations are linear in φ, φ. using Cramer’s rule to obtain a single coupler curve equation in just p, p¯. Using |α β| to denote the determinant of a 2 × 2 matrix having columns α and β, we may write the coupler curve equations succinctly as

where

fcc (p, p¯; q) = |v u ¯||v u| + |u u ¯|2 = 0,

(3.20)

p−a ¯1 ) + b1¯b1 − 21 (p − a1 )(¯ , v= (p − a2 )(¯ p−a ¯2 ) + b2¯b2 − 22 ¯ p−a ¯1 ) b1 (¯ b1 (p − a1 ) , u= and u ¯= ¯ b2 (¯ p−a ¯2 ) b2 (p − a2 )

(3.21)

and where q is the set of linkage parameters: ¯1 , a ¯2 , ¯b1 , ¯b2 , 1 , 2 ). q = (a1 , a2 , b1 , b2 , a Inspection of equations (3.20) and (3.21) shows that the coupler curve is degree six in p, p¯, and in fact, the highest exponent on p or p¯ is three. In other words, the coupler curve is a bi-cubic sextic. The number of monomials in a bi-cubic equation in two variables is 16, which is sparse compared to the possible 28 monomials for a general sextic. The bi-cubic property implies that a coupler curve can intersect a circle in at most 6 distinct points, one of the important properties of coupler curves. The coupler curve equation answers the workspace analysis question for four-bar path generation. For example, the question of where the coupler curve crosses a given line comes down to solving a sextic in one variable, a process that can be repeated for a set of parallel lines to sweep out a picture of the whole curve in the plane. (This is not the simplest procedure that can be applied for this mechanism, but it is a workable one.) A much tougher kind of question inverts the workspace analysis problem. These are synthesis problems of finding four-bars whose coupler curve meets certain specifications. For example, we might request that the coupler curve interpolate a set of points. The most extreme case is the nine-point synthesis problem of finding all four-bars whose coupler curve interpolates nine given

486


points.1 Essentially, this means solving for the parameters q that satisfy the system: fcc (pj , p¯j ; q) = 0, j = 0, . . . , 8, (3.22) for nine given points (pj , p¯j ) ∈ C2 , j = 0, . . . , 8. This problem, first posed by Alt (1923), finally succumbed to solution by polynomial continuation in 1992 (Wampler, Morgan and Sommese 1992, 1997), where it was demonstrated that 1442 coupler curves interpolate a generic set of nine points, each coupler curve generated by three cognate linkages (Roberts’ cognates (Roberts 1875)). The system of equation (3.22) consists of nine polynomials of degree eight. Roth and Freudenstein (1963) formulated the problem in a similar way, but ended up with eight polynomials of degree seven. Since this version has become one of our test problems, we give a quick re-derivation of it here. The key observation is that since we are given the nine points, we may choose the origin of coordinates to coincide with the first one: (p0 , p¯0 ) = (0, 0). Moreover, the initial rotations of the links can be absorbed into the link parameters, so we may assume that Θ1 , Θ2 , Φ = 0 at the initial precision point. Accordingly, at that point equation (3.19) becomes 2 = (b1 − a1 )(¯b1 − a ¯1 ), (3.23a) 1

22 = (b2 − a2 )(¯b2 − a ¯2 ).

(3.23b)

Substituting from these into the expression for v in equation (3.21), one may eliminate 21 and 22 . Accordingly, the Roth–Freudenstein formulation of the nine-point problem is fcc (pj , p¯j ; q) = 0,

j = 1, . . . , 8,

(3.24)

with the precision points (pj , p¯j ) given and with variables a1 , a2 , b1 , b2 , a ¯1 , a ¯2 , ¯b1 , ¯b2 . Although the equations appear to be degree eight, the highest-degree terms all cancel, so the equations are actually degree seven. Hence, the total degree of the system is 78 . The original Roth–Freudenstein formulation did not use isotropic coordinates, but their equations are essentially equivalent to the ones presented here. Various other synthesis problems can be posed to solve engineering design problems. The nature of the synthesis problem depends on the use to which the linkage is being applied. For example, if the four-bar is being used for function generation, one may seek those mechanisms whose motion interpolates angle pairs (Θ1,j , Θ2,j ) ∈ R2 (for up to five such pairs). For body 1

Nine is the maximum possible number of generic points that can be interpolated exactly, because that is the number of independent parameters. Although we have 10 parameters in q, there is a one-dimensional equivalence class. This could be modded out by requiring that b1 be real, i.e., ¯b1 = b1 . We enumerated a count of nine independent parameters earlier in this section.


487

guidance, one may specify triples (pj , p¯j , Φj ) ∈ SE (2) for up to five poses. Instead of just interpolating points, synthesis problems can also include derivative specifications (points plus tangents). Kinematicians sometimes distinguish between these as ‘finitely separated’ versus ‘infinitesimally separated’ precision points. Every synthesis problem has a maximum number of precision points that can be exactly interpolated. If more points are specified, one may switch from exact synthesis to approximate synthesis, in which one seeks mechanisms that minimize a sum of squares or other measure of error between the generated motion and the specified points. While this subsection has focused on the four-bar planar linkage, it should be appreciated that similar questions will arise for other arrangements of links and joints, and like the 6R spatial generalization of the 3R planar robot, some of these move out of the plane into three-space. 3.3. Platform robots Suppose you are given the task of picking up a long bar and moving it accurately through space. While it may be possible to do so by holding the bar with one hand, for a more accurate motion, one naturally grasps it with two hands spread a comfortable distance apart. This reduces the moments that must be supported in one’s joints. On the down side, a two-hand grasp limits the range of motion of the bar relative to one’s body due to the finite length of one’s arms. Similar considerations have led engineers to develop robotic mechanisms where multiple chains of links operate in parallel to move a common endeffector. Generally, this enhances the robot’s ability to support large moments, but this comes at the cost of a smaller workspace than is attained by a serial-link robot of similar size. While many useful arrangements have been developed, we will concentrate on two ‘platform robots’: one planar and one spatial. In each case, a moving end-effector link is supported from a stationary base link by several ‘legs’ having telescoping leg lengths. As SE (2) and SE (3) are 3- and 6-dimensional, respectively, it turns out that the number of legs that make the most useful mechanisms is also 3 and 6, respectively. We begin with the planar 3-RPR platform robot and then address the related 6-SPS spatial platform robot. (These notations will be explained shortly.) 3.3.1. 3-RPR planar robot Recall that R indicates a rotational joint and P indicates a prismatic joint. An RPR chain is a set of four links in series where links 1 and 2 are connected by an R joint, 2 and 3 by a P joint, and 3 and 4 by another R joint. A 3RPR planar robot has three such chains wherein the first and last links are common to all three chains. This arrangement is illustrated in Figure 3.7(a).

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C. W. Wampler and A. J. Sommese P C1 C2 B2

B1

C3

A3

A2

A1 (a)

(b)

Figure 3.6. Pentad (3-RR) linkage.

Before working further with the 3-RPR robot, we first consider what happens to a four-bar linkage if we add a fifth link between its coupler point and a new ground pivot, A3 , as shown in Figure 3.6(a). As it has five links, this new linkage is known as the pentad linkage. Due to the new constraint imposed on the motion of P by the link A3 P , the pentad becomes an immobile structure. The possible assembly configurations can be found by intersecting the four-bar coupler curve (branches C1 and C2 ) that we first met in Figure 3.4(b) with the circle C3 centred on A3 with radius |AP |. As shown in Figure 3.6(b), this particular example has six real points of intersection, one of which gives the illustrated configuration. We already noted in the previous section that due to the bi-cubic nature of the coupler curve, there are generically six such intersection points over the complexes. In this example all of them are real. The pentad could be called a 3-RR mechanism, as the three links (legs) between ground and the coupler triangle have R joints at each end. If we add ‘prismatic’ slider joints on each leg, it becomes the 3-RPR platform robot shown in Figure 3.7(a). The prismatic joints allow the leg lengths 1 , 2 , 3 to extend and retract, driven by screws or by hydraulic cylinders. Thus, the input to the robot are the three leg lengths (1 , 2 , 3 ) ∈ R3 , and the output is the location of the coupler triangle, a point in SE (2). We have already done most of the work for analysing the 3-RPR robot when we formulated equations for the four-bar linkage. The vector diagram in Figure 3.7(b) is identical to Figure 3.5 except for the addition of vector a3 to the new ground pivot and the new link length 3 . Using the same conventions for isotropic coordinates as before, we simply add the new equations 3 θ3 = p − a 3 ,

3 θ¯3 = p¯ − a ¯3

and

θ3 θ¯3 = 1.

(3.25)

489


P

φb2 3

φb1

3 2

Θ3 2

Φ

Φ

1

p Θ2 a3

1 a2

Θ1 a1 (a)

(b)

O

Figure 3.7. 3-RPR planar platform robot.

If we wish to eliminate Θ3 , we replace these with the circle equation p−a ¯3 ). 23 = (p − a3 )(¯

(3.26)

The inverse kinematics problem is to find the leg lengths (1 , 2 , 3 ) given ¯ ∈ SE (2). Since the leg a desired location of the coupler triangle (p, p¯, φ, φ) lengths must be positive, they can be evaluated uniquely from equations (3.19) and (3.26). In the opposite direction, the forward kinematics problem comes down to computing the simultaneous solution of equations (3.20) and (3.26). This is the same as solving the pentad problem illustrated in Figure 3.6(b), which has up to six distinct solutions. Notice that once we have found (p, p¯) from ¯ from equation (3.19), where they this intersection, we may recover (φ, φ) appear linearly. Using the notation of equation (3.21) for 2×2 determinants, this solution reads φ = |¯ u v|/|u u ¯|, φ¯ = |v u|/|u u ¯|. (3.27) 3.3.2. 6-SPS ‘Stewart–Gough’ platform A natural generalization of the 3-RPR planar platform robot is the 6-SPS spatial platform robot, where the rotational R joints are replaced by spherical S joints and the number of legs is increased to six. This arrangement, illustrated in Figure 3.8, is usually referred to as a ‘Stewart–Gough2 platform’. The device has a stationary base A and a moving platform B connected by six legs of variable length. The six legs acting in parallel can 2

For historical reasons, the arrangement first became widely known as a Stewart platform, although D. Stewart’s invention actually had a different design. As it appears that E. Gough was the true first inventor of the device, current usage (Bonev 2003) now credits both contributors.

490


B b1 B1 p L1

a1

A1

A

Figure 3.8. General Stewart–Gough platform robot.

support large forces and moments, which makes the platform robot ideal as the motion base for aircraft simulators, although it has many other uses as well. Most practical versions of the device resemble the arrangement shown in Figure 3.9, where the six joints on the base coincide in pairs, forming a triangle, and those on the moving platform are arranged similarly. The edges of these triangles along with the six connecting legs form the twelve edges of an octahedron. In more general arrangements, such as the one shown in Figure 3.8, the joints have no special geometric configuration. It is easy to see that if the legs of the platform are allowed to extend and retract freely without limit, the moving platform can be placed in any pose, say (p, C) ∈ SE (3), where p ∈ R3 is the position of a reference point on the platform and C ∈ SO(3) is a 3 × 3 orthogonal matrix representing the orientation of the platform. Let aj ∈ R3 , j = 1, . . . , 6 be vectors in reference frame A to the centres A1 , . . . , A6 of the spherical joints in the base, and let bj ∈ R3 , 1, . . . , 6 be the corresponding vectors in reference frame B for the joint centres B1 , . . . , B6 in the moving platform. Figure 3.8 labels the joints and joint vectors for leg 1. The length of the jth leg, denoted Lj is just the distance between its joint centres: L2j = |Aj Bj |2 = p + Cbj − aj 22 = (p + Cbj − aj )T (p + Cbj − aj ),

j = 1, . . . , 6.

(3.28)

This spatial analogue to equation (3.19) solves the inverse kinematic prob-


491

Figure 3.9. Octahedral Stewart–Gough platform robot.

lem for the 6-SPS robot. There is only one inverse solution having all positive leg lengths. Another way to view equation (3.28) is that when leg length Lj is locked, it places one constraint on the moving platform: joint centre Bj must lie on a sphere of radius Lj centred on joint centre Aj . Since SE (3) is sixdimensional, it takes six such constraints to immobilize plate B, turning the robot into a 6-SS structure. The forward kinematics problem is to find the location of B given the six leg lengths, that is, to solve the system of equations (3.28) for (p, C) ∈ SE (3) given the leg lengths (L1 , . . . , L6 ) and the parameters of the platform (aj , bj ), j = 1, . . . , 6. Notice that for the platform robots, the inverse kinematics problem is easy, having a unique answer, while the forward kinematics problem is more challenging, having multiple solutions. This is the opposite of the situation for serial-link robots. It can happen that both directions are hard; for example, consider replacing the simple SPS-type legs of the Stewart–Gough platform with ones of type 5R. It is somewhat inconvenient to solve the forward kinematics problem as posed in equation (3.28), because we must additionally impose the conditions for C to be in SO(3), namely C T C = I and det C = 1. In the planar 3-RR case, isotropic coordinates simplified the treatment of displacements in SE (2). Although isotropic coordinates do not generalize directly to SE (3), there is an alternative applicable to the spatial case: Study coordinates. ¯ ∈ C4 are restricted to the bilinear Just as isotropic coordinates (p, p¯, φ, φ) ¯ quadric φφ = 1, Study coordinates [e, g] ∈ P7 are restricted to the bilinear quadric S62 = V(Q), where Q(e, g) = eT g = e0 g0 + e1 g1 + e2 g2 + e3 g3 = 0.

(3.29)

The isomorphic mapping between Study coordinates [e, g] ∈ S62 ⊂ P7 and (p, C) ∈ C3 × SO(3) is discussed in the Appendix. In Study coordinates,

492


translation p is rewritten in terms of [e, g] via quaternion multiplication and conjugation as p = g ∗ e /(e ∗ e ) while rotation of vector v by rotation matrix C is rewritten as Cv = e ∗ v ∗ e /(e ∗ e ). In Study coordinates, the leg length equations become: L2i = (g ∗ e + e ∗ bj ∗ e )/(e ∗ e ) − aj 22 ,

j = 1, . . . , 6.

(3.30)

Using the relations v2 = v ∗ v , and (a ∗ b) = b ∗ a , one may expand and simplify equation (3.30) to rewrite it as 0 = g ∗ g + (bj ∗ bj + aj ∗ aj − L2j )e ∗ e + (g ∗ bj ∗ e + e ∗ bj ∗ g ) − (g ∗ e ∗ aj + aj ∗ e ∗ g ) − (e ∗ bj ∗ e ∗ aj + aj ∗ e ∗ bj ∗ e ) = g T g + g T Bj e + eT Aj e,

j = 1, . . . , 6.

(3.31)

In the last expression, g and e are treated as 4 × 1 matrices and the 4 × 4 matrices Aj (symmetric) and Bj (antisymmetric) contain entries that are quadratic in aj , bj , Lj . Expressions for Aj and Bj can be found in Wampler (1996). In the forward kinematics problem, Aj and Bj are considered known, so these equations along with the Study quadric equation, equation (3.29), are a system of seven quadrics on P7 . The forward kinematics problem is to solve these for [e, g] ∈ P7 given the leg lengths (L1 , . . . , L6 ). The forward kinematics problem for general Stewart–Gough platforms received extensive academic attention leading up to its effective solution in the early 1990s. It can be shown that a generic instance of the problem has 40 isolated solutions in complex P7 . The first demonstration of this was numerical, based on polynomial continuation (Raghavan 1991, 1993), while around the same time others published demonstrations based on Gröbner bases over a finite field (Lazard 1993, Mourrain 1993) and a proof by abstract algebraic geometry (Ronga and Vust 1995). The formulation of the problem in Study coordinates was arrived at independently by Wampler (1996), who gave a simple proof of the root count of 40, and Husty (1996), who showed how to reduce the system to a single degree 40 equation in one variable. Using current tools in numerical algebraic geometry, the solution of seven quadrics is a simple exercise, requiring from a few seconds to a minute or so of computational time, depending on the computer applied to the task, to track 27 homotopy paths. As we shall discuss further below, once the first general example has been solved to find just 40 solutions, all subsequent examples can be solved by tracking only 40 homotopy paths.

4. Algebraic kinematics Now that we have discussed a variety of problems from kinematics, let us examine how we may fit them all into a common framework. This framework will help show why so many problems in kinematics are algebraic at their core.


493

4.1. Rigid-body motion spaces In the examples, we have already dealt in some detail with SE (3), the sixdimensional set of rigid-body transformations in three-space. The most useful representations of SE (3) are as follows. • (p, C) ∈ R3 × SO(3), where SO(3) = {C ∈ R3×3 | C T C = I, det C = 1}. This acts on a vector v ∈ R3 to transform it to u = Cv + p. • The 4×4 homogeneous transform version of this, where (p, C) are placed in a matrix so that the transform operation becomes C p v u . = 0 1 1 1 • Study coordinates [e, g] ∈ S62 ⊂ P7 , where S62 = V(eT g) and the transform operation is u = (e ∗ v ∗ e + g ∗ e )/(e ∗ e ). In all three cases, the representations live on an algebraic set and the transform operation is also algebraic. There are several subgroups of SE (3) that are of interest. Most prominent is SE (2), the set of planar rigid-body transformations, with representations as follows. • (p, C) ∈ R2 × SO(2), where SO(2) = {C ∈ R2×2 | C T C = I, det C = 1}. The transform rule looks identical to the spatial case: u = Cv + p. • The unit circle form {(x, y, s, c) ∈ R4 | c2 + s2 = 1}. This is the same as the former with p = xi + yj and c −s . C= s c • The tangent half-angle form (x, y, t) ∈ R3 , in which rotations become 1 1 − t2 −2t C= . 2t 1 − t2 1 + t2 ¯ ∈ C4 | θθ¯ = 1}. Real transforms must • Isotropic coordinates {(p, p¯, θ, θ) ∗ ∗ ¯ The action of transform (p, p¯, θ, θ) ¯ on a vector satisfy p = p¯ and θ = θ. given by isotropic coordinates (v, v¯) is the vector (u, u ¯) given by ¯v ). (u, u ¯) = (p + θv, p¯ + θ¯ Again, each of these representations lives on an algebraic set and has an algebraic transform operation. Clearly, SE (2) is a three-dimensional space. Another subspace of interest is the set of spherical transforms, that is, just SO(3), another three-dimensional space. This is SE (3) with the translational portion set identically to zero. The terminology ‘spherical’ derives from the fact that this is the set of motions allowed by a spherical ball set in a spherical socket of the same diameter.

494


At root, all of these are algebraic because an essential property of a rigidbody transform is the preservation of distance between any two points. Squared distances are, of course, algebraic. The second property is that the transform must preserve handedness, as we do not wish to transmute a rigid body into its mirror image. It is this latter consideration that restricts the rotational portion to SO(3), which is the component of the set of all orthogonal matrices, O(3), that contains the identity matrix. 4.2. Algebraic joints A mechanism is a collection of rigid bodies connected by joints. Without the joints, each body could move with six degrees of freedom anywhere in SE (3). Typically, we declare one body to be ‘ground’ and measure the locations of all the other bodies relative to it, so a collection of n bodies lives in SE (3)n−1 . Joints are surfaces of contact between bodies that constrain the motion of the mechanism to a subset of SE (3)n−1 . Algebraic joints are those which constrain a mechanism to algebraic subsets of SE (3)n−1 . The most important joints for building mechanisms are the lower-order pairs. These are pairs of identical surfaces that can stay in full contact while still allowing relative motion. In other words, they are formed by a surface that is invariant under certain continuous sets of displacements. The lower-order pairs form six possible joint types, having the following standard symbols. R ‘Revolute’. A surface of revolution is invariant under rotation about its axis of symmetry. A general R pair allows a 1DOF rotational motion equivalent to SO(2). An example is a door hinge. P ‘Prismatic’. A surface swept out by translating a planar curve along a line out of its plane. A general P pair allows a 1DOF translational motion. An example is a square peg in a matching square hole. H ‘Helical’, also known as a ‘screw joint’. A surface swept out by a curve that simultaneously rotates and translates about a fixed axis, with the two rates in fixed proportion. This yields a 1DOF motion for a general H pair. The ratio of the linear rate to the rotational rate is called the ‘pitch’ of the screw. An example is a nut on a screw. C ‘Cylindrical’. A round peg in a matching round hole can both rotate around the axis of symmetry and translate along it independently, generating a 2DOF motion. E ‘Plane’, from the German ‘ebener’ since P is already taken. Plane-toplane contact allows a 3DOF motion, namely the set SE (2). S ‘Spherical’. A spherical ball in a matching spherical socket allows a 3DOF motion, the set SO(3).


495

The importance of the lower-order pairs derives from the fact that surfaceto-surface contact spreads forces of contact over a larger area, reducing stresses that might wear out the machinery. Fortunately – from the viewpoint of an algebraic geometer – five of these six joint types are algebraic. The exception is the H joint. The relative motion of a helical joint of pitch ρ with axis through the origin along the z direction is given by 4 × 4 transforms of the form 

cos θ − sin θ  sin θ cos θ   0  0 0 0

 0 0 0 0  . 1 ρθ 0 1

(4.1)

The mixture of θ with cos θ and sin θ makes the motion truly trigonometric, hence non-algebraic. An alternative line of reasoning is to observe that a helix and a plane containing its symmetry axis intersect in an infinite number of isolated points. Any algebraic curve in R3 intersects a plane in at most a finite number of isolated points. Even more fortuitously, helical joints are rarely used as a direct motion constraint in a manner that impacts kinematic analysis. Instead, screws are usually used to transmit power along a prismatic joint. Consequently, the geometric motion of a great many mechanisms is governed by algebraic joints. Not all joints in mechanism work are lower-order pairs. Line contact between two surfaces can be adequate to spread stresses, and it is typical of many rolling contacts (e.g., wheel on plane). With sufficient lubrication, even sliding line contacts can endure, such as happens in some cam contacts. To demonstrate that a joint type is algebraic, one may write down the constraint conditions it imposes between the transforms for the two bodies in contact. Suppose (p1 , C1 ), (p2 , C2 ) ∈ SE (3) are the rigid-body transforms for bodies 1 and 2. To impose constraints equivalent to the lower-order joints, we need to specify geometric features that are associated with the joints. To this end, let a be a point of body 1, given in body 1 coordinates. ˆ = p1 + C1 a. Similarly, let b The world coordinates of the point are then a ˆ = p2 + C2 b. Let be a point in body 2, so that its world coordinates are b (u1 , u2 , u3 ) be a dextral set of mutually orthogonal unit vectors in body 1, that is, the matrix formed with these as its column vectors is in SO(3). Let (v1 , v2 ) be orthogonal unit vectors in body 2. The unit vectors transform ˆ i = C1 ui , i = 1, 2, 3, and v ˆ i = C2 vi , i = 1, 2. into world coordinates as u With these features, the constraints imposed by lower-order pairs R, P, C, E, and S can be modelled with equations that are algebraic in the transforms (p1 , C1 ) and (p2 , C2 ) and the features a, b, u1 , u2 , u3 , v1 , v2 , as follows.

496


ˆ and u ˆ=b ˆ1 = v ˆ 1 , which after substituting from the expressions above R: a become p1 + C1 a = p2 + C2 b, and C1 u1 = C2 v1 . For brevity, in the remaining cases we only give the constraints on the world coordinates, which the reader may easily expand to get equations in the transforms and features. ˆ −a ˆ −a ˆ T2 (b ˆ) = 0, u ˆ T3 (b ˆ) = 0, u ˆ1 = v ˆ1, u ˆ2 = v ˆ2. P: u ˆ −a ˆ −a ˆ T2 (b ˆ) = 0, u ˆ T3 (b ˆ) = 0, u ˆ1 = v ˆ1. C: u ˆ −a ˆ) = 0, u ˆ1 = v ˆ1. ˆ T1 (b E: u ˆ ˆ = b. S: a Equating two points imposes three constraints, whereas equating two unit vectors imposes only two independent constraints. From this, we see that the R joint imposes 5 constraints, the C joint imposes 4, and the E and S joints each impose 3 constraints. For the P joint, we are equating a pair of orthogonal unit vectors to another such pair, which together imposes three constraints. With the other two scalar equations, this brings the total number of constraints imposed by a P joint to 5. These models are not minimal with respect to the number of parameters involved, so in practice we usually write down equations in other ways.3 Nevertheless, this suffices to show that the lower-order pairs R, P, C, E, and S are, indeed, algebraic. Joints can be described either extrinsically in terms of the constraints they impose or intrinsically in terms of the freedoms they allow. The foregoing description is extrinsic. We have already seen how to model R joints intrinsically in the discussion of spatial 6R serial-link robots (see Section 3.1.2). Note that this was written in terms of 4 × 4 transforms Rz (θ) from equation (3.12), which is equivalent to the transform for a helical joint in equation (4.1) with zero pitch, ρ = 0. The lower-order pairs can be modelled intrinsically using matrices of the form   cos θ − sin θ 0 a  sin θ cos θ 0 b   . (4.2)  0 0 1 c 0

0

0 1

R: Use (4.2) with a = b = c = 0, leaving θ as the joint variable. P: Use (4.2) with θ = a = b = 0, leaving c as the joint variable. 3

The Denavit–Hartenberg formalism is a minimal parametrization. (2000) or any modern kinematics textbook for a definition.

See McCarthy


497

Table 4.1. Algebraic lower-order pairs: constraints and freedoms.

Constraints Freedoms

R

P

C

E

S

5 1

5 1

4 2

3 3

3 3

C: Use (4.2) with a = b = 0, leaving θ and c both as joint variables. E: Use (4.2) with c = 0, leaving θ, a, and b all as joint variables. S: Use

C 0 0 1

with C ∈ SO(3) as the joint freedom. In an intrinsic formulation, the link geometry is encoded in link transforms that are interposed between the joints, such as the matrices Aj in the formulation of the 6R kinematics in Section 3.1.2. If c is the number of independent constraints imposed by a joint, then the number of freedoms of body 2 relative to body 1 allowed by the joint is 6 − c, as summarized in Table 4.1. When modelling the joints with a low number of freedoms (R, P, C) it is usually more convenient to use an intrinsic formulation, while S joints are best modelled extrinsically. 4.3. Mechanism types, families, and spaces To fit a wide variety of kinematics problems into a common format, we need the definitions of a mechanism type and a mechanism family. Definition 4.1. A mechanism type is defined by the number of links, nL , and a symmetric nL ×nL adjacency matrix T whose (i, j)th element denotes the type of joint between links i and j, one of R, P, H, C, E, S, or ∅, where ∅ indicates no connection. By convention, all diagonal elements are ∅. (Each joint appears twice in the matrix: Ti,j = Tj,i are the same joint.) We assume here that the joints are limited to the lower-order pairs, but the list of possibilities could be extended. The enumeration of all possible mechanism types for each value of nL without double-counting mechanisms that are isomorphic under renumbering of the links is a problem in discrete mathematics. Choosing a prospective mechanism type is the first step in a mechanism design effort, and methods for guiding the enumeration of promising alternatives fall into the category of type synthesis. In this paper,

498


we assume that this crucial step is already done so that we begin with a mechanism type. Each mechanism type has an associated parameter space. We have seen in Section 4.2 one way to model each of the algebraic lower-order pairs, R, P, C, E, and S, in terms of feature points and unit vectors. The crossproduct space of all these geometric features forms a universal parameter space for the mechanism type. One may choose to model the joints in a more parsimonious way, but we assume that in the alternative model there still exists a parametrization for each joint and an associated parameter space for all the joints taken together. Definition 4.2. A universal mechanism family (T, Q) is a mechanism type T with an associated parameter space Q describing the geometry of the joints. We assume that Q is irreducible. If one has a parameter space Q that is not irreducible, each irreducible component should be considered to define a separate universal mechanism family. Definition 4.3. A mechanism family (T, Q ) is a subset of a universal mechanism family (T, Q) restricted to an irreducible algebraic subset Q ⊂ Q. Examples of the common sorts of algebraic restrictions that define a mechanism family include the condition that the axes of two R joints in a certain link must be parallel, perpendicular, or intersecting, etc. As a particular example, consider that the universal family of spatial 3R serial-link chains includes the family of 3R planar robots of Section 3.1.1 wherein the R joints are all parallel. One should appreciate that there can be subfamilies within families, and so on. For certain mechanisms, all points of the links move in parallel planes, hence the links move in SE (2) and the mechanism is said to be planar. In particular, a mechanism family wherein all joints are either rotational R with axis parallel to the z-direction or prismatic P with axis perpendicular to the z-direction is planar. Definition 4.4. The link space Z for an n link mechanism is SE (3)n−1 , where one of the links is designated as ground (p, C) = (0, I). Any of the isomorphic representations of SE (3) from Section 4.1 can be used as models of SE (3). If the mechanism family is planar, then Z = SE (2)n−1 in any of its isomorphic representations from Section 4.1. Definition 4.5. The mechanism space M of a mechanism family (T, Q) is the subset of Z × Q that satisfies the joint constraints. Proposition 4.6. If a mechanism family is built with only the algebraic joints R, P, C, E, and S, then its mechanism space is algebraic.


499

Proof. Section 4.1 with the Appendix shows that Z is algebraic and Q is algebraic by assumption. That is, Z and Q are sets defined by algebraic equations. Section 4.2 shows that the algebraic joints impose algebraic constraints on the coordinates of Z and Q, and hence all the defining equations for M are algebraic. Definition 4.7. A mechanism is a member of a mechanism family (T, Q) given by a set of parameters q ∈ Q. 4.4. Kinematic problems in a nutshell In this section, we present an abstract formulation that summarizes all the main types of geometric problems that arise in kinematics. In the next section, we will discuss more concretely how to map a mechanism into this formulation. The key to our formulation is the following diagram: X π1 6 X ×Q

J K M Y H ˆ ˆ Jπ H K 6 π4 HH M

π2

? - Q

j H

π3

(4.3)

Y ×Q

The main elements of the diagram are four sets X, M, Y, Q and three maps J, K, πM . The four sets are as follows. • X is the input space of the mechanism. In robotics, it is usually called the ‘joint space’. Its coordinates are typically quantities that we command by controlling motors or other actuators. • Y is the output space, often called the ‘operational space’ in robotics. Its coordinates are the final output(s) we wish to obtain from the mechanism, such as the location of a robot’s hand. • Q is the parameter space of a family of mechanisms. It is the set of parameters necessary to describe the geometry of the joints in each link. Each point in Q is therefore a specific mechanism with designated link lengths, etc. The whole set Q constitutes a family of mechanisms, such as the set of all 6R robot arms, with the coordinates of Q representing all possible link lengths, etc. We assume that Q is an irreducible algebraic subset of some Cm , that is, it is an irreducible component of V(G) for some system of analytic functions G : Cm → Cm . If V(G) has more than one irreducible component, then each such component is considered a different family of mechanisms. • M is the mechanism space, which describes all possible configurations of the mechanism for all possible parameters. Let Z be the space of all possible locations of the links when they are disconnected. That

500


is, for an N -link spatial mechanism with one link designated as ground, Z = SE (3)N −1 . Then, M is the subset of Z × Q where the link locations satisfy the constraints imposed by the joints between them. Let F : Z × Q → Cc be a set of polynomials defining the joint constraints. Then, M = V(F ) ∩ V(G) is an extrinsic representation of M . Each point (z, q) ∈ M is a specific mechanism q ∈ Q in one of its assembly configurations z ∈ Z. In some cases, it is more natural to describe M intrinsically via an irreducible set, say Θ, that parametrizes the freedoms of the joints of the mechanism, so that Z becomes Θ × SE (3)N −1 . We will use this, for example, to describe M for 6R serial-link robots. In such a representation, F includes the equations that define Θ along with the equations relating link poses to joint freedoms and equations for the constraints imposed by closing kinematic loops. Formulating such equations is part of the art of kinematics, and we will not delve into it in this paper beyond what is necessary to present specific examples. After choosing a representation for SE (3), and if present, for the joint freedom space Θ, the space Z is a subspace of some Euclidean space, Z ⊂ Cν , and z ∈ Z has coordinates z = (z1 , . . . , zν ). Three maps are defined on M , as follows. • J : M → X is the input map, which extracts from M the input values. The symbol J acknowledges that the inputs are usually a set of joint displacements. • K : M → Y is the output map, which extracts from M the output values. • πM : M → Q is a projection that extracts the parameters from M . It is the natural projection operator on Z × Q restricted to M given by πM : (z, q) → q. For the moment, we assume only that F, G, J, K are analytic so that the spaces X, M, Y are analytic sets. Although later we will restrict further to algebraic maps and algebraic sets, the analytic setting allows a somewhat wider range of mechanisms into the framework. Specifically, H joints are analytic but not algebraic. The commutative diagram is completed by defining Jˆ := (J, πM ) and ˆ K := (K, πM ) and the associated natural projections π1 , π2 , π3 , π4 . It should be understood that M characterizes a family of mechanisms, such as the family of spatial 6R serial-link robots, the family of planar fourbar linkages, or the family of Stewart–Gough platforms. Maps J and K are tailored to an application of the mechanism. For a four-bar function generator, J gives the input angle and K gives the output angle, while for a four-bar path generator, K gives instead the position of the coupler point.


501

Using the diagram of equation (4.3), we may succinctly recast all the problems mentioned in the motivating examples of Section 3. The problems broadly classified into three types of problems: • analysis (mobility analysis, forward and inverse kinematics, workspace analysis), • synthesis (precision point problems), and • exceptional mechanisms. We describe each of these in more detail next. 4.4.1. Analysis In analysis problems, one has a specific mechanism, say q ∗ ∈ Q, and one wishes to analyse some aspect of its motion. Definition 4.8. The motion of a mechanism given by parameters q ∗ ∈ Q −1 ∗ in a family with mechanism space M is πM (q ) = M ∩V(q −q ∗ ) where (z, q) are coordinates in Z × Q. This can also be called the motion fibre over q ∗ . In the following, it is also convenient to define the inverses of J and K: J −1 (x) = {(z, q) ∈ M | J(z, q) = x}, K −1 (y) = {(z, q) ∈ M | K(z, q) = y}. These are defined for x ∈ X and y ∈ Y , respectively. In the set J −1 (x) for a particular x ∈ X, q is not fixed, so this inverse applies across a whole mechanism family. When we wish to address just one particular mechanism, q ∗ , we want to consider the inverse of Jˆ instead: ˆ q) = (x, q ∗ )}. Jˆ−1 (x, q ∗ ) = {(z, q) ∈ M | J(z, Similarly, we have ˆ q) = (y, q ∗ )}. ˆ −1 (y, q ∗ ) = {(z, q) ∈ M | K(z, K The basic problems in analysis are as follows. −1 ∗ • Motion decomposition of a mechanism breaks πM (q ) into its irreducible components, often called assembly modes by kinematicians. (See Section 7 for a description of irreducible components.) The numerical −1 ∗ irreducible decomposition of πM (q ) (Section 7.5) finds the dimension and degree of each assembly mode and provides a set of witness points on each.

• Motion decomposition of a mechanism family breaks M into its irreducible components. If A ⊂ M is one of these components, then πM (A) ⊂ Q is the subfamily of mechanisms that can be assembled in that mode, dim πM (A) is the dimension of the subfamily, and dim A − dim πM (A) is the mobility of that mode.

502


• Mobility analysis seeks to find the degrees of freedom (DOFs) of the mechanism, which is equivalent to −1 ∗ (q ). Mobility := dim πM

(4.4)

As the dimension of an algebraic set is always taken to be the largest dimension of any of its components, this definition of mobility picks out the assembly mode (or modes) having the largest number of DOFs. There are simple formulas, known as the Gruebler–Kutzbach formulas, that correctly estimate the mobility for a wide range of mechanisms, and even more mechanisms submit to refined analysis based on displacement group theory, but there exist so-called ‘paradoxical’ mechanisms that have higher mobility than these methods predict. To handle all cases, one needs to analyse the equations defining M in more detail, taking into account that q ∗ may be on a subset of Q having exceptional mobility. • Local mobility analysis finds the mobility of a mechanism in a given assembly configuration. That is, given (z ∗ , q ∗ ) ∈ Z × Q, one wishes to find −1 ∗ Local mobility := dim(z ∗ ,q∗ ) πM (q ).

(4.5)

A mechanism can have more than one assembly mode, corresponding −1 ∗ (q ). The local mobility is the to the irreducible components of πM dimension of the assembly mode that contains the given configuration, z ∗ , or if there is more than one such mode, the largest dimension among these. • Forward kinematics seeks to find the output that corresponds to a given input x∗ for a mechanism q ∗ . That is, for x∗ ∈ X and q ∗ ∈ Q, one wishes to find (4.6) F K(x∗ , q ∗ ) := K(Jˆ−1 (x∗ , q ∗ )). Example: given the joint angles of a particular 6R serial-link robot, find its hand pose. • Inverse kinematics is similar to forward kinematics but goes from output to input. For y ∗ ∈ Y and q ∗ ∈ Q find ˆ −1 (y ∗ , q ∗ )). IK(y ∗ , q ∗ ) := J(K

(4.7)

Example: given the hand pose of a particular 6R serial-link robot, find all sets of joint angles that reach that pose. • Singularity analysis finds configurations where the maps lose rank. If we have found a motion decomposition of the mechanism, then for −1 ∗ each assembly mode A ⊂ πM (q ) there is an associated input space J(A) and an output space K(A). The input and output maps have


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Jacobian matrices ∂J/∂z and ∂K/∂z. Assume for the moment that A is a reduced algebraic set. (For example, V(x − y) is a reduced line in the (x, y)-plane, while the double line V((x − y)2 ) is non-reduced.) For generic points (z, q ∗ ) ∈ A, the Jacobian matrices have a constant rank, say rank[∂J/∂z(z, q ∗ )] = rJ and rank[∂K/∂z(z, q ∗ )] = rK . Then, there may be input and output singularities, as follows. ∗ Input singularities: {(z, q ∗ ) ∈ A | rank ∂J ∂z (z, q ) < rJ }. In the common case that ∂J/∂z is square and generically full rank, these are the special configurations where, to first order, the mechanism can move without any change in its input. ∗ Output singularities: {(z, q ∗ ) ∈ A | rank ∂K ∂z (z, q ) < rK }. In the common case that ∂K/∂z is square and generically full rank, these are the special configurations where, to first order, the mechanism can move without any change in its output.

If A is a non-reduced assembly mode, a kinematician might consider the whole set to be singular, as the defining equations for A are degenerate in the sense that the column rank of the Jacobian for the system of −1 ∗ equations that define πM (q ) is less than the codimension of the set A. Alternatively, one might wish to consider the input and output singularities of the reduction of A, which can be analysed via a deflation of A. (See Section 6.3 for a discussion of deflation.) • Workspace analysis seeks to find all possible outputs of a robot or −1 ∗ mechanism. Ignoring limits on the inputs, this is just the set K(πM (q )). The main concern in practice is the set of outputs for real assembly configurations, so letting AR denote the real points in an assembly mode A, the corresponding workspace is K(AR ). As we discussed in Section 3.1.1, the boundaries of the real workspace K(AR ) are given by the real output singularities. When joint limits are included, these may also induce boundaries in the workspace. If B ⊂ X is a joint limit boundary, then −1 ∗ K(J −1 (B) ∩ πM (q )) may form part of the workspace boundary. Positional workspaces and orientational workspaces fit into the same picture by reformulating K to output just position or orientation. Example 1: for a 6R serial-link robot, find all possible poses that the hand can reach. Example 2: for a given four-bar linkage with output defined as the position of its coupler point, find the coupler curve. Example 3: for a 6-SPS (Stewart–Gough) platform robot with limits on the leg lengths, find all possible poses of the moving platform.

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The motion of a mechanism over the complexes contains its real motion, but the extraction of the real motion from the complex one can be difficult, all the more so as the dimensionality of the motion grows. See Section 11 for a discussion. The real motion may break up into smaller pieces than the complex one: for example, the coupler curve illustrated in Figure 3.4(b) has two real branches but they both belong to the same irreducible curve in the complexes. Some branches of a real motion may even have a smaller dimension than the complex motion that contains them. See Wampler, Hauenstein and Sommese (2011) for further discussion. The problems presented above mainly concern the geometry of a mechanism’s motion, where principally angles, positions, and poses enter the picture. As indicated by the questions of singularity analysis, one may also be concerned with differential relations between these, so that joint rates, linear velocity, and angular velocity may become objects of study, as might quantities related to acceleration. Once one has a point on M , differentiation is a straightforward process as is the treatment of the linear maps defined by the Jacobian matrices for the input and output maps. Even so, it can be of interest to find computationally efficient procedures for evaluating the derivative relations, particularly in real-time control of robots. Singularity analysis is important as the robot must either be banned from moving too near its singularities or else special control methods must be employed to avoid degenerate, and potentially dangerous, behaviour.

4.4.2. Synthesis While analysis determines how a mechanism moves, synthesis finds mechanisms that move in a specified way. Synthesis problems begin with a set of desired outputs or a set of input/output pairs and seek to find the mechanisms that will produce these. Synthesis tends to be harder than analysis because one must consider the ability of the mechanism to reach each desired state. In essence, we must consider multiple copies of M simultaneously. The relevant construction in algebraic geometry is called the fibre product. Definition 4.9. For algebraic sets M and Q with map π : M → Q, the fibre product of M over Q is M ×Q M := {(u, v) ∈ M × M | π(u) = π(v)}. This just means that u and v are two assembly configurations of the same mechanism q, where q = π(u) = π(v). If F (z; q) = 0 is a set of polynomial equations that defines M , that is, M = V(F (z; q)), then M ×Q M simply repeats the equations keeping the parameters the same in both copies: M ×Q M = V(F (z1 ; q), F (z2 ; q)).


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Clearly, the fibre product operation can be extended to triple fibre products and higher, with the k-fold fibre product defined as k Q

M = M ×Q · · · ×Q M := {(u1 , . . . , uk ) ∈ M k | π(u1 ) = · · · = π(uk )}.

k times

We may define a map Π :

k

QM

→ Q as the composition of

· · × π

π × · k times

with the map (q, . . . , q) → q. The input and output maps J and K also extend naturally to fibre products, and we can also define a combined input/output map JK as follows: Kk :

k

M → Y k,

Q

where Kk (u1 , . . . , uk ) = (K(u1 ), . . . , K(uk )), JKk :

k

and

(4.8)

M → (X × Y )k ,

Q

where JKk (u1 , . . . , uk ) = {(J(u1 ), K(u1 )), . . . , (J(uk ), K(uk ))}. Note that K1 ≡ K. With these maps, we may define several kinds of synthesis problems. The following problems are known as precision point problems, since there is a set of specified points which the mechanism must interpolate exactly. • Output synthesis seeks mechanisms that can reach a set of specified outputs. For (y1 , . . . , yk ) ∈ Y k , we wish to find the set {q ∈ Q | Kk (Π−1 (q)) = (y1 , . . . , yk )}. Kinematicians distinguish between different types of output synthesis. Path synthesis finds mechanisms where the path of a point of the mechanism interpolates a set of given points. In this case, K is defined on M such that Y ⊂ C3 . Body guidance. In this case, the output is the pose of one body of the mechanism, that is, Y ⊂ SE (3). The purpose of the mechanism is to guide that body through a set of specified poses. • Input/output synthesis seeks mechanisms that produce a coordinated input/output relationship specified by a set of input/output pairs. For ((x1 , y1 ), . . . , (xk , yk )) ∈ (X × Y )k , we wish to find the set {q ∈ Q | JKk (Π−1 (q)) = ((x1 , y1 ), . . . , (xk , yk ))}.

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A common case is a 1DOF mechanism, such as a four-bar, with the input being the angle of one link with respect to ground. Then, with K defined as in a path synthesis problem, the input/output problem becomes path synthesis with timing. Similarly, one can have body guidance with timing. (The nomenclature derives from an assumption that the input moves at a constant rate.) If the input and output are both angles of the mechanism, then input/output synthesis becomes function generation, as the precision points approximate some desired functional relationship between input and output. What makes these problems difficult is that the whole system of equations defining M is repeated k times, increasing the total degree of the system exponentially in k. For any precision point problem, there is a maximum number of precision points that can be specified. Roughly speaking, this is the total number of independent parameters in the mechanism family under consideration divided by the number of constraints placed on the parameters by each precision point. If more than the maximum number of precision points is specified, then there will in general be no mechanism that interpolates them exactly. One may then reformulate the problem by defining an error metric and seek mechanisms whose motion best fits the specified approximation points. This is analogous to finding a best-fit line that approximates three or more points. We should note that all these synthesis problems have been formulated only at the geometric level. It is also possible to specify motions at the level of velocity or acceleration or to mix specifications at several levels. For a 1DOF motion, differential relations can be approximated by limits as precision points approach each other. For this reason, classical synthesis theory sometimes distinguishes between finitely separated precision points and infinitesimally separated precision points. We will not discuss synthesis problems involving differential relations further here. 4.4.3. Exceptional mechanisms −1 ∗ While M describes the motion of an entire family of mechanisms, πM (q ) is the motion of a particular mechanism in the family. For any generic q ∈ Q, attributes such as the mobility of the mechanism or the local mobilities of its assembly modes all stay constant. However, there may be algebraic subsets of Q where mobilities increase. These exceptional mechanisms are often called ‘overconstrained mechanisms’, as a slight perturbation of the parameters off of the exceptional set into a generic position suddenly brings in extra constraints that reduce mobility. One may define subsets of M where the local mobility is constant, that is, −1 Dk∗ = {(z, q) ∈ M | dim(z,q) πM (q) = k}.

(4.9)


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The closures of these, Dk = Dk∗ , are algebraic sets. When Dj ⊂ Dk , j > k, we say that Dj is an exceptional set of mechanisms, a family of overconstrained mechanisms. The discovery of exceptional mechanisms is perhaps the most difficult kind of kinematics problem. One may think of these as a kind of synthesis problem where the only thing that is specified about the motion is its mobility. As in the precision-point synthesis problems, it turns out that fibre products play a central role. We leave further discussion of this to Section 12. 4.5. Fitting into the nutshell The examples of Section 3 illustrated how a range of problems from kinematics can be written as systems of algebraic equations. Moreover, Sections 4.1 and 4.2 show that any mechanism composed of n rigid links connected by any combination of R, P, C, E, or S joints leads to a set of constraint equations that is algebraic in the link locations (pj , Cj ) ∈ SE (3), j = 1, . . . , n and is also algebraic in the parameters defining the joints. In Sections 4.3 and 4.4, we put forward a schema that formulates a wide variety of kinematics problems in terms of spaces X, M, Y, Q and maps J, K, π between them. In this section, we will detail how the example mechanism types of Section 3 fit into this schema. 4.5.1. Planar 3R robots Consider first the 3R planar serial-link robots. These have nL = 4 links, one of which is ground. The adjacency matrix has R in each element of the super- and sub-diagonals and ∅ everywhere else. Since the mechanism is planar, the link space is Z = SE (2)3 . Using the reference frames as indicated in Figure 4.1, we have coordinates for Z as z1 = (Px , Py , x1 , y1 ),

z2 = (Qx , Qy , x2 , y2 ),

z3 = (Rx , Ry , x3 , y3 ), (4.10)

where (Px , Py ) are the coordinates of point P , etc., and xj = cos φj , yj = sin φj , j = 1, 2, 3. (Recall from Figure 3.1(c) that φ1 , φ2 , φ3 are the absolute rotation angles of the links.) Accordingly, the algebraic equations defining the link space Z are x2j + yj2 − 1 = 0,

j = 1, 2, 3.

(4.11)

The parameters of the mechanism are just the link lengths (a, b, c), so the parameter space Q is C3 . In the plane, the constraint imposed on two links by a rotational joint is the coincidence of the point of connection. Point O = (0, 0) in the ground must coincide with point (−a, 0) in the reference frame of link 1: (0, 0) = (Px − ax1 , Py − ay1 ).

(4.12)

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C. W. Wampler and A. J. Sommese x3 y3 y2 y1

R c

x2 Q

b x1

y0 O

a

P

x0

Figure 4.1. Planar 3R robot with reference frames.

Similarly, the other two joints impose the constraints (Px , Py ) = (Qx −bx2 , Qy −by2 ) and (Qx , Qy ) = (Rx −cx3 , Ry −cy3 ) (4.13) Accordingly, equations (4.11)–(4.13) define the mechanism space M . To complete the picture, we need the maps π, J, K. The projection π : Z → Q simply picks out the parameters: π : (z1 , z2 , z3 , a, b, c) → (a, b, c).

(4.14)

T3

Assuming the input space X = is the relative rotation angles (θ1 , θ2 , θ3 ) (see Figure 3.1(b)) represented by cosine/sine pairs, the difference formulas for cosine and sine give J : (z1 , z2 , z3 , a, b, c) (4.15)

→ ((x1 , y1 ), (x2 x1 + y2 y1 , y2 x1 − x2 y1 ), (x3 x2 + y3 y2 , y3 x2 − x3 y2 )). Finally, assuming the output space Y = SE (2) is the location of reference frame 3, the output map is K : (z1 , z2 , z3 , a, b, c) → (z3 ).

(4.16)

If instead the robot is applied to just positioning point R in the plane, we have Y = C2 with the output map K : (z1 , z2 , z3 , a, b, c) → (Rx , Ry ).

(4.17)

With these definitions, all the problems posed in Section 3.1.1 fit neatly into the nutshell schema. 4.5.2. Spatial 6R robots The case of spatial 6R robots is quite similar to the 3R planar case, but we shall choose to handle the joint constraints by introducing variables


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implicitly modelling the freedom of the joints rather than explicitly writing constraint equations. A 6R serial-link chain has nL = 7 links, one of which is ground. The adjacency matrix has entries of R on the super- and subdiagonals and ∅ elsewhere. Let the link space be Z = T 6 ×SE (3)6 , with unit circle representations of the joint angles and 4 × 4 homogeneous transforms for the link locations, so that Z is represented as z = {(cj , sj , Tj ), j = 1, . . . , 6} with c2j + s2j − 1 = 0, j = 1, . . . , 6. (4.18) The first factor of Z is precisely the joint space X = T 6 and the output is the location of the ‘hand’, the last frame in the chain, T6 . The link parameters are 4 × 4 transforms Aj ∈ SE (3), j = 0, 1, . . . , 6. One can use general transforms, but the Denavit–Hartenberg formalism shows that by choosing reference directions aligned with joint axes and their common normals as indicated in Figure 3.3, it suffices to parametrize the Aj as   1 0 0 aj 0 αj −βj 0   Aj =  (4.19) 0 βj αj dj , j = 0, . . . , 6. 0 0 0 1 In this expression, (αj , βj ) are a cosine/sine pair for the twist of link j, aj is the length of the link (along its x-direction), and dj is the link offset distance (along its z-direction). To keep Aj in SE (3), these must satisfy αj2 + βj2 − 1 = 0,

j = 0, . . . , 6.

(4.20)

With this parametrization, the parameter space is Q = T 7 × C14 with coordinates q = (αj , βj , aj , dj ), j = 0, . . . , 7. Joint rotations Rz (cj , sj ) of the form   c −s 0 0 s c 0 0  (4.21) Rz (c, s) =  0 0 1 0 0 0 0 1 alternate with relative link displacements Aj to give the transforms of the link locations as T1 = A0 Rz (c1 , s1 )A1 ,

Tj = Tj−1 R(cj , sj )Aj ,

j = 2, . . . , 6.

(4.22)

It is easy to see how equation (4.22) leads to equation (3.13) when one eliminates all but the last transform, T6 . Equations (4.18)–(4.21) define the mechanism space M in terms of coordinates (z, q). The associated maps from M to Q, X = T 6 , and Y = SE (3) are π : (z, q) → q,

J : (z, q) → ((cj , sj ), j = 1, . . . , 6),

K : (z, q) → T6 . (4.23)

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4.5.3. Four-bar linkages The four-bar has four links with four R joints. If we call the ground link 0, the two links connected to ground as links 1 and 2, and the coupler as link 3, then the adjacency matrix has entries T1,3 = T2,3 = T0,1 = T0,2 = R. We have already detailed all the necessary equations to describe the four-bar mechanism family in Section 3.2 using isotropic coordinates. The link space ¯ θ1 , θ¯1 , θ2 , θ¯2 ) subject to the unit Z is given by coordinates z = (p, p¯, φ, φ, ¯ length conditions of equation (3.18) (φφ = θ1 θ¯1 = θ2 , θ¯2 = 1). The parame¯ 1 , a2 , a ¯2 , b1 , ¯b1 , b2 , ¯b2 , 1 , 2 ). ter space is Q = C10 with coordinates q = (a1 , a Equations (3.16)–(3.17) express the conditions for connecting the coupler link and thereby define the mechanism space M . The input and output spaces and their associated maps depend on the application of the mechanism. For path generation, we have output Y = C2 with map K : (z, q) → (p, p¯). For body guidance, Y = SE (2) with K : ¯ If timing along the coupler curve or timing of the body (z, q) → (p, p¯, φ, φ). motion are of concern, we may name the angle of one of the links connected to ground as input, say X = T 1 given by J : (z, q) → (θ1 , θ¯1 ). For function generation, the input is the angle of link 1 and the output is the angle of link 2, so J : (z, q) → (θ1 , θ¯1 ) and K : (z, q) → (θ2 , θ¯2 ). 4.5.4. Planar 3-RPR platforms Coordinates for the mechanism space M of the 3-RPR planar platform are an extension of those for the four-bar with a3 , a ¯3 , θ3 , θ¯3 , 3 appended. However, there is a shuffle in which coordinates are parameters and which are variables of the motion. The new maps are ¯ 1 , a2 , a ¯ 2 , a3 , a ¯3 , b1 , ¯b1 , b2 , ¯b2 ), π : (z, q) → (a1 , a J : (z, q) → (1 , 2 , 3 ), ¯ K : (z, q) → (p, p¯, φ, φ).

(4.24)

4.5.5. Spatial 6-SPS platforms For the forward and inverse kinematics problems of the 6-SPS platform, we do not need to explicitly represent transforms for the upper and lower leg segments. It is enough to use the leg lengths and the transform for the moving platform. Hence, the link space is Z = C6 × SE (3), and if we use Study coordinates for SE (3), the space is Z = C6 ×S62 , where S62 ⊂ P7 is the Study quadric given by equation (3.29). By the notation of Section 3.3.2, the coordinates of Z are (L1 , . . . , L6 ), [e, g]. The parameter space Q consists of the vectors aj , bj ∈ C3 , j = 1, . . . , 6. The mechanism space M is given by the Study quadric along with the leg length equations (3.31). The input space X = C6 is the set of leg lengths L1 , . . . , L6 , and the output space is Y = S62 is the Study quadric for the transform of the moving platform. The maps J, K are the associated natural projections.


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5. History of numerical algebraic geometry and kinematics The interaction between algebraic geometry and kinematics dates back to the nineteenth century, when many leading mathematicians were quite familiar with kinematics. In this section, we briefly trace the arc from those beginnings to the current state of the interaction between numerical algebraic geometry and kinematics. Symbolic computer algebra with application to kinematics, often called computational algebraic geometry,4 shares the same roots, but the scope of this account is limited to the numerical branch of developments. As our goal is to give a sense of the history, not a thorough review of the literature, we cite only representative articles here. In the late nineteenth century, mathematicians such as Chebyshev and Cayley (see Cayley (1876)), made direct contributions to kinematics, while Sylvester is known to have followed developments, having presented news of the Peaucellier straight-line mechanism to the Royal Institution in January 1874 (Sylvester 1874). Offshoots of his investigations into straight-line motion are the Sylvester plagiograph and the Sylvester quadruplanar inversor (Dijksman 1976, Kempe 1877). Sylvester’s interest in kinematics was reportedly spurred by Chebyshev urging him to ‘Take to kinematics, it will repay you; it is more fecund than geometry; it adds a fourth dimension to space’ (Halsted 1895). J. de Groot’s Bibliography on Kinematics (de Groot 1970) lists 36 entries by Chebyshev, one of which, Théorie des Mécanismes Connus Sous le Nom de Parallélogrammes (Chebyshev 1854), contains the first appearance of what are now known as Chebyshev polynomials. Another famous contribution is the Chebyshev approximate straight-line mechanism (Kempe 1877, Dijksman 1976). Other mathematical notables with an interest in kinematics include Kempe (1877), Schönflies (see Schönflies and Greubler (1902)), Study (1891), and Clifford (1878). The strong interaction between kinematics and algebraic geometry was suspended in the early twentieth century, when algebraic geometry took a turn towards abstraction under the Hilbert programme. At this time, the consideration of concrete examples of polynomial systems fell out of favour as mathematicians clamoured to obtain results using the power of the new abstract formalism. In addition, the shift in focus was at least partially spurred on by the increasing difficulty of carrying out by hand the calculations required to study new and more challenging systems. However, kinematicians persevered, relying on the old methodologies, such as the Sylvester ‘dialytic elimination’ method, a generalization of the Sylvester resultant. Bottema and Roth (1979) and Hunt (1978) give thorough discourses on the theoretical results in algebraic kinematics up to the late 1970s, 4

Good general references in the area are Cox, Little and O’Shea (1997, 1998). While most work in the area assumes exact (rational) coefficients, Stetter (2004) has pioneered the consideration of numerical issues introduced by inexact (floating-point) data.

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while Dijksman (1976) covers more details of the planar case. The mathematical techniques used in these classic books would have been familiar to mathematicians from the turn of the century. The same can also be said of most later contributions such as those of Angeles (2007) and McCarthy (2000), although over time emphasis has shifted from hand derivations and graphical methods to computer algorithms based on the same underlying mathematics. As computers first became available, kinematicians showed a strong interest in applying them to solving polynomial systems, a drive that would eventually bring the kinematics community back in touch with the applied mathematics community. One of the first such publications, Freudenstein and Sandor (1959), derived a quartic equation for synthesizing a four-bar coupler curve to interpolate five precision points. The expressions were derived by hand and solved on a computer using the quartic formulas. Just a few years later, Roth and Freudenstein (1963) attacked the much more difficult problem of interpolating nine general points, the most possible with a four-bar coupler curve. This work introduced the technique of continuation, which they called the ‘bootstrap method’ and subsequently the ‘parameterperturbation procedure’ (Freudenstein and Roth 1963), and applied it to finding solutions to the associated system of eight seventh-degree equations. From that point onwards, problems from kinematics have played a strong role in motivating the development of continuation into the powerful tool that it has become today. Roth and Freudenstein were motivated by the fact that Newton’s method from a very rough initial guess rarely succeeded in the nine-point problem. Continuation overcomes this problem by defining a parametrized family of systems wherein there exists a starting system with a known solution. This solution may then be updated by Newton’s method as the parameters are stepped in small increments to arrive at the target system. In Roth and Freudenstein’s formulation, there was no guarantee that the solution path would safely arrive at a solution to the target system. In particular, they worked over the real numbers, hence there was always the possibility that the solution path would encounter a singularity where the real solution branches out into the complexes. Although they introduced procedures to evade such trouble spots, these still gave no guarantee of finding a solution. Even so, they unquestionably advanced the art by showing that they could compute some solutions to a problem that had previously seemed intractable. The development of continuation into a technique that could find all solutions to a system of polynomials required new mathematical ideas. The first such algorithm was published by Drexler (1977), followed soon by Garcia and Zangwill (1979), Garcia and Li (1980), and Chow, Mallet-Paret and Yorke (1979). These latter works all used what today is known as a total degree homotopy (see Section 6.1).


513

In the beginning, continuation was rather computationally expensive, so its wider acceptance as a convenient tool depended on more efficient algorithms and more powerful computers. However, early on it was recognized as being more robust than techniques based on algebraic elimination, since an elimination procedure more readily fails on encountering special cases, such as when certain leading coefficients are zero or nearly so. (Consider that Gaussian elimination without pivoting will fail on encountering a zero leading coefficient.) Due to its robustness, Morgan implemented continuation as a back-up method to elimination for finding intersections in GMSOLID, an early solid-modelling program (Boyse and Gilchrist 1982). Morgan’s work attracted the attention of co-worker L.-W. Tsai, a kinematician who thought that continuation might provide a solution to the problem of the inverse kinematics of general six-revolute (6R) robot arms (see Section 3.1.2). At the time, this was considered the most important unsolved problem in the area, having once been declared the ‘Mount Everest of kinematics’ by F. Freudenstein. Although an elimination procedure had been found in 1980 that reduced the problem to a single univariate polynomial of degree 32 (Duffy and Crane 1980), Tsai and Morgan’s (1985) solution by continuation was considered to be the first robust algorithm for solving the problem across all special cases. Their work established that the problem has at most 16 isolated solutions, and ASME recognized its importance to the robotics community by awarding them the Melville Medal. Although Tsai was a student of Roth and therefore knew about the earlier bootstrap method, the Tsai–Morgan paper was the one which alerted the wider kinematics community that continuation is a powerful technique for solving their problems. This message was further cemented in Morgan’s book (Morgan 1987), which made continuation accessible to the engineering community. Around the same time, Morgan teamed up with one of us (Sommese) to incorporate ideas from algebraic geometry into the formation of homotopies for continuation. This allowed the field to progress beyond the total degree homotopy to a multihomogeneous formulation that in many instances reduces the number of continuation paths required (Morgan and Sommese 1987a). Another major advance was the coefficient-parameter technique (Morgan and Sommese 1989; see Section 10.1), which justifies homotopies in the parameter space of a family of problems. Once a single generic problem in the family has been solved, using the total degree homotopy, a multihomogeneous homotopy, or any other method that gives all isolated solutions, coefficient-parameter homotopy solves all subsequent problems in the same family with the number of continuation paths equal to the generic root count of the family. This idea lies behind the justification of all continuation methods. These works were at least in part motivated as improvements to the application of continuation to the 6R problem.

514


To test the power of the new techniques, Morgan, Sommese and Wampler went back to the nine-point synthesis problem that Roth had worked on nearly twenty years earlier. Instead of the 78 = 5 764 801 paths that would be required in a total degree homotopy, they used a multihomogeneous homotopy and symmetry to reduce the number of paths to be tracked to only 143 360. In the end, it was found that generic instances of the problem have 1442 triplets of solutions. These triplets obey a three-way symmetry discovered by Roberts (1875) through an indirect algebraic argument, and first explained by a geometric construction due to Cayley (1876). Since we only need one solution in a triplet to generate all of them, the problem can be solved for any subsequent set of nine precision points using a parameter homotopy that has just 1442 paths. Some other notable applications of continuation to kinematics include Raghavan’s (1991) early demonstration that the forward kinematics problem for general Stewart–Gough parallel-link platforms has 40 roots; see Section 3.3.2. Su, McCarthy and Watson (2004) solved synthesis problems for all of the basic two-degree-of-freedom spatial mechanisms. The synergy between kinematics and numerical algebraic geometry continues today. Problems from kinematics continue to serve as test cases for further developments in algorithms, while those algorithms are increasingly becoming a standard tool for kinematics research.

PART TWO Numerical algebraic geometry The fundamental problem in numerical algebraic geometry is to numerically compute and manipulate the solution set of a system of polynomials   f1 (x)   f (x) :=  ... , (5.1) fn (x) where x = (x1 , . . . , xN ) ∈ CN . In this survey we have seen that many kinematics questions reduce to questions about systems of polynomial equations. Systems of polynomials naturally arise in many other engineering and science disciplines, e.g., chemical systems, game theory, and the discretization of differential equations. Algebraic geometry, the mathematical discipline that investigates systems of polynomials, concerns itself not only with polynomials with real and complex coefficients, but polynomials with coefficients from arbitrary commutative rings. Algebraic geometry over the complex numbers, which is classically called transcendental algebraic geometry, is special because the


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powerful tools of topology, complex analysis, and differential equations are applicable and yield deep results about systems of polynomials. In fact, the development of these tools drew motivation in large part out of transcendental algebraic geometry. Furthermore, many of the tools for algebraic geometry over general commutative rings have been developed by analogy with the geometric results developed for complex algebraic geometry. The books by Hartshorne (1977) and Griffiths and Harris (1994) are excellent introductions for abstract algebraic geometry and transcendental algebraic geometry respectively. Different approaches to algebraic geometry engender different computational tools. Corresponding to the abstract approach to algebraic geometry is the symbolic approach (including in particular resultants and Gröbner bases). Corresponding to transcendental algebraic geometry is numerical algebraic geometry. Historically, within numerical algebraic geometry, the problem of finding isolated solutions for square systems, i.e., systems such as equation (5.1) in the case n = N , came first. The most basic tool is homotopy continuation (or continuation for short), which consists of studying a square system of polynomials f (x) by starting with a simpler system g(x) that we know how to solve and deforming g(x) and the solutions of g(x) = 0 to f (x) and the solutions of f (x) = 0. The book by Morgan (1987) is an excellent explication of the numerical solution of square systems by continuation, as practised right at the end of the period when the main tools were from differential topology with few ideas from algebraic geometry. Li (1997) and Sommese and Wampler (2005) are good references for the numerical approach to solving systems of polynomials up to the end of 2004. The solution of non-square systems (n = N in equation (5.1)) came considerably later than methods for square systems. The techniques we employ always reformulate such problems to reduce them once again to finding isolated solutions. At first sight, the case of n > N (more equations than unknowns) would seem to be numerically unstable, as a small perturbation of the system can obliterate a solution. Even for square systems, singular solutions and near-singular solutions raise serious questions about stability with respect to perturbations in the parameters of the homotopy. To address numerical stability concerns, our approach to the numerical solution of algebraic problems is based on two intertwined principles: (1) extra digits may be used as needed; and (2) the formulation of the problem is exact. We dedicate the next few paragraphs to explaining these principles.

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Though the first assumption will be discussed further below (see Section 6.5), we would like to discuss next several situations where mere double precision does not suffice. Consequently, to obtain reliable results, enough extra digits must be used to ensure that computational processes in floatingpoint accurately reflect the underlying principles of algebraic geometry. The first situation concerns the concepts of generic, general , and probability one. The numerical approach to the solution of polynomial systems relies heavily on these notions. Extended discussions of these concepts may be found in Sommese and Wampler (1996) and (2005, Chapter 4). The fundamental idea is that for algebraic problems depending on parameters, with variables and parameters taken over the complex numbers, there is usually a dense open subset of the parameter space on which some sort of generic behaviour occurs. For example, for most choices of coefficients, two polynomials of degree d1 and d2 on C2 have exactly d1 d2 non-singular isolated solutions. We can often use genericity, which in practice is implemented using a random number generator, to very quickly check something that would otherwise be computationally expensive. For example, let X ⊂ CN be a solution set of a polynomial system. Assume X is irreducible, i.e., that the manifold points of X are connected. Let p(x) denote a polynomial on CN . Either p(x) is identically zero on X or else the set B = V(p) ∩ X is very thin. In the latter case, B is contained in a set of real dimension two less than that of X, and in particular, it is of measure zero. Thus, with probability one, we can check if p(x) is identically zero on X by evaluating p(x) at a random point of X. The problem is that even if a point x∗ is a solution of p(x) accurate to a large number of decimal points, p(x∗ ) may be far from zero; see, e.g., Sommese and Wampler (2005, Section 5.3). One needs to evaluate p(x) with enough digits of precision to reliably decide if p(x∗ ) is zero. Another basic example of this comes up in path-tracking. The gamma trick (Morgan and Sommese 1987b), which is discussed briefly in Section 6.1, is a probability-one method of guaranteeing that different paths in continuation do not cross. In practice, two paths, say A and B, may come so close to crossing that the path-tracking algorithm jumps from A to B. This is often not a problem for finding isolated solutions, since any permutation of the solution list is acceptable (for example, a pair of jumps, one from path A to path B and one from path B to path A, still gives all end-points). However, for modern efficient algorithms, this does not suffice: the algorithms need to be sure which start-point leads to which end-point. Even for finding isolated solutions, a path-tracker that is highly resistant to path jumping minimizes the need to check and correct for such faults. Consider the nine-point path synthesis problem for four-bars mentioned in Section 3.2. The homotopy used to solve this problem in Wampler, Morgan and Sommese (1992) requires 143 360 paths to be tracked, of which all but 4326 end on various


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degenerate sets. The 4326 roots of interest appear in a three-way symmetry known at the outset. The original computations for this problem were conducted in double precision followed by a check for any points missing from the expected symmetry groups. Reruns in extended precision cleaned up any paths having questionable numerical stability, filling in the missing points and thereby establishing with high confidence that the solution list was complete. Experiments with a more sophisticated path-tracker having adaptive multiprecision found that in order to be tracked accurately, 0.83% of the paths required precision higher than double precision somewhere in the middle of the paths before returning to double precision (see Bates, Hauenstein, Sommese and Wampler (2009b, Section 5.3)). This approach consistently finds the entire solution set without requiring any reruns or other corrective actions. Although in the early 1990s, this was an extraordinarily difficult computation, we now use this problem as a moderately difficult test system. Now, let us turn our attention to the second principle, the assumption that the formulation of the problem is exact. At first sight, this would seem to run counter to the approximate nature of problems from engineering and science. However, for the majority of polynomial systems that arise in actual problems, the coefficients are small integers when the natural parameters are incorporated into the systems. The uncertainties are in the parameter values not in the coefficients. Consider a system of polynomials   f1 (x, q)   .. f (x, q) :=  (5.2) , . fn (x, q) where x = (x1 , . . . , xN , q1 , . . . , qM ) ∈ CN +M . We regard this as a family of polynomial systems on CN with q ∈ CM regarded as parameters: completely general complex analytic parameter spaces easily fit within this framework. We have already seen in the motivating kinematics examples of Part 1 that in addition to complex Euclidean space, CM , other typical parameter spaces from that field are SE (3) and the unit circle, for which the defining polynomials have coefficients of ±1. We may not know the twist angle of a link exactly, but we do know that its cosine/sine pair must lie on the unit circle. When discussing parameter spaces, it is useful to have the concept of a Zariski open set. By a Zariski open set of CM we mean a set that is the complement of a subset defined by the vanishing of a system of polynomials on CM . A Zariski open set of CM is (except for the case when it is empty) dense and path-connected. The same is true for a Zariski open set of any irreducible algebraic set, such as a parameter space Q associated to a mechanism family in Part 1. For any parametrized family of systems as in equation (5.2), there is a Zariski open set U ⊂ CM of points q ∗ such that the solution sets of

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f (x, q ∗ ) = 0 are very similar: see Morgan and Sommese (1989) and Sommese and Wampler (2005, Section A14). For example, the systems f (x, q ∗ ) = 0 have the same number of isolated solutions with the same multiplicities and the same number of positive-dimensional components with corresponding multiplicities and degrees. In essence, these are all structural properties that have integer values and must remain constant unless some extra algebraic conditions are met. In particular, if parameter values q ∗ ∈ U are changed slightly, a multiple solution moves just slightly to a solution of the same multiplicity. On the complement of U , that is, on the algebraic set, say A, whose complement defines U = CM \ A, the integers may change in specific ways. For example, along a path in U that terminates at a point in A, several isolated solutions might merge to a single isolated solution whose multiplicity is the sum of those for the incoming solutions. Now, A is a new parameter space and the structural properties (isolated root count, etc.) remain constant on a Zariski open set of A. To make ideas concrete, consider the inverse kinematics problem for 6R serial-link robots from Section 3.1.2, formulated into the kinematics nutshell schema in Section 4.5.2. From this starting point, one can derive a square system of equations, say f (x; y, q) = 0, whose solutions are sought as in Tsai and Morgan (1985) or Morgan and Sommese (1987b). In the nutshell ˆ −1 (y, q), y ∈ SE (3), q ∈ Q, so the schema, the problem is to compute K parameter space of the problem is SE (3) × Q, where y ∈ SE (3) is the desired location of the robot hand and where the definition of Q includes the unit circle relations associated to the link twists, equation (4.20). If one lets the hand orientation be a general 3 × 3 matrix instead of a member of SO(3) and lets the cosine/sine pairs (αj , βj ) for the twist angles range over all of C2 instead of the unit circle, one creates a new family of systems with a Euclidean parameter space that contains the family of 6R inverse kinematics problems. This larger family, whose equations are still f (x; y, q) = 0 but with (y, q) in Euclidean space instead of restricted to SE (3) × Q, has a bigger root count than the inverse kinematics problem. As the parameters approach SE (3) × Q, some solutions diverge to infinity. (We will mention in Section 6.2 how one may compute the end-points of such paths even as they diverge.) When one tries to solve the 6R inverse kinematics problem numerically, one must use finite precision, and so numerical round-off almost inevitably perturbs the parameters off of SE (3) × Q into the ambient Euclidean space. Standard methods of numerical analysis can be used to evaluate how far the solutions of the inverse kinematics problem will be moved from their true values by the parameter perturbations. What might seem more problematic, however, is that the perturbations also make extra solutions appear – the ones that would diverge to infinity as the perturbations away from the unit circle go to zero. If improperly interpreted, one might erroneously attribute too large a root count to the problem.


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The saving grace is that we know the conditions for the true parameter space exactly, and we can compute using values for the parameters as close as we wish to it. It is just a matter of using enough digits of precision. Increasing the precision of both the parameters and the end-point computations pushes the extra solutions ever nearer to infinity, providing clear evidence that they will be at infinity for the nearby exact problem. If one does not take the knowledge of the parameter space into account, one might indeed come to the wrong conclusion. In particular, an incautious person might take a version of the problem in which coefficients have been computed (or printed) in limited precision as being a true member of the kinematics problem. No matter how high a precision is applied subsequently in solving the problem, the parameter perturbations have already been locked in and the extra solutions persist without approaching infinity. This clouds the correct interpretation of the solution set. By properly applying the knowledge of the true parameter space, one can avoid this pitfall. For example, when solving the inverse kinematics problem for an actual 6R robot, one cannot know the twist angles of the links exactly, but one does know that whatever these angles may be, the associated cosine/sine pairs fall exactly on the unit circle. Similarly, if one takes an example ‘IPP’ problem from Morgan and Sommese (1987b) (‘inverse position problem’ is another moniker for the 6R inverse kinematics problem) using a hand orientation as printed there to only single precision, one gets extra solutions that would disappear to infinity if one corrected the data to make it lie on SE (3). For a very concrete example of this, one may compare solutions obtained from the version of the IPP problem posted at www.math.uic.edu/˜jan/, which uses the data from Morgan and Sommese (1987b) without correction, to the solutions obtained from the version of the problem posted at www.nd.edu/˜sommese/bertini/ExampleSystems.html, which with a bit of overkill has coefficients computed from parameters that lie on the requisite parameter space to within 300 digits of precision. As an aside, we note that the fact that systems with parameters often have small integer coefficients makes it possible to use exotic characteristic p methods to solve some real world problems, e.g., the construction in Geiss and Schreyer (2009) of a family of exceptional Stewart–Gough mechanisms. With our two basic assumptions in place, we are ready to survey the techniques that make up the field of numerical algebraic geometry. In this survey, we follow the historical order of developments. We start with a quick review of the basic mechanics of continuation for finding isolated solutions of square systems in Section 6. This starts with an introduction to forming homotopies in Section 6.1 and a review of how to track solution paths in Section 6.2. For non-singular roots, basic pathtracking methods follow all the way to the end of the homotopy paths and report the solution there. For singular solutions, it is beneficial to bring

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in a deeper understanding of multiplicities and to introduce the notion of deflation, as discussed in Section 6.3. This leads to special methods, called endgames, for computing path end-points to obtain both the non-singular and the singular solutions (Section 6.4). Path-tracking and endgames are often impossible to carry out with only double precision, especially when systems become large or the degrees of the equations become high. Adaptive precision, an important advance that has happened since 2004, addresses this problem (Section 6.5). In Section 7, we discuss the computation of positive-dimensional sets. We begin in Section 7.1 by discussing how the solution set of a polynomial system compares to the solution set of a randomization of the polynomial system. In Section 7.2, we discuss the behaviour of the solution set of a polynomial system under intersections with general affine linear subspaces of the ambient Euclidean space. These results are important because the data structure we use to represent an irreducible component X of the solution set of V(f ) of a polynomial system f is a triple (f, W, L), where L is a random affine linear subspace of dimension N − dim X and W = X ∩ L. In Section 7.3, we discuss membership tests that decide which irreducible component of V(f ) a given solution point of f (x) = 0 lies on. In Section 7.4, we discuss how the deflation operation of Section 6.3 applies to an irreducible component: often this operation allows us to reduce to the case of a multiplicity-one component. In Section 7.5, we discuss the numerical irreducible decomposition of V(f ). This decomposition of V(f ) is often the starting point in applications.

6. Finding isolated roots In this section we briefly discuss the continuation method of finding isolated solutions of a polynomial system   f1 (x)   f (x) :=  ... , fN (x) where x = (x1 , . . . , xN ) ∈ CN . We refer to Li (1997) and Sommese and Wampler (2005) for more details. 6.1. Homotopy By a homotopy for f (x) = 0, we mean a system of polynomials   H1 (x, t)   .. H(x, t) :=   . HN (x, t)

(6.1)


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in (x1 , . . . , xN , t) ∈ CN +1 with H(x, 0) = f (x). One of the most classical homotopies is the total degree homotopy     f1 (x) g1 (x)     H(x, t) := (1 − t)  ...  + t  ... , (6.2) fN (x) gN (x) where gj is a polynomial of degree deg gj = deg fj = dj . For example, gj d might be xj j − 1. Assume that we have an isolated solution y ∗ of H(x, 1) = 0. Imagine for a moment that there is a function x(t) : (0, 1] → CN such that (1) x(1) = y ∗ ; (2) H(x(t), t) = 0; and (3) the Jacobian Jx H(x, t) of H(x, t) with respect to the variables x is of maximal rank at (x(t), t) for t ∈ (0, 1]. If this happens then the path satisfies the Davidenko differential equation N ∂H(x, t) dxj (t) j=1

∂xj

dt

+

∂H(x, t) = 0, ∂t

and we say that we have a good path starting at y ∗ . Given the initial condition x(1) = y ∗ , we can solve the differential equation numerically. A typical approach is to move an appropriate predictor step in the t variable and then use Newton’s method on H(x, t) = 0 in the x variables to make a correction step. This sort of procedure, called pathtracking, has been thoroughly examined. Given the tracking of the path, some estimation procedure may be attempted to compute a solution x∗ := lim H(x(t), t) t→0

of f (x) = 0. Several things can go wrong: (1) there might be no such path x(t) for the values t ∈ (0, 1]; (2) the path x(t) might exist, but the Jacobian Jx H(x, t) might not be of maximal rank at all points (x(t), t) for t ∈ (0, 1]; and (3) it might be that limt→0 H(x(t), t) is not finite. Bézout’s theorem says that given general polynomials gj of degree dj , the solution set of the system (g1 (x), . . . , gN (x)) = 0 consists of d1 · · · dN non-singular isolated solutions. A classical result (Morgan 1987) is that the total degree homotopy starting at a system g(x), with the gj being sufficiently general degree dj polynomials, has good paths for the d1 · · · dN

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non-singular isolated solutions of (g1 (x), . . . , gN (x)) = 0 and that the finite limits limt→0 H(x(t), t) of the d1 · · · dN paths contain all the isolated solutions of f (x) = 0, though they frequently also contain some points lying on positive-dimensional components of the solution set of f = 0. Finding the exact set of isolated solutions requires considerably deeper theory: the first algorithm to do this is given in Sommese, Verschelde and Wampler (2001a). The most general polynomials of degree dj in N variables can have a large +dj different possible monominumber of terms; to be exact, there are N N als. The γ trick of Morgan and Sommese (1987b) allows one to define a good homotopy without introducing that level of complexity. Suppose that polynomials gj of degree dj are such that the solution set of the system (g1 (x), . . . , gN (x)) = 0 consists of d1 · · · dN non-singular isolated solutions. Then the γ trick states that with probability one, the homotopy     f1 (x) g1 (x)     H(x, t) := (1 − t)  ...  + γt  ... , (6.3) fN (x) gN (x) with γ a random complex number, satisfies the properties: (1) (x, t) | t ∈ (0, 1]; x ∈ CN ; H(x, t) = 0 is a union of d1 . . . dN good paths x1 (t), . . . , xd1 ···dN (t) starting at the solutions of H(x, 1) = 0; and (2) the set of limits limt→0 xj (t) that are finite include all the isolated solutions of H(x, 0) = 0. This theory justifies the use of the very simple start system defined by d gj (x) = xj j − 1, j = 1, . . . , N . If we have an isolated solution x∗ of H(x, 0) = 0 and if (1) there is at least one good path with x∗ as a limit as t → 0; and (2) every path over (0, 1] with limit x∗ as t → 0 is good, then we say that H(x, t) is a good homotopy for x∗ . As will be discussed in Section 6.3, general theory tells us that if H(x, t) is a good homotopy for a solution x∗ of H(x, 0) = 0, then the number of paths going to x∗ equals the multiplicity of x∗ as a solution of H(x, 0) = 0. Constructing good homotopies with the number of paths not too different from the number of isolated solutions of f was an important research topic at the end of the twentieth century. There is detailed discussion of this topic in Sommese and Wampler (2005, Chapter 8). For systems that are not too large, which includes many mechanism systems, polyhedral homotopies are excellent (see Li (2003) for a report on this by the inventor of the leading algorithm for this method). For larger systems, the regeneration approach


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discussed in Section 10.2 is more favourable (Hauenstein, Sommese and Wampler 2010, 2011). For a hint of what difference the selection of homotopy can make, consider the 3-RPR forward kinematics problem of Section 3.3.1, which is equivalent to intersecting a four-bar coupler curve with a circle, as shown in Figure 3.6. We saw at equation (3.20) that a four-bar coupler curve fcc (p, p¯, q) = 0 is degree six in (p, p¯) but only degree three in p or p¯ considered independently. That is, it is a bi-cubic sextic. On the other hand, the circle is degree two, but bi-linear in p and p¯: see equation (3.26). Accordingly, for this problem, a total degree homotopy has 6 · 2 = 12 paths, while a twohomogeneous homotopy has only 6 paths, the same as the generic number of solutions. (Multihomogeneous homotopies were first proposed in Morgan and Sommese (1987a) and discussed in Sommese and Wampler (2005, Section 8.4).) For a given system, a polyhedral homotopy will never have more paths than the best possible multihomogeneous one. In some cases, the differences can be extreme. For example, the generalized eigenvalue problem (A + λB)v = 0 for given N × N matrices A and B has total degree 2N but a two-homogeneous root count of just N . Due to the existence of highly efficient software specialized for eigenvalue problems, one would not use homotopy to solve these, but the root count is illustrative of why much effort has gone into finding efficient formulations of good homotopies. 6.2. Path-tracking Path-tracking for not necessarily polynomial systems is highly developed: see Allgower and Georg (1993, 1997, 2003). An overarching advantage of homotopy methods is that each path may be followed on a different node of a computer cluster, leading to highly parallel algorithms well designed for modern supercomputers. Not all algorithms used in continuation are inherently parallel. In particular, polyhedral methods have so far not parallelized well: for systems of moderate size, the computation of the polyhedral start system (done of necessity on a single node) dwarfs the actual path-tracking. Equation-byequation methods, on the other hand, not only surpass polyhedral methods on a single core for moderate systems, but are highly parallel (see Sections 8 and 10.2). As will be discussed below, multiple paths go to singular solutions. Since singular solutions are considerably more expensive than non-singular isolated solutions, it is lucky that the Cauchy endgame allows us to only have to do an endgame for a fraction of the paths going into the singular solution (see Section 6.4). Polynomial systems are much better behaved than general systems. As mentioned above, the γ trick ensures that paths of a homotopy are well

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behaved. Even so, there are numerical difficulties if the path gets near a branchpoint. The Bertini package (Bates et al. 2008) circumvents these difficulties by means of adaptive multiprecision (Bates, Hauenstein, Sommese and Wampler 2008b, 2009b; see Section 6.5). Another difficulty is paths going to infinity. Tracking them may be computationally expensive as the path in CN is infinitely long, and numerical conditioning may be poor as the magnitudes of the solution variables grow. Morgan’s projective transformation trick (Morgan 1986) is to work on a random coordinate patch in the projective space containing CN . This means that instead of solving a given system f (x) = (f1 (x), . . . , fN (x)), of polynomials in x = (x1 , . . . , xN ) ∈ CN , one shifts to solving the related system on CN +1 ,   f1 ( x0 , . . . , x N )   ..   . (6.4)   = 0,  fN ( x0 , . . . , x N )  a0 x 0 + · · · aN x N − 1 where

x 1 x N deg fj x0 , . . . , x N ) = x 0 fj ,..., fj ( x 0 x 0

is the homogenization of fj and (a0 , . . . , aN ) are random constants (near 1 in absolute value). Note that the solutions ( x∗0 , . . . , x ∗N ) of the system equation (6.4) with x ∗0 = 0 correspond to the finite solutions ( x∗1 / x∗0 , . . . , x ∗N / x∗0 ) of f = 0. The random coordinate patch trick works generally: there is a straightforward prescription (Bates, Hauenstein, Peterson and Sommese 2010b) for how to write down the random coordinate patch system for homogeneous compactifications of CN , e.g., products of Grassmannians. 6.3. Multiplicities and deflation For polynomials in one variable, the multiplicity of a solution z ∗ of a polynomial p(z) = 0 is easy to understand. It is the positive integer µ such that p(z) = (z − z ∗ )µ g(z) with g(z) a polynomial not zero at z ∗ . Of course, numerically the question of whether a polynomial is zero at a point is a real issue, but this is discussed in Section 6.5. In several variables, multiplicity still makes sense, but it is a much weaker measure of the local structure of a solution of a polynomial system than multiplicity is in one dimension. In this subsection we consider only isolated solutions of a polynomial system. The much more difficult question of how to decide if a solution is isolated is put off to Section 9.2. The definition of multiplicity for a system of polynomials in several variables is more technical than in the single-variable case. From the defi-


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nition above of multiplicity in the single-variable case, one can see that derivatives at a solution z ∗ of multiplicity µ vanish up to order µ − 1, i.e., dj p(z ∗ )/dz j = 0, j = 0, 1, . . . , µ − 1. The somewhat technical way to state the equivalent property in the multivariate case is as follows. Let z ∗ be an isolated solution of a polynomial system f as in equation (5.1). Let Iz ∗ denote the ideal generated by f1 , . . . , fn in the ring Oz ∗ of convergent power series in variables z1 , . . . , zN centred at z ∗ . The multiplicity µ(f, z ∗ ) of z ∗ as a solution of the system f is defined as the dimension of A := Oz ∗ /Iz ∗ considered as a complex vector space. Note that A contains much more information than µ(f, z ∗ ). Indeed, taking into account the vanishing derivatives, the polynomial system vanishes on an infinitesimal ‘chunk’ of the CN . As a set this ‘chunk’ consists of just the single point z ∗ , but the fine structure of this point is defined by A. One may regard A as the set of algebraic functions defined in an infinitesimal neighbourhood of z ∗ . The local ring A is the non-reduced structure of the algebraic set defined by Iz ∗ at z ∗ . If n = N and we computed z ∗ using a good homotopy in the sense of Section 6.2, then the number of paths ending in z ∗ equals the multiplicity; e.g., Sommese and Wampler (2005, Theorem A14.11). When the multiplicity µ of solution z ∗ is greater than one, z ∗ is said to be a singular solution. Such solutions are difficult to work with numerically. A primary problem is that the vanishing derivatives ruin the convergence properties of Newton’s method near the singular point. For this reason, tracking paths to z ∗ from a good homotopy for z ∗ is computationally expensive and often impossible in double precision. To deal with these points effectively, we use endgames (see Section 6.4) and adaptive precision (see Section 6.5). Deflation is another approach for dealing with singular points (Ojika, Watanabe and Mitsui 1983, Ojika 1987, Leykin, Verschelde and Zhao 2006, Leykin, Verschelde and Zhao 2008, Hauenstein, Sommese and Wampler 2011). Consider that in the univariate case, z ∗ is a non-singular root of dµ−1 p(z)/dz µ−1 = 0. If we can determine the correct multiplicity, we can restore the quadratic convergence of Newton’s method by searching for zeros of this derivative instead of zeros of p. Deflation is a generalization of this manoeuvre to the multivariate case. For reasons we discuss at the end of this subsection, this approach is not very beneficial for treating isolated singular solutions, but it turns out to be important for components, as we shall discuss later in Section 7.4. To prepare for that eventuality, we discuss deflation for isolated solutions next. Let z ∗ denote an isolated solution of a system f (z) = 0 of N polynomials in the N variables z = (z1 , . . . , zN ). Let F (z, ξ) denote a system of N + k polynomials in the variables z and ξ = (ξ1 , . . . , ξk ). F (z, ξ) = 0 is called a

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deflation of f (z) = 0 at z ∗ if there is a non-singular isolated solution (z ∗ , ξ ∗ ) of F (z, ξ) = 0. The usual deflations are linear in the variables ξ. Leykin, Verschelde and Zhao (2006) presented a procedure to find a deflation of f (z) = 0 at a multiplicity µ isolated solution z ∗ that terminates in less than µ steps. A step in the procedure consists of replacing f (z) = 0 with   f (z)    ξ1    ..    Df (z) ·  .        = 0, ξN       ξ1 1    .   .     Ik A ·  ..  −  ..  ξN 1 where Df is the Jacobian matrix of f (z), N − k is the rank of Df evaluated at z ∗ , Ik is the k × k identity matrix, and A is a random k × N complex matrix. Dayton and Zeng (2005) made a significant improvement in the understanding of how many steps it requires for the procedure to terminate. The main difficulty with this procedure lies in determining the rank of Df evaluated at z ∗ . For a specific root z ∗ one may investigate and make good guesses, but a polynomial system of interest may have millions or even billions of solutions. In the face of this, one needs to have a highly reliable, automated procedure to determine the rank. The trouble is that the rank of Df (z) is falling as z approaches z ∗ , so one needs an accurate value of z ∗ to get the correct rank of Df (z ∗ ). This leads to a vicious circle, since computing z ∗ accurately is the initial objective anyway. The upshot is that for isolated solution points the cost of computing a deflation system dwarfs the cost of computing the point accurately using the endgame methods in the next subsection. 6.4. Endgames Let H(x, t) = 0 be a homotopy as in equation (6.1) and let z(t) with t ∈ (0, t] be a path as in Section 6.2 satisfying H(z(t), t) = 0. Endgames refer to the process of computing x∗ := limt→0 z(t). Assume (as we may, by using Morgan’s projective transformation trick, equation (6.4)), that x∗ is finite, i.e., x∗ ∈ CN . The key to endgames is to realize that regarding t as a complex variable, there is an algebraic curve C ⊂ CN × C such that (z(t), t) ∈ C for all t ∈ (0, 1]. Note that C is an irreducible component of V(H) and thus by the uniformization theorem (Sommese and Wampler 2005, Theorem A.2.2), there is a closed disc ∆(0, δ) := {s ∈ C | |s| ≤ δ}


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of radius δ centred at the origin and a holomorphic map φ : ∆ → C, such that (1) φ(0) = x∗ ; and (2) φ is an isomorphism on the punctured disc ∆∗ (0, δ) := ∆(0, δ) \ 0. By reparametrizing, we can assume that sc = t for a positive integer c, called the winding number of the path z(t). We set = δ c . Let π : CN × C → C be the projection (x, t) → t. Note that c is at least 1. Theory tells us that in a small enough open ball B around (x∗ , 0), the one-dimensional components of V(H) ∩ B that surject onto C under π have winding numbers adding up to the multiplicity of x∗ as a solution of H(x, 0) = 0. Note that the coordinate functions of φ expanded in power series in fractional powers of t are classically called Puiseux series. The first endgames using the structure inherent in the above uniformization were in articles by Morgan, Sommese and Wampler (1991, 1992a, 1992b), where the term ‘endgame’ was given for such algorithms. Of the three related endgames from those articles, we only discuss the Cauchy endgame (Morgan, Sommese and Wampler 1991), which has certain special properties making it stand above the others. By Cauchy’s integral theorem z(s) 1 ∗ x = ds, 2πi |s|=δ s or in terms of sc = t = eiθ 1 x = 2πc ∗

2πc

z 1/c eiθ/c dθ.

0

Note that this last integral can easily and accurately be computed using continuation on t = eiθ , going around the circle c times. One does not need to know c ahead of time, because it is the lowest integer such that along the continuation path z( e2πic ) = z( ), that is, one continues around the circle until z(t) repeats its initial value. There might be a worry that one could have numerical trouble deciding when the repeat has occurred, but this is not a problem. √ Indeed the distances between the values of z(t) over are of the order c . There are several difficulties with the Cauchy endgame that will prevent it (or any endgame) from working if c is large enough and only double precision is available. Indeed for double precision, even c = 4 can be numerically challenging. Happily these difficulties disappear when we have adaptive precision.

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To understand the difficulty with such solutions, consider x −t =0 H(x, t) = 16 x2 − t and the path z(t) = (t, t1/6 ). At t = 10−6 , (x1 , x2 ) = (10−6 , 10−1 ) is quite far from (0, 0) = limt→0 z(t). Putting aside the cost of tracking z(t) to still smaller values of t, we find the condition number of Dx H is of the order of 105 . By Wilkinson’s theorem, we may potentially be losing 5 digits of accuracy. The closer we go to t = 0, the larger the condition number and the more digits we lose. Thus we need to stay far from t = 0. But we might not be able to stay as far from t = 0 as we would like! There may be multiple branch points near 0, i.e., the for which the uniformization theorem holds may need to be chosen very small. Thus we have two competing demands. The needs to be chosen small enough so that the uniformization theorem holds, but not so small that ill-conditioning of Dx H precludes path-tracking. The region where this is possible is called the endgame operating region. Unfortunately, that region may be empty in double precision. However, using adaptive precision, this region always exists, because one can carry enough digits to overcome the ill conditioning. Paying attention to how the values of the path should behave when t is in the region where the uniformization theorem holds yields heuristics that signal when one is in the endgame operating region. A big advantage of the Cauchy endgame is that it parallelizes nicely (Bates, Hauenstein and Sommese 2010a). Indeed, each time we complete a path-tracking loop around the small circle, we find another point over t = , giving c points in all. Tracking these back to t = 1, we find which starting points of the homotopy lead to the same end-point. Having already calculated that end-point, we need not do any further computations, and thereby save running an endgame for c − 1 other start-points. Each trip around the endgame circle can cost on the same order as tracking from t = 1 to the endgame region, so as c grows, the endgame begins to dominate the computation. By running the endgame on only one of c start-points, the computation is reduced considerably, and the net cost for the group of c paths is approximately the same as if they each had cycle number 1. 6.5. Adaptive precision Using extra digits of accuracy solves many problems, but comes with real costs. The first cost is the programming effort to implement high-precision arithmetic on machines with different architectures. Happily, two packages, the ‘GNU MPFR Library’ (www.mpfr.org) and the ‘GNU Multiple Precision Arithmetic Library’ (gmplib.org), which come with standard Linux distributions, remove the need to develop high-precision arithmetic on different


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machines and make it straightforward to port software (such as Bertini (Bates et al. 2008)) to different machines. The price for this standardization is that high-precision arithmetic makes little use of hardware doubleprecision arithmetic, so there is a significant cost to using it. Even if the computational cost of high precision matched the theoretical cost – O(n ln n), where n is the number of digits – the cost would be significant. For these reasons, it is important to use at least as many digits as needed, but not too many more than that. In homotopy algorithms, the most crucial area where high-precision may be needed is in path-tracking. However, the number of digits needed may fluctuate along a path, so it is desirable to scale the precision up and down in response. Bates, Hauenstein, Sommese and Wampler (2008b) introduce an adaptive multiprecision algorithm that tries to ensure that the Newton corrector part of a path-tracker has enough precision. In Bates et al. (2009b) they combine adaptive multiprecision and adaptive step size under the assumption that the predictor part of the path-tracking uses the Euler ODE method. In Bates, Hauenstein and Sommese (2011) they make modifications to combine the algorithms when higher-order ODE methods are used for the predictor step. The basic idea (without the technical details that make it a practical method) recognizes that path-tracking solves linear systems involving the Jacobian matrix in both the ODE predictor and in the Newton corrections. The condition number of the matrix gives an estimate on how many significant digits these processes yield. There must be enough digits to make Newton’s method converge within the tolerance set for path-tracking. As the accuracy of the step also depends on the accuracy with which the polynomials are evaluated, the sizes of the coefficients of the polynomial system need to be part of the estimates (see Bates et al. (2008b, 2009b)). (As discussed in the introduction to Part 2, it also means that the polynomials themselves must be given with sufficient accuracy to match the precision required in their evaluation.) To do all this with enough margin for calculations to be secure, but not enough to make the cost prohibitive, requires a facility for scaling precision up and down on the fly along with fast estimates of how much precision is needed at any time. As noted earlier, using this sort of algorithm to solve the nine-point path synthesis problem for four-bars, we found that nearly one per cent of the paths required extra digits beyond double precision somewhere in the middle of the t-interval (0, 1] before returning back to double precision. There are other uses of multiprecision besides path-tracking. At times, one may wish to check the vanishing of a polynomial at a given point or test the rank of the Jacobian matrix. It can be useful to do these checks at several different precisions to see if the magnitude of the polynomial or of the putatively zero singular values of the Jacobian matrix actually approach

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zero in proportion to the precision used. Though no numerical method is absolutely certain, these moves are effective. Yet another use of multiprecision is to produce points of a witness set (see Section 7.2) to a very high number of digits so that various manoeuvres to construct exact equations may be carried out (Bates, Peterson and Sommese 2008c).

7. Computing positive-dimensional sets As we saw in Section 6.1, the polynomial systems to which continuation naturally applies are square, i.e., they have the same number of equations as variables. What can we say about non-square systems? How do we deal with them numerically? The answers to these questions lead quickly to positive-dimensional algebraic sets. But before we go any further, we need a few definitions from algebraic geometry. A complex affine algebraic set is defined to be the solution set X := V(f ) of a system of polynomials as in equation (5.1) on some complex Euclidean set CN . Unless stated otherwise, algebraic sets in this article are complex affine algebraic sets. Here we drop all multiplicity information, e.g., V(x) = V(x2 ) where x is the coordinate on the complex line. We say that X is irreducible if the set of smooth points Xreg of X is connected. In this case we define dim X = dim Xreg , where, unless stated otherwise, we are using complex dimensions, which are one-half of the real dimension, e.g., CN is N -dimensional. Theory tells us that Xreg is dense in X and that a single polynomial p(x) being irreducible, i.e., not factorizable into polynomials of strictly lower degrees, is equivalent to V(p) being irreducible. Let X = V(f ) for a system of polynomials f = 0 on CN . The irreducible decomposition of X is the decomposition of X into the union of closures of the connected components of Xreg . Each of these closures is an algebraic set in its own right. We index the i-dimensional irreducible components of this decomposition by Xij for j ∈ Ii , where Ii is a possibly empty finite set. Thus the i-dimensional algebraic subset of X is defined to be Xi := ∪j∈Ii Xij , and X is the union of these: X = ∪N i=0 Xi . The irreducible decomposition is the typical starting point in most computations involving algebraic sets. The analogue of the irreducible decomposition is the numerical irreducible decomposition discussed in Section 7.5. We say that an algebraic set X ⊂ CN is pure i-dimensional if all the irreducible components in the irreducible decomposition of X are i-dimensional, i.e., X = Xi . Note that such an X has a well-defined integer deg X


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defined as the number of points in the set X ∩ Li where Li is a generic i-codimensional linear space, in other words, Li is a general (N − i)-dimensional linear subspace of CN . Remark 7.1. We can define a complex quasi-algebraic set more generally as a solution set of a system of homogeneous polynomials on PN minus the solution set of a different such system. There is an analogous irreducible decomposition. These concepts are discussed extensively in Sommese and Wampler (2005), but for the sake of simplicity we restrict ourselves to affine algebraic sets. Before we consider positive-dimensional solution sets, let us consider nonsquare systems when we want to compute isolated solutions. To this end, fix a system f of n polynomials in N variables. The system is said to be overdetermined if n > N and underdetermined if n < N . If n ≤ N , then general theory tells us (e.g., Sommese and Wampler (2005, Corollary A.4.7) and Sommese and Wampler (2005, Corollary A.4.12)) that if V(f ) is not empty, then the complex dimension of V(f ) is at least N − n. Regarding f as defining a map from CN to Cn , then the image is Zariskidense in an algebraic set B of a dimension b such that (1) for y in the complement of an algebraic subset of B of lower dimension than B, dim f −1 (y) = N − b; and (2) for all y ∈ f (CN ), dim f −1 (y) ≥ N − b. In other words, for almost all points in the image of f , the pre-image is a set of dimension N − b, but there may be an algebraic subset of the image for which the pre-image has a higher dimension. By Sard’s theorem (Sommese and Wampler 2005, Theorem A.4.10), b equals the rank of the Jacobian of f evaluated at a general point of CN . The integer b is called the rank of the polynomial system. Thus if we are interested in isolated solutions, we need the rank of the polynomial system to be N , which requires that n ≥ N . Suppose that we were interested in isolated solutions of a rank N system with n > N equations. In a numerical approach, where equations are only evaluated approximately, this situation can present some challenges. Consider the polynomial system  2 x  y 2  = 0. (7.1) xy This system has a unique root, (0, 0). However, if we change the equations slightly, say we perturb the last one to xy − = 0, the root disappears. In this sense, the root is unstable under numerical perturbation. We observe that the multiplicity of the root (0, 0) is 3.

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One way to stabilize the root is to replace the system in equation (7.1) with the system 2 x + αxy = 0, (7.2) y 2 + βxy where α and β are random, preferably of magnitude near 1. This new system has a unique root (0, 0) of multiplicity 4. Moreover if the equations in the system equation (7.2) are changed slightly, the isolated solutions of the new system will lie in a small neighbourhood of (0, 0), and their multiplicities will add up to 4. Although this manoeuvre does increase the root’s multiplicity, it has the substantial benefit that it stabilizes the root. Let us consider another simple example: (x − 1)y = 0. (7.3) (x − 1) The solution set consists of the vertical line x = 1. This time, if we perturb one of the equations slightly, e.g., if we replace the second equation with x − 1.00001, the solution set changes from a one-dimensional set to just the point (x, y) = (1.00001, 0). As in the previous example, the solution set is numerically unstable. One could say that the root cause is the same, because in a sense a system of two equations that defines a one-dimensional solution set in C2 is overdetermined in the same way a system of three equations is overdetermined for defining a zero-dimensional set (a point). To stabilize the solution, we may take a similar action as before: use randomization. To do so, we form the single equation (x − 1)y + β(x − 1) = 0,

(7.4)

with β a random number near 1 in absolute value. The solution set of this new equation consists not just of x = 1, but also of an extra component y = −β. Changing the system in equation (7.3) slightly gives a perturbed version of equation (7.4) whose solution set is a curve lying near the union of these two lines. So the randomization has stabilized the solution of the original system, but it has introduced an extraneous component. In either of the two preceding examples, it is tempting to find the solution set by solving the system with one equation omitted, and then checking if these preliminary solutions satisfy the extra equation. This works in these examples, but not for the system   (x + y − 1)y (x + y − 1)x = 0. (7.5) xy This system has three isolated roots, {(0, 0), (1, 0), (0, 1)}, but any system formed by two out of three of these equations has just one of these as an


isolated root. In contrast, the randomized system (x + y − 1)y + αxy = 0, (x + y − 1)x + βxy

533

(7.6)

with random α and β, has four isolated roots. These consist of the three roots of the original system and one extraneous root that depends on α and β. Randomization is our approach to stabilizing solutions in any overdetermined situation, and it is key to dealing with positive-dimensional sets. Accordingly, we devote the next section to understanding this procedure better. 7.1. Randomization Fix a system f of n polynomials in N variables. As we saw above, a natural move is to replace f by a new system consisting of k linear combinations of the equations making up f , e.g.,   f1  ..  Rk (M, f )(x) := M ·  . , (7.7) fn where M is a k × n matrix. For reasons that will be made clear in what follows, we suppress M and write Rk f (x) instead of Rk (M, f )(x) when M is chosen generically, i.e., when the entries of M are chosen randomly. For numerical stability, it is typically advantageous to choose M close to unitary. We refer to Rk f (x) as a randomization of f . How does V(Rk (M, f )) compare with V(f )? Without some restriction on M , all we can say is that V(f ) ⊂ V(Rk (M, f )). A remarkable consequence of Bertini’s theorem (Sommese and Wampler 2005, Chapter 13.5) is that for most M , the relation is strong and simple. Theorem 7.2. (A version of Bertini’s theorem) Fix a system f of n polynomials in N variables. Fix an integer k ≤ N . There is a non-empty Zariski open set U ⊂ Ck×n , such that for M ∈ U , every irreducible component of V(f ) of dimension equal to N − k is an irreducible component of V(Rk (M, f )); and the irreducible components of V(Rk (M, f )) of dimension greater than N − k coincide with the irreducible components of V(f ). In other words, a general randomization of f to k ≤ N equations preserves all solution components of dimension N − k or higher, but may introduce some extraneous components at dimension N − k. Bertini’s theorem implies some additional useful facts concerning general randomizations V(Rk f (x)) for k ≤ N .

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• The extraneous solutions V(Rk f (x)) \ V(f ) are smooth of dimension N − k. • The multiplicity of an irreducible component of V(f ) of dimension N −k has multiplicity with respect to f = 0 no greater than its multiplicity with respect to Rk f = 0. • If the multiplicity of an irreducible component of V(f ) of dimension N −k has multiplicity one with respect to f = 0, then it has multiplicity one with respect to Rk f = 0. Using the fact that a general k × k matrix is invertible, it follows that if k ≤ n, then we may use M of the form [Ik A], where Ik is the k × k identity matrix and A is a random k × (n − k) matrix, i.e., we may use Rk f (x) of the form   f1 . Ik A ·  .. . fn 7.2. Slicing As we saw from Section 7.1, if we want to investigate the k-dimensional components of V(f ), it is natural to randomize down to N − k equations. Moreover, there will be no k-dimensional components unless we have a system f whose rank is at least N − k. In algebraic geometry, a classical way to understand a positive-dimensional set is to intersect it with linear spaces (Beltrametti and Sommese 1995). This leads us to the approach to positive-dimensional solution sets of Sommese and Wampler (1996), where we coined the term numerical algebraic geometry as the area that corresponds to algebraic geometry in the same way that numerical linear algebra corresponds to linear algebra. The critical theory underlying the approach is a variant of Bertini’s theorem (Sommese and Wampler 2005, Chapter 13.2) that covers slicing. In the following theorem, any set of negative dimension is the empty set. Theorem 7.3. Let V(f ) be the solution set of a system of n polynomials CN . Let L be a generic (N − k)-dimensional affine linear subspace of CN . Then, given any r-dimensional irreducible component X of V(f ), it follows that: (1) dim X ∩ L = r − k and dim Sing(X ∩ L) = dim Sing(X) − k; and (2) the algebraic set X ∩ L is irreducible if r > k. Many other properties of X are inherited by X ∩ L. As mentioned in the introduction to Section 7, the numerical irreducible decomposition is the starting point for many computations with polynomial


535

systems. The numerical irreducible decomposition was first constructed in Sommese et al. (2001a). Given a k-dimensional irreducible component Xkj of V(f ), we define a witness set for Xkj to be a triple (f, Wkj , Lk ), where Lk is a set of k random linear equations and Wkj denotes the set of points on Xkj satisfying the equations Lk = 0. Note that Wkj = Xkj ∩ V(Lk ), thus there are deg Xkj points in the set Wkj . Note that Wkj are isolated solutions of the polynomial system    f1   .  A ·  ..  RN −k (f )  I =  N −k (7.8)  = 0, Lk  fn  Lk where IN −k is the (N − k) × (N − k) identity matrix and A is a random (N − k) × (n − N + k) matrix. Note that the isolated solutions Wkj do not have to be of multiplicity one. If they are, then Xkj is said to be a generically reduced component of V(f ). We can always reduce to this case (see Section 9), and so for the rest of this subsection, we will assume we are in this case. A witness set Wkj is a good proxy for Xkj . Indeed, let B be a set of k general linear equations on CN . By using continuation (for t going from 1 to 0 with Wkj as our start-points) on the system of equation (7.8) with Lk replaced with (1 − t)B + γtLk , we get the solutions of B = 0 on Xkj . This process lets us construct many widely separated points on Xkj . Interpolation of a sufficiently large set of points on Xkj constructs polynomials vanishing on it. This was used in Sommese et al. (2001a) to give a probability-one membership test to determine whether a point x∗ is on a k-dimensional irreducible component Xkj of the solution set V(f ). Crucially, in the whole process only a finite number of points S on CN arise. Unfortunately, for a degree d component, the number of points required to perform the interpolation is one less than the number of monomials of degree at most d in N variables, NN+d − 1, which grows quickly. Although some measures can be taken to reduce this number (Sommese, Verschelde and Wampler 2001b), a completely different approach is needed to avoid the combinatorial complexity inherent in any interpolation scheme. For this reason, interpolation methods have been supplanted by a much more efficient approach based on monodromy, and we visit this next. 7.3. The monodromy membership test The main membership tests used in practice are variants of the following test from Sommese, Verschelde and Wampler (2001c) (see Sommese and Wampler (2005, Chapter 15.4) for details).

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Let f be a system on n polynomials on CN . Let X be a k-dimensional irreducible component of V(f ) and let (f, W, L) be a witness set for X associated to the system equation (7.8). Problem 7.4.

Decide if x∗ ∈ V(f ) is a point of X.

To decide this, choose a set B of k linear polynomials that are general except for the constraint that they vanish on x∗ . This is easily done by choosing a random matrix M ∈ Ck×N and setting B = M · (x − x∗ ). Use continuation with the homotopy    f1  .  I  N −k A ·  ..  (7.9)  =0  fn  (1 − t)B + tL to track the points for t going from 1 to 0 starting with W and ending with a set E. Then x∗ ∈ X if and only if x∗ ∈ E. Moreover if x∗ ∈ X, the multiplicity of x∗ as a point of X equals the number of paths ending in x∗ . 7.4. Deflation revisited The monodromy membership test for a component X ∈ V(f ) and its historical predecessor based on interpolation both require tracking points on X as the system of linear slicing equations is moved in a homotopy. To do this efficiently, the points sliced out of X should be reduced so that the homotopy function has full rank. In general, however, X might be non-reduced, that is, it might be a component of multiplicity greater than one. In such cases, it is desirable to deflate X by operating on f to produce a new system for which X is a reduced component. We already saw in Section 6.3 how this can be done when X is just an isolated point. This subsection describes a procedure to handle the case when X is positive-dimensional. Let f be a system on n polynomials on CN , let X be a k-dimensional irreducible component of V(f ), and let (f, W, L) be a witness set for X associated to the system equation (7.8). Similar to the definition of multiplicity for an isolated point described in Section 6.3, the multiplicity of components may be defined using the local ring associated to the prime ideal of polynomials vanishing on X. This is the same number as the multiplicity of any one of the points x∗ ∈ W with respect to the restriction of f to V(L), that is, the multiplicity of x∗ ∈ W as an isolated point of V(f, L). Tracking such a point x∗ as the equations L change in equation (7.9) is quite difficult when the multiplicity of X is greater than one (Sommese, Verschelde and Wampler 2002a). Fortunately, the deflation procedure of Section 6.3 when done at x∗ serves to generically deflate the component X (see Sommese and Wampler (2005, Chapter 13.3.2)). To see what the


537

systems for components look like, let us consider the first deflation step as applied to equation (7.8). We need to take the Jacobian with respect to coordinates on V(Lk ), so letting B and b be matrices such that Lk = B · x − b and letting B ⊥ be the N × (N − k) matrix with column vectors an orthonormal basis of the kernel of x → B · x = 0, we have     f1      IN −k A ·  ...       fn       ξ1    .  = 0,  IN −k A · Df (x) · B ⊥ ·   ..     ξN −k            ξ1 1    .  .   . . C I · −  .  .   ξN −k 1 where N − k − is the rank of IN −k A · Df (x∗ ) · B ⊥ , Df (x∗ ) is the Jacobian matrix of f (x) evaluated at x∗ , Ik is the k × k identity matrix, and C is a generic × (N − k − ) matrix. With Lk and therefore B fixed, this system is a deflation step for a non-empty Zariski set of points x ∈ X. When this procedure yields a system that deflates x∗ as a solution of f restricted to Lk , it will also deflate a non-empty Zariski open set of X. When one step of deflation does not suffice to deflate x∗ , one may apply a second step of deflation to the above system, and so on until a full reduction to multiplicity one has been obtained. For a multiplicity µ component, at most µ − 1 deflation stages are required. A more controlled deflation is given in Hauenstein et al. (2011). No matter which deflation procedure is used, the end result is as recorded next. Theorem 7.5. Let X be an irreducible component of V(f ) ⊂ CN , where f (x) is a polynomial system on CN . Let (f, W, L) be a witness set for X. There is polynomial system F (x, u) on CN × CM and an irreducible component X of V(F ), such that (1) X is of multiplicity one; (2) under the linear projection π : CN × CM → CN , a Zariski open set of X maps isomorphically onto a Zariski open set of X containing W; and (3) (F, π −1 (W) ∩ X , L) is a witness set of X .

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7.5. Numerical irreducible decomposition Let f : CN → Cn be a polynomial system. Let dim V(f )

V(f ) := ∪i=0

dim V(f )

Xi = ∪i=0

∪j∈Ii Xij

denote the irreducible decomposition. By the numerical irreducible decomposition, we mean a set of witness sets (f, Wij , Li ), one for each Xij . Note that we have the same set of linear equations Li := Bi · x + b for each j ∈ Ii , where B ∈ Ci×N and b ∈ Ci×1 . Since the multiplicities of any two witness points for the same component are the same, and since a deflation of a component works on a set containing the witness set of an irreducible component, we can assume after renaming that the Wi are non-singular solutions of the system equation (7.8). The computation of the numerical irreducible decomposition proceeds in three main steps: • computing a witness superset; • removing ‘junk’ points to obtain a composite witness set; and • decomposing the composite witness set into irreducible pieces. First, witness supersets are computed for the Xi . By a witness superset i , Li ) such that Li is a set of i general linear for Xi we mean a triple (f, W equations and i Xi ∩ V(Li ) ⊂ W and i ⊂ ∪V(f ) Xk . W k=i This may be done by using continuation to solve the polynomial system given in equation (7.8). This was done first in Sommese and Wampler (1996) and improved in Sommese and Verschelde (2000). The most efficient algorithm for this is presented in Hauenstein, Sommese and Wampler (2010). The witness superset may contain some extra points that we designate as i = Wi + Ji , where Wi is the witness point set for Xi ‘junk’. Specifically, W i does not and Ji is contained in ∪k>i Xk . The homotopy that generates W distinguish between the desired witness points and the junk, so this requires an extra step. The first approaches to junk removal were based on membership tests (Sommese et al. 2001a, 2001b). This hinges on the fact that the junk points for Xi all lie on the components of dimension greater than i. Thus, if we start at the top dimension and work down, we can test which points in i belong to higher-dimensional sets and expunge them. This works with W


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either the interpolation membership test or the more efficient monodromy membership test, both described above. Since one must use witness sets for all the dimensions greater than i before one can attack dimension i, we say that these are global methods. This is in contrast to a local method, which can decide whether a point is junk without any knowledge about solution sets of other dimensions. Although the global methods are now obsolete, the original one in Sommese et al. (2001a) is notable for being the first route discovered for computing the numerical irreducible decomposition. Because of that, it was also the first algorithm for finding X0 , which is the exact set of isolated solutions of f (x) = 0. Without junk removal, all previous homotopies provided a superset of X0 . The global approach to junk removal has been supplanted by a more efficient method based on using local dimension tests (Bates et al. 2009a; see Section 9.2). The idea is that one does not really need to know which higher-dimensional set a junk point is on, or even know the precise dimension of the containing set; one only needs to know that a point is on some higherdimensional set to conclude that it is junk. A local dimension test provides this minimal information with much less computation than is required by global membership testing. With the junk points removed, the next step is the decomposition of Wi into Wij , the witness point sets for the irreducible components of Xi . Recall that irreducible components are the path-connected pieces of the solution set after singularities have been removed. Thus, if we can show that two points are connected by a path that avoids singularities, we know that they are in the same irreducible set. We have Wi = Xi ∩ V(Li ) for Li being a set of i random linear equations. Let us regard Li (x, y) := Bi · x + y as a family of linear equations in x ∈ CN with parameters y ∈ Ci . For a Zariski open set U of Ci , the set Xi ∩ V(Li (x, y)) consists of deg Xi nonsingular points. Any appropriately random one-real-dimensional path in Ci will lie in U with probability one. Tracking the points Wi around a closed path in U , we obtain a permutation of the set Wi . Points that are interchanged lie on the same component. Theory tells us that there are enough paths to achieve the break-up into irreducible components (Sommese, Verschelde and Wampler 2001c, 2002b). This is the basis for the monodromy approach to decomposing the witness set into irreducibles.5 Although we know that paths exist to achieve the break-up, there is no a priori way to know which paths or how many of them are needed. The way we proceed is to take random paths and form groups of points 5

Though in theory monodromy loops in y alone suffice, in practice looping in both Bi and y is more robust.

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Figure 7.1. Griffis–Duffy platform of Type I.

known to belong to the same irreducible component. We then check each group to see if it is in fact a complete witness point set for a component. The test for completeness is based on traces (Sommese, Verschelde and Wampler 2002b), discussed in Sommese and Wampler (2005, Chapter 15). If the monodromy loops have not yet found a complete decomposition but the number of incomplete groups is not too big, exhaustive trace testing can determine which incomplete groups need to be merged to complete the decomposition. In this way, the algorithm is guaranteed to terminate in a finite number of steps. One interesting example of the application of the numerical irreducible decomposition is a special case of the Stewart–Gough platform called the Griffis–Duffy Type I architecturally singular platform. These have base and moving platforms that are equilateral triangles, with legs connecting vertices of the base to midpoints of the moving platform and vice versa in a cyclic pattern (Husty and Karger 2000, Sommese, Verschelde and Wampler 2004a). No matter what the leg lengths are, a general case of this type of platform has a motion curve in Study coordinates of degree 28. This is illustrated in Figure 7.1, where the path of a small sphere attached to the moving plate is shown. This path has degree 40 in R3 . For a special case of this in which the two triangles are congruent and the leg lengths are equal, this curve factors into five pieces: four sextics and one quartic. The numerical irreducible decomposition is able to find all of these, providing a witness set on each (Sommese et al. 2004a).


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8. Software There are several software packages that compute isolated solutions of polynomial systems: Bertini (Bates et al. 2008), HOM4PS-2.0 (Lee, Li and Tsai 2008), Hompack90 (Watson et al. 1997) and its extensions (Su, McCarthy, Sosonkina and Watson 2006, Wise, Sommese and Watson 2000), and PHCpack (Verschelde 1999). Hompack90 has general parallel tracking facilities. HOM4PS-2.0 has the best implementation of polyhedral methods, but is not a parallel code. Only Bertini and PHCpack implement algorithms of numerical algebraic geometry for positive-dimensional solution sets. PHCpack and Bertini both allow the user to define their own homotopy and prescribed start-points, but HOM4PS-2.0 currently does not. HOM4PS-2.0 uses only double-precision arithmetic to perform computations. To varying degrees, both PHCpack and Bertini have the capability of using higher-precision arithmetic. PHCpack does not currently have the capability of adapting the precision based on the local conditioning of the homotopy path. This means that more human interaction is needed to verify that the precision is set appropriately to accurately and reliably perform the requested computations. The more advanced algorithms of numerical algebraic geometry (including the powerful equation-by-equation methods for finding isolated solutions) place strong requirements on the underlying numerical software (Bates, Hauenstein, Sommese and Wampler 2008a). For example, without secure path-tracking and adaptive precision, computing the numerical irreducible decomposition for systems that involve more than a few variables is not possible. Only Bertini gives the numerical irreducible decomposition directly. Exceptional features of Bertini include: • secure path-tracking; • adaptive multiprecision (Bates et al. 2008b, 2009b, 2011); • utilities for working with polynomial systems given as straight-line programs; • parallel high-precision linear algebra (in a forthcoming release); • the numerical irreducible decomposition (Sommese et al. 2001a, Sommese and Wampler 2005; see Section 7.5); • equation-by-equation methods such as regeneration (Hauenstein et al. 2010, 2011; see Section 10.2), which require secure path-tracking for all but the smallest systems; • local dimension testing (Bates et al. 2009a; see Section 9.2); and

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• various endgames (see Section 6.4) including the power series endgame (Morgan, Sommese and Wampler 1992b), the Cauchy endgame (Morgan et al. 1991), and a parallel endgame based on it (Bates et al. 2010a). To put some of this in perspective, consider a sequence of polynomial systems arising from discretizing the Lotka–Volterra population model that is presented in Hauenstein et al. (2011). HOM4PS-2.0 and PHCpack both implement the polyhedral homotopy method for solving sparse polynomial systems, while Bertini implements regeneration (discussed in Section 10.2) for solving such systems (Hauenstein et al. 2011). For these systems, the number of solutions over C is the number of paths tracked by the polyhedral homotopy method. Even so, the polyhedral homotopy becomes impractical as the number of variables increases due to the computational complexity of computing a start system via the mixed volume. For example, consider solving the n = 24, 32, and 40 variable instances of the Lotka–Volterra model using a single core running √ 64-bit Linux. Here the number of nonsingular isolated solutions equals 2n (which equals the mixed volume). For the 24 variable system, PHCpack took over 18 days while HOM4PS-2.0 and Bertini both took around 10 minutes. For the 32 variable system, PHCpack failed to solve the system in 45 days, HOM4PS-2.0 took over 3 days and Bertini took roughly 5 hours. For the 40 variable system, PHCpack and HOM4PS-2.0 both failed to solve the system in 45 days, but Bertini solved the system in under a week. Since regeneration is parallelizable, we also solved the 32 and 40 variable systems using 64 cores (8 nodes each having dual 2.33 GH Xeon 5410 quad-core processors). In parallel, Bertini took less than 8 minutes and 4 hours to solve the 32 and 40 variable polynomial systems, respectively.

PART THREE Advanced topics 9. Non-reduced components At several points, we have noted that a component of the solution set V(f ) can be non-reduced, meaning that it has a multiplicity greater than one. How do we compute the multiplicity of a k-dimensional irreducible component of V(f ) ⊂ CN , where f is a polynomial system (see equation (5.1)) of n polynomials? As mentioned earlier, by slicing with a general (N − k)dimensional affine linear space, this comes down to computing the multiplicity of an isolated solution of a polynomial system. We will see in Section 9.1 that the Macaulay matrix gives a powerful method for solving this problem. Modifying this method gives us an algorithm in Section 9.2 for computing the local dimension of an algebraic set.


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9.1. Macaulay matrix The Macaulay matrix gives a powerful method to compute multiplicities and lay bare the local structure of an algebraic set. This approach to the numerical computation of the multiplicity was presented by Dayton and Zeng (2005) (see also Dayton, Li and Zeng (2011)). An alternative approach, which played a role in the development of local dimension test (Bates et al. 2009a) of Section 9.2, was given by Bates, Peterson and Sommese (2006). We need multi-index notation. Given an s-tuple of non-negative integers α = (α1 , . . . , αs ) and an s-tuple of complex numbers u = (u1 , . . . , us ), let uα := uα1 1 · · · uαs s , α! := α1 ! · · · αs !, ∂ α1 ∂ αs 1 α ··· , D := α! ∂x1 ∂xs |α| := α1 + · · · + αs . Let x∗ denote a solution of f (x) = 0, a system of n polynomials in the variables x1 , . . . , xN . From this we can construct the kth Macaulay matrix: (9.1) Mk (f, x∗ ) := Dα (x − x∗ )β fj (x) (x∗ ) , where the rows are indexed by (β, j) with |β| ≤ k −1, j ≤ n and the columns are indexed by α with |α| ≤ k. Letting Pd,N denote the dimension of the vector space of polynomials of degree at most d in N variables, Mk (f, x∗ ) is an (nPd,N ) × Pd,N matrix. The rows are Taylor series expansions of generators of approximations to the ideal generated by f in the ring of convergent power series at x∗ . Theory (see Dayton and Zeng (2005), Bates et al. (2009a), Dayton et al. (2011)) implies that the dimensions µk (f, x∗ ) of the null space ξ ∈ CPd,N | Mk (f, x∗ ) · ξ = 0 are either strictly increasing with no limit (in which case V(f ) is positivedimensional at x∗ ) or strictly increasing until the first non-negative integer k with µk (f, x∗ ) = µk+1 (f, x∗ ) (in which case x∗ is an isolated solution of f (x) = 0 of multiplicity µk (f, x∗ )). In particular, assume that we know a positive number µ ˆ, such that if x∗ is an isolated solution of f (x) = 0, then µ ˆ is a bound for the multiplicity of x∗ as an isolated solution of f (x) = 0. In this case, we can check whether x∗ is isolated or not, because the µk (f, x∗ ) must either stabilize or eventually surpass µ ˆ. Since µk (f, x∗ ) is strictly increasing, if it does not stabilize, it must pass µ ˆ no later than by k = µ ˆ + 1. This last observation leads to the local dimension test (Bates et al. 2009a) explained in the next subsection. The Macaulay matrices Mk (f, x∗ ) grow quickly in size, but to compute the µk (f, x∗ ), Mk (f, x) may be reduced to a much smaller matrix (Zeng 2009).

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9.2. Local dimension In computing the numerical irreducible decomposition, we noted that compared with the use of membership tests, a local dimension test can be used to improve the speed of removing junk points. Even beyond this application, the topic of determining local dimension is of interest in its own right. The problem to be addressed is as follows. Problem 9.1. Given a solution x∗ of a polynomial system f (x) = 0, determine the local dimension at x∗ of the solution set V(f ). By local dimension at x∗ , we simply mean the maximum of the dimensions of the irreducible components of V(f ) that pass through x∗ . In junk removal, we do not even need the full determination of the local dimension; it is i has a local dimension enough to determine whether or not a point in W higher than i. A global way to find the local dimension is to carry out a full irreducible decomposition of V(f ) and use membership tests to determine which component or components contain x∗ . This is a global method requiring a very expensive computation with much of the output of the computation unused. The test described in the previous section which uses the Macaulay matrix to determine whether or not a point is isolated was put forward in Bates et al. (2009a) as an efficient method for junk removal in the numerical irreducible decomposition of V(f ). The test requires a positive integer µ ˆ which would be a bound for the multiplicity of the point in question. In the irreducible decomposition, suppose we have a point, say x∗ , in the witness i for dimension i. If x∗ is an isolated point of V(f (x), Li (x)), superset W where Li (x) = 0 is the set of i linear equations that slice out the witness set, then it belongs in the witness set. Otherwise, if it is not isolated, it is a junk point to be expunged. In witness superset generation, the bound µ ˆ is just the number of incoming paths to x∗ in the homotopy that computes the witness superset. With this bound, the Macaulay matrix test for isolation can eliminate junk points. Since we no longer need all the higherdimensional sets (dimensions j > i) to do junk removal at dimension i, the local dimension test allows us to do the numerical irreducible decomposition for each dimension independently. While junk removal only needs one test for isolation, a procedure described in Bates et al. (2009a) uses a sequence of such tests to exactly determine local dimension. The procedure uses the following fact. Let (f (x), L(x)) = 0 be a new system formed by appending to f a single linear equation L(x) that is general except for the restriction that it vanishes at x∗ . If x∗ ∈ V(f ) is not isolated, then the local dimension of V(f, L) at x∗ is precisely one less than the local dimension of V(f ) at x∗ . Thus we can compute the dimension by repeatedly slicing until x∗ is isolated. For this to


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work, each time we add another linear equation to the system, we need to have a bound µ ˆ on what would be the multiplicity of x∗ if it were isolated. The article by Bates et al. (2009a) leaves open the question of how to find these bounds. A natural way to provide the multiplicity bounds needed to complete the local dimension procedure is to construct a good homotopy H(x, t) = 0 (see the end of Section 6.1) for finding isolated solutions of (f (x), Li (x)) = 0 and count the number of paths coming into x∗ . (Here Li (x) is a system of i general linear polynomials that vanish at x∗ .) If there are no paths with limit x∗ , then x∗ is not an isolated solution of (f (x), Li (x)) = 0. If paths do arrive at x∗ , their number gives the requisite bound and we may check for isolation. If x∗ is an isolated solution of this system, then the local dimension of V(f ) at x∗ is i. If it is not isolated, then the local dimension must be greater than i and we may start over, appending some greater number of linear polynomials. Eventually, all the possible dimensions (from 0 to N ) will be exhausted, so at some point we must terminate with the correct local dimension. The homotopies used here to get the multiplicity bounds are in fact generating witness supersets dimension-by-dimension, so if one is charged with finding the local dimension at several test points, one might be just as well off computing an entire witness superset for V(f ). At present, there is no known way to find the local dimension without resorting to these homotopies. Suppose that we do not have a multiplicity bound in hand and we are unwilling to run homotopies to find one. It can still be useful to investigate local dimension by testing the ranks of the Macaulay matrices. Limiting the tests to Mk (f, x), for k ≤ k ∗ , gives the k ∗ -depth-bounded local dimension (Wampler et al. 2011). For large enough k ∗ , this is the correct local dimension, but without a multiplicity bound, one cannot make a definitive conclusion. Still the depth-bounded local dimension yields much of what one needs to know in practice, where the difference between a very high multiplicity and increased dimension can become academic. For example, in the case of a mechanism, a high multiplicity isolated root in the rigid-body model may in fact exhibit substantial motion when small elastic deformations of the links, which are always present in a physical device, enter the picture.

10. Optimal solving There is a basic and ever-present problem. Problem 10.1. Assume we have a large polynomial system f (x) = 0 with few solutions. How do we find the solutions without doing the amount of work that the size of the system would suggest?

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The words large and few and the problem itself may seem imprecise, but the problem is quite real. For simplicity we consider only a square system f , and we seek to find a finite set of points in V(f ) that contains all the isolated solutions. Most problems in numerical algebraic geometry may be reduced to this problem. In the mid-1980s, a polynomial system with 8 equations and 8 variables could be quite challenging. For example, to solve the inverse kinematics problem for 6R robots (see Sections 3.1.2 and 4.5.2), Tsai and Morgan (1985) used a total degree homotopy with 256 paths. Each path took several minutes on a large mainframe, and paths that went to infinity were computationally much more expensive than finite isolated non-singular solutions. The use of multihomogeneous structure (Morgan and Sommese 1987a) allowed the construction of a homotopy tracking 96 paths, which were overall not only fewer in number, but better behaved. Polyhedral homotopies are even better behaved. The nine-point path synthesis problem for four-bars (see Section 3.2 and Wampler et al. (1992, 1997)) has only 1442 solutions, though a total degree homotopy for the Roth–Freudenstein formulation of the problem would require tracking 78 = 5 764 801 paths. This would have required much more computational power than was available around 1990, when the problem was solved using a multihomogeneous homotopy. Even that calculation pushed the limits of its day: it tracked 143 360 paths and required over 100 hours of computation on a large mainframe computer. Over and beyond numerics, finding the homotopy used in that first solution of the nine-point problem required a great deal of insight. It is interesting to consider whether there is a solution method that could proceed efficiently to solve the original Roth–Freudenstein formulation of the system. We will return to this question shortly. As computers advance, the meanings of ‘large’ and ‘few’ change, but the problem remains. For example, polyhedral methods are remarkable, but when the number of variables grows, the computation to construct the polyhedral homotopy eventually precludes solving the system (see the examples in Hauenstein et al. (2011)). This is because the construction involves a combinatorial calculation, called the mixed volume, that grows quickly with the number of variables and the number of monomials in the problem. Another aspect of this problem is that many polynomial systems are given in an efficient straight-line form, meaning that each polynomial is evaluated using a sequence of expressions, with the later expressions being functions of the earlier expressions. Evaluation for such systems may be much quicker and more stable than if the same polynomials were expanded out into a sum of terms. Consider, for example, the four-bar coupler curve equation as written in equation (3.20). The efficient way to evaluate this is to first evaluate each of the column matrices of equation (3.21), then the 2 × 2


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determinants in equation (3.20), and then finally evaluate the polynomial using these. Polyhedral homotopies work with the expanded monomials of the problem, so may destroy the straight-line structure. Computational work is roughly proportional to the number of paths tracked. Optimally, one hopes for a method of solution of the problem whose computational cost is a small multiple of the computational cost of tracking, for a system of similar characteristics, a number of paths equal to the number of solutions. We shall see in Section 10.1 that this hope can be attained for a parametrized family of problems f (x; q) = 0, if only one can first solve any one general example in the family. In Section 10.2, we will discuss a class of methods which promise to achieve near-optimal computational cost, even for the first example problem. 10.1. Coefficient parameter homotopies Fix a parametrized family of polynomial systems   f1 (x, q)   .. f (x, q) :=  , .

(10.1)

fN (x, q) where (x, q) ∈ CN × CM with M ≥ 1. The basic result of Morgan and Sommese (1989) (see also Li, Sauer and Yorke (1989) and Sommese and Wampler (2005, Chapter 7)) is that, though it might be computationally expensive to solve f (x, q) = 0 for a generic q ∗ , the cost of solving f (x, q) = 0 for subsequent q is the optimal cost. Let us make this precise. Theorem 10.2. (Morgan and Sommese 1989) Let f (x, q) = 0 be a system of polynomials   f1 (x, q)   .. f (x, q) :=  , . fN (x, q) where (x, q) ∈ CN × CM . There is a non-empty Zariski open set U ⊂ CM such that for every positive integer µ, the set Zµ := (x, q) ∈ CN × U | x is a multiplicity µ isolated solution of f (x, q) = 0 is either empty or a manifold with the map πZµ : Zµ → U , under the restriction of the projection π : CN × CM → CM , a finite covering. Given any q ∗ ∈ CM \ U and any isolated solution x∗ of f (x, q ∗ ) = 0, for any open ball B around x∗ , there is an open ball D around q ∗ , such that, for every q ∈ D ∩ U , f (x, q) = 0 has solutions in B × {q} and they are all isolated.

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Set Z := ∪µ≥1 Zµ . Fix a generic q ∗ ∈ CM . Then U may be chosen with the further property that the deflation of f (x, q ∗ ) at an isolated solution of f (x, q ∗ ) = 0 deflates the whole component of Z that (x, q ∗ ) belongs to. Thus, at the expense of computing the isolated solutions of f (x, q ∗ ) = 0, and carrying out a set of deflations one time, we can find the isolated solutions of f (x, q) = 0 for any other q ∈ CM at the cost of tracking a number of paths equal to the sheet number of πZ : Z → U . This means that if we solve enough examples of the problem family, the amortized cost of solving f (x, q) = 0 approaches the cost of tracking a number of solutions equal to the number of isolated solutions of f (x, q) for a generic q ∗ . Finally we note that all the results above are true with CM replaced by any irreducible and reduced algebraic set which is locally irreducible (Sommese and Wampler 2005). The coefficient-parameter homotopy is especially simple when one only wishes to find Z1 , the multiplicity-one isolated solutions. Suppose we have already solved by some means the system f (x, q ∗ ) = 0, with q ∗ general. Call the set of multiplicity-one isolated solutions Z1∗ . Now suppose that we wish to solve for Z1∗∗ for the problem f (x, q ∗∗ ) = 0. If the parameter space is Euclidean, then the path γτ , τ ∈ (0, 1], φ(t) := tq ∗∗ + (1 − t)q ∗ , t = 1 + (γ − 1)τ leads from q ∗ to q ∗∗ as τ goes from 1 to 0, and it stays general with probability one for random choices of γ ∈ C (Sommese and Wampler 2005, Chapter 7). Accordingly, the coefficient-parameter homotopy is H(x, τ ) = f (x, φ(t(τ ))).

(10.2)

If the parameter space is not Euclidean, one must arrange a path that stays in the parameter space. This is easily done in most cases. For example, if the parameter space is a unit circle, we just replace φ(t) by φ(t)/|φ(t)|. As we saw in Part 1, problems from kinematics automatically come with parameter spaces so coefficient-parameter homotopy becomes a powerful tool in the area. As a first example, consider the forward kinematics problem for general Stewart–Gough (6-SPS) platforms, given by equations (3.29) and (3.31). These are 7 equations in [e, g] ∈ P7 , all quadratic. The parameter space for the problem may be taken as the entries in the 4×4 matrices Aj , Bj , j = 1, . . . , 6. One can solve a general member of this family using a total degree homotopy having 27 = 128 paths. If one takes general matrices Aj and Bj , the problem has 84 finite solutions. But recall that in the kinematics problem each Bj is an antisymmetric matrix. On this parameter space, the problem has just 40 solutions. One can solve any other example in the family


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with a coefficient-parameter homotopy that has just 40 paths. Moreover, there are several different subfamilies of interest wherein some of the S joints coincide. One of these is the octahedral family illustrated in Figure 3.9, for which the problem has only 16 roots, appearing in a two-way symmetry. (Reflection of the mechanism through the plane of the base does not alter the geometry.) Since a coefficient-parameter homotopy respects this symmetry, only eight paths need to be tracked. As discussed in Sommese and Wampler (2005, Section 7.7), after solving one general member of any Stewart–Gough subfamily, the remaining ones can be solved with an optimal number of paths by coefficient-parameter homotopy. Although these problems are all simple enough that a elimination approach can be devised – and this has been done for most cases – each special case requires a new derivation. In contrast, homotopy methods cover all the cases seamlessly. A more extreme illustration of the power of the coefficient-parameter homotopy technique is provided by the nine-point path synthesis problem for four-bars. As we mentioned earlier, the best multihomogeneous formulation found for the problem has 143 360 paths. Of these, only 4326 paths have finite end-points. So after a one-time execution of that homotopy for a general example, all subsequent examples can be solved with a coefficientparameter homotopy having only 4326 paths. But the story gets even better, because the 4326 solutions appear in a three-way symmetry called Roberts’ cognates (Roberts 1875). The coefficient-parameter homotopy respects this symmetry, so only one path in each symmetry group needs to be tracked, resulting in a homotopy with only 1442 paths. This is nearly a 100-fold decrease in the number of paths compared to the original multihomogeneous homotopy. 10.2. Equation-by-equation methods The idea behind equation-by-equation methods is to process the system adding one equation at a time, that is, intersecting the solution of the new equation with the solution set for all the preceding equations. The hope is that the computations at any stage will remain comparable to what one might expect from an optimal homotopy having one path to each of the final solutions. Specifically, one hopes that the number of paths in each stage of the equation-by-equation approach is at most a small multiple of the number of solutions at the final stage. If that number is small compared to the total degree, or – even better – small compared to the mixed volume (the number of paths in a polyhedral homotopy), then the multistage equation-byequation approach will pay off. Although a coefficient-parameter homotopy is efficient for all but the first example in a family, the equation-by-equation approach tries to improve the efficiency of solving the very first example. Also, one might hope that this can be done for a problem in its most basic

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presentation, without requiring extensive work up front to simplify it. In fact, the equation-by-equation approaches often go a long way towards these goals. For example, the regeneration method below solves the original unaltered Roth–Freudenstein formulation of the nine-point problem (Hauenstein et al. 2011, Section 9.3) with computational effort similar to the extensively reworked multihomogeneous version from Wampler et al. (1992). We discuss the two equation-by-equation methods below. The diagonal method (Sommese, Verschelde and Wampler 2004b, 2005, 2008) is important in its own right because it applies to the problem of intersecting general algebraic sets. For the problems that both methods apply to, the regeneration method (Hauenstein et al. 2010, 2011) typically runs several times more quickly than the diagonal method. For simplicity we present the special case of both algorithms that apply to find the non-singular isolated solutions of a system f of N equations f1 (x), . . . , fN (x) in N variables. Since V(f1 , . . . , fN ) = V(f1 ) ∩ · · · ∩ V(fN ), an isolated non-singular solution x∗ ∈ V(f ) must be a smooth point of all the V(fi ) and they must be transversal at x∗ , i.e., have linearly independent normals at x∗ . Thus the component of V(f1 , . . . , fk ) must be smooth of codimension k at x∗ . Fix N linear functions L1 (x), . . . , LN (x), each of the form Li (x) = ai,1 x1 + · · · + ai,N xN − 1

(10.3)

with the coefficients ai,j random. We will use the notation Li:j (x) to mean the subset of these linears from index i to j:   Li (x)   Li:j (x) :=  ... . Lj (x) It is to be understood that if i > j, then Li:j (x) is empty. For each k ≥ 1, we proceed from V(f1 , . . . , fk ) to V(f1 , . . . , fk )∩V(fk+1 ) = V(f1 , . . . , fk+1 ). Let Fk (x) := {f1 (x), . . . , fk (x)}, and let Wk denote the non-singular isolated solutions of Fk (x) . Lk+1:N (x) The number of paths in stage k of the equation-by-equation approaches is proportional to the number of points in Wk . Typically, the sizes of these sets grow as one proceeds towards WN , so that the total computational work is governed by the final number of solutions. This is not a hard-and-fast rule; it is possible for the sizes of the intermediate sets to grow large and fall again, but this is not generally seen in practice.


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Diagonal homotopy. Let Sk+1 denote the non-singular isolated solutions of Gk+1 (x) = 0, where   L1:k (x) Gk+1 (x) :=  fk+1 (x) . Lk+2:N (x) Note that the solution of Gk+1 (x) = 0 is V(fk+1 ) ∩ L, where L denotes the line L := V(Li:k−1 (x), Lk+1:N (x)). After solving for this line using standard linear algebra, one can restrict fk+1 to it, turning the solution of Gk+1 = 0 into a polynomial in one variable. In this way, one finds Sk+1 easily, and it has at most deg fk+1 points. Given Wk and Sk+1 , the goal is to construct the non-singular isolated solutions Wk+1 of {Fk+1 (x), Lk+2:N (x)} = 0, and by progressing in this incremental way arrive at WN , the non-singular isolated solutions of FN (x) = f (x) = 0. Consider the homotopy   Fk (x)      fk+1 (y)    L1 (y) x1 − y1        .. ..       . . Hk (x, t) := t  (10.4)  + (1 − t)     Lk (y)   xk − yk     Lk+1 (x) xk+1 − yk+1      Lk+2:N (x) Lk+2:N (y) on CN × CN . Note that (a, b) ∈ Wk × Sk+1 are non-singular solutions of Hk (x, y, 1) = 0. The isolated non-singular solutions of Hk (x, y, 0) = 0 must satisfy x = y and may be identified with the isolated non-singular solutions of Fk+1 (x) = 0 using the map (x, y) → (x). In this manner, we have generated Wk+1 , a result proved in Sommese, Verschelde and Wampler (2008). All that remains to be done is to recur this procedure until we arrive at the final answer WN . Notice that in equation (10.4), the trailing equations Lk+2:N (x) = 0 and Lk+2:N (y) = 0 do not change during the continuation. Therefore, one can use linear methods to compute a basis for the null space of these at the beginning (they both have the same null space, of course), and use this to substitute for x and y in the other equations. In this way, the homotopy is effectively reformulated to work on Ck+1 ×Ck+1 . When k is small compared to N , this manoeuvre reduces the computation significantly. Regeneration. Regeneration is an equation-by-equation method, whose ancestry includes the m-homogeneous homotopies of Morgan and Sommese (1987a), the set structure homotopies of Verschelde and Cools (1993), and the product homotopies of Morgan, Sommese and Wampler (1995).

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The set-up for regeneration uses functions Fk (x), Li (x) and the set Wk in the same way as the diagonal homotopy, but we increment from Wk to Wk+1 in a new way. To do this, we introduce additional random linear functions of the same form as equation (10.3), but with deg fk of them associated to the kth polynomial. Let us denote these linear functions as ˆ i,j (x), i = 1, . . . , N , j = 1, . . . , deg fi . (It is acceptable to take L ˆ i,1 ≡ Li , L but the others must be distinct random linear functions.) Regeneration proceeds in two stages. First, for j = 1, . . . , deg fk+1 , we move from Wk to Wk,j , being respectively the non-singular isolated solutions ˆ k+1,j (x), Lk+2:N (x)} = 0. This is of {Fk (x), Lk+1:N (x)} = 0 and {Fk , L accomplished with the homotopies   Fk (x) ˆ k+1,j (x) + tLk+1 (x) = 0, j = 1, . . . , deg fk+1 . Hk,j (x, t) = (1 − t)L Lk+2:N (x) (10.5) After completing these, we may gather all the solutions into one set, deg fk+1

Sk = ∪j=1

Wk,j .

The second stage of the regeneration procedure uses Sk as the start-points for the homotopy   Fk (x) ˆ k+1,1 (x) · · · L ˆ k+1,deg f (x). (10.6) Rk (x, t) = (1 − t)fk+1 (x) + tL k+1 Lk+2:N (x) The non-singular end-points of this homotopy as t goes from 1 to 0 are the set Wk+1 . As in the diagonal homotopy method, we recursively apply the two stages of regeneration until we arrive at the final solution set WN . The main result of Hauenstein et al. (2011) justifies this procedure for finding all the non-singular isolated solutions of f (x) = 0. In both stages, the trailing equations, Lk+2:N (x) = 0, do not change during the continuation, so, as before, one can work on the associated null space and thereby reduce the number of variables to k + 1. This is half of the number of variables used in the diagonal homotopy. To appreciate the power of regeneration, note that solving the Roth– Freudenstein system for the nine-point problem using regeneration is comparable in timing to solving it using a four-homogeneous formulation and faster on one processor than the best polyhedral code (Hauenstein et al. 2011). This happens even though the total degree of the Roth–Freudenstein system is about twenty times the number of paths in the four-homogeneous formulation and sixty-five times the number of paths in the polyhedral homotopy. (When run using a parallel computer with 65 processors to track the homotopy paths, the regeneration method was proportionately faster.)


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More tests comparing regeneration to polyhedral methods are reported in Tari, Su and Li (2010), along with a complementary method that helps sort the solution sets more automatically.

PART FOUR Frontiers 11. Real sets Throughout this article we have dealt almost exclusively with complex solutions, even though isolated and positive-dimensional real solutions are the main interest for most applications. For isolated solutions, we may simply pick the isolated real solutions out of the isolated complex solutions. Typically most complex solutions are not real, but we do not know any algorithms that find all the isolated real solutions without finding all the isolated complex solutions. Indeed, the majority of systems arising in applications depend on parameters, and the continuation of a real isolated solution as one goes from one point in the parameter space to another, e.g., using the methodology in the coefficient parameter theorem 10.1, very often does not end up at a real solution. Similarly, the path that leads to a real solution might begin at a complex one. Consequently, continuation of just the real solutions over a real parameter space is problematic and generally will not produce a complete solution list, so we work over the complex numbers instead. For positive-dimensional sets, there is no direct method of computing the real solution set when given the complex solution set. There is a classical description, using symbolic computation, called the cylindrical decomposition (Basu, Pollack and Roy 2006), which may be approached numerically. Let VR (f ) ⊂ RN denote the set of real solutions of the system f (x) = 0. Throughout this article, we have used V(f ) to mean the complex solution set, though for emphasis here, we may use VC (f ). Real and complex solution sets are not as closely related as one might expect, and this shows up when considering the dimension of VR (f ) versus VC (f ). For example, consider the equation x2 + y 2 = 0. Over the complex numbers the solution set consists of two lines meeting at the origin, i.e., a set of complex dimension one. The real solution set is the origin, an isolated singular point, i.e., a set of real dimension 0. This difference of dimension only happens at real solutions contained in the singular set of the complex solution set. One-dimensional sets are relatively straightforward (Lu, Bates, Sommese and Wampler 2007). Let f (x) = 0 be a polynomial system with real coefficients. Assume that the solution set is one complex dimension, i.e.,

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On a double line

On a quadric

On the quartic

Figure 11.1. Selected poses of the foldable Stewart–Gough platform.

dim V(f ) = 1. We know that dimR VR (f ) ≤ 1. We assume for simplicity that all the positive-dimensional components are of multiplicity one (by using deflation one can always make this so). Choose a general linear projection π : CN → C. The projection π is of the form   x1  ..  A ·  .  + b, xN where A is a 1 × N generic complex matrix and b is a random complex constant. If we restrict A and b to be real, the set VR (f ) will map to the real line. We can now compute the set of points S where πV(f ) is not of rank 1. Away from these points πV(f ) is a covering map. We know that VR (f ) is a union of isolated solutions (which must lie in S) plus intervals meeting and ending at points of S. This information is enough to give a decomposition of VR (f ) that may be used in applications. The case when dim VR (f ) = 2 is more difficult. It is natural to choose a random linear projection to C2 which has real coefficients so that VR (f ) maps to R2 . A decomposition of VR (f ) into isolated points, intervals, and cells may be done if all components are multiplicity one. When some components have higher multiplicity, one must introduce deflation, but this brings in some technical issues that are not easily dispatched. Thus the treatment of the two-dimensional case is at present only partially complete. In Section 7.5, we illustrated the Griffis–Duffy Type I platform robot, a special case of the Stewart–Gough (6-SPS) platform, and mentioned that the motion for the Griffis–Duffy Type II subcase factors into five pieces: four sextics and one quartic. Lu et al. (2007) considered an even more special example of a Griffis–Duffy Type II robot, one whose leg lengths are all


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equal to the altitude of the base and moving triangles (which are congruent equilateral triangles). This robot is unusual in that it can fold up into the plane with both triangles coinciding. Its motion is a curve that factors even more finely than general Type II cases into three double lines, three quadrics, and four quartics. (The sum of the degrees 3·2+3·2+4·4 = 28 is the same as the degree of the irreducible curve in the Type I case.) Numerical irreducible decomposition finds this factorization, and the technique sketched above extracts the real curves inside these complex factors. It turns out that three of the quartics have no real points, but each of the others gives a real curve. One pose on each type of curve is shown in Figure 11.1. Notice that the double lines are non-reduced, so deflation is needed to sample points on them.

12. Exceptional sets Many problems may be rephrased as a problem of finding the set of parameters where some exceptional behaviour occurs. An important example that motivated much of our work is finding overconstrained mechanisms, i.e., mechanisms of a given family that have more degrees of freedom than most of the other mechanisms in the family. Fibre products give a powerful handle on the computation of exceptional sets. Before we say more, let us give a definition and an example. Let p : X → Y and q : W → Y be two algebraic maps between algebraic sets. Then the fibre product X ×Y W is the set (x, w) ∈ X × W | p(x) = q(w) . Note that X ×Y W is an algebraic set and has natural maps to X, W , Y . If Y is a point, the fibre product is the usual product. On the level of systems of polynomials, let f (x, y) = 0 and g(w, y) = 0 be two systems of polynomials on CN1 × CM and CN2 × CM respectively. Letting X = V(f (x, y)) and W = V(g(w, y)), we have the maps p : X → Y and q : W → Y induced by the natural projections to CM . The fibre product X ×Y W equals V(f (x, y), g(w, y)). Letting X = V(f (x, y)), the fibre product X ×Y X of X with itself over Y is the solution set of (f (x1 , y), f (x2 , y)) = 0. The striking property of the fibre product is that exceptional sets correspond to irreducible components of fibre products. As an example, consider the algebraic set X = V(x − u, xy − v), which we would like to consider as the solution set of a family of systems having (x, y) as variables and (u, v) as parameters. Note that the map (x, y, u, v) → (u, v) restricted to X has a zero-dimensional fibre over general (u, v), given by (x, y, u, v) = (λ, µ/λ, λ, µ), but it has a one-dimensional fibre X1 over (0, 0) given by (x, y, u, v) = (0, λ, 0, 0), where λ, µ ∈ C. (Also, the fibre over (0, λ)

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for λ = 0 is empty, but this is not our main concern at the moment.) Since X1 has larger dimension than the fibre over general points, it is an exceptional set. Since the whole set X can be parametrized as (λ, µ, λ, λµ), it is isomorphic to C2 . It is therefore one irreducible set that contains X1 , and so X1 does not stand out as a component on its own in an irreducible decomposition of X. However, the fibre product of X with itself is given by V(x1 −u, x1 y1 −v, x2 −u, x2 y2 −v), and it can be shown that the solution set of this system has two components. One is the two-dimensional set of the form (x1 , y1 , x2 , y2 , u, v) = (λ, µ, λ, µ, λ, λµ) that maps onto C2 , and the other is the two-dimensional set of the form (x1 , y1 , x2 , y2 , u, v) = (0, λ, 0, µ, 0, 0) that maps onto (0, 0). This latter component is the fibre product of X1 with itself, and it now stands out as an irreducible component. In the example above, one fibre product was sufficient to promote the exceptional set to irreducibility. In general, it may take several fibre products, e.g., X ×Y X ×Y X, and so on, to make the exceptional set stand out. The main results of Sommese and Wampler (2008) show that taking higher and higher fibre products will eventually promote any exceptional set to irreducibility. It makes precise how exceptional sets become irreducible components and gives some bounds on how many successive fibre products are needed for different exceptional sets. The polynomial systems that arise from fibre products grow large quickly, but they have much internal structure with many components that are irrelevant for finding exceptional sets. There is promise that this approach will yield effective numerical algorithms to compute overconstrained mechanisms. We have already touched on several overconstrained mechanisms in this paper. First, and simplest, is the planar four-bar linkage of Figure 3.4. This is overconstrained in the sense that a 4R spatial closed-loop mechanism (four links connected by rotational joints 1 to 2, 2 to 3, 3 to 4, and 4 to 1) cannot move. For 4 × 4 link transforms Aj ∈ SE (3), j = 1, 2, 3, 4, the condition for closing the 4R loop is Rz (Θ1 )A1 Rz (Θ2 )A2 Rz (Θ3 )A3 Rz (Θ4 )A4 = I4 ,

(12.1)

where Rz (Θ) is as in equation (3.12) and I4 is the 4 × 4 identity matrix. When the Aj are general, this system is equivalent to six independent equations (the dimension of SE (3)) in only four unknowns, so there are no solutions. However, when the Aj are constrained to SE (2), the mechanism becomes planar and we have just three independent equations (the dimension of SE (3)) in four unknowns, which allows in general a one-dimensional motion (1DOF). Similarly, with Aj restricted to SO(3), we obtain the spherical four-bar linkages, which also have 1DOF. These cases are easy to see by merely considering the subgroups SE (2) and SO(3) in SE (3), but these do not tell the whole story. Bennett (1903) made the surprising discovery that there is another family of moveable four-bar linkages, now called


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Figure 12.1. Double-parallelogram linkage.

Bennett four-bars, that have special relations between the four link transforms Aj . Delassus (1922) proved that there are no other families of moveable four-bars. Apart from four-bar linkages, the theory of overconstrained mechanisms is not so complete. The Griffis–Duffy Type 1 family of 6-SPS mechanisms illustrated in Figure 7.1 is an example of what is known as an architecturally singular Stewart–Gough mechanism. Such mechanisms have the property that if one locks the six leg lengths at the values corresponding to any pose of the upper platform, the resulting 6-SS mechanism still moves, whereas a general 6-SS platform can only be assembled in 40 isolated configurations. Karger (2003, 2008) has classified all the architecturally singular Stewart–Gough platforms. However, there also exist 6-SS platforms (Geiss and Schreyer 2009) that move but are not covered by Karger’s analysis, because they are not architecturally singular: the movement of the associated 6-SPS platform only occurs when locking the leg lengths at special poses of the upper platform. While Geiss and Schreyer found such an example by generalizing to the reals from a search conducted over finite fields of small characteristic, there is not yet any general theory for classifying all such mechanisms. For a simpler example, consider the planar analogue of the spatial 6SS mechanism, which is the planar 3-RR mechanism, also known as the planar pentad, of Figure 3.6. It can be shown that the only non-trivial moving pentads (that is, ignoring cases with some link length equal to zero) are the double-parallelogram linkages, as illustrated in Figure 12.1. These have upper and ground links that are congruent triangles, and legs that are all equal length. The upper triangle moves in a circular fashion without rotating. In the figure, two poses of the linkage are shown in gray and a final one in black. The question of the existence of a moving pentad is equivalent

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to asking if a four-bar coupler curve can have a circle as one factor. (See Figure 3.4(b).) The relationship between moving pentads and four-bars that trace a circle illustrates the fact that a search for a mechanism with special motion characteristics can often be cast as a search for a related overconstrained mechanism. In this way, we see that methods for finding overconstrained linkages actually have a wider applicability to finding non-overconstrained mechanisms that have special motions. One classical question of this type, discussed briefly in Section 5, was the search in the late 1800s to find mechanisms with only rotational joints that draw an exact straight line. The modern version of this is the search for 3DOF spatial parallel-link robots whose output link translates without rotating (Carricato and ParentiCastelli 2003, Di Gregorio and Parenti-Castelli 1998, Gogu 2004, Huang et al. 2004, Kong and Gosselin 2002, Kong and Gosselin 2004, Li and Xu 2006, Tsai, Walsh and Stamper 1996). To date, investigations of overconstrained mechanisms tend to employ specialized arguments for the specific mechanism family under consideration. The fibre product approach to finding exceptional sets has the potential to provide a general approach applicable to many mechanism families.

PART FIVE Conclusions From its roots in methods to find isolated roots of polynomial systems by continuation, numerical algebraic geometry has matured into a set of tools for finding and manipulating solution sets of any dimension. The key step is to use generic linear slices to isolate a finite number of points on a positivedimensional set, thereby forming a finite witness set that becomes a proxy for the entire set. Of central importance are methods to compute the numerical irreducible decomposition, which consists of a witness set for each irreducible component of an algebraic set. Through the use of witness sets to represent intermediate results, a large system can be solved introducing one equation at a time, an approach that often reveals simplifications as the computation progresses and thereby reduces the computations required in the most expensive final stages. While advanced formulations of homotopies reduce the number of continuation paths to be tracked, the numerical techniques required to implement them are of equal importance. Adaptive multiprecision is an important factor in obtaining high reliability, and the use of parallel computing greatly increases speed. These advances in methods for solving polynomial systems are quite useful in treating problems from algebraic kinematics. We have reviewed some of the specific problems from that field which have provided useful information

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to kinematicians while also motivating the development of better algorithms. Furthermore, we have shown how problems in kinematics can be formulated as algebraic systems, thereby introducing the concept of a mechanism space and its associated input and output maps. This provides a framework for understanding the definitions of a variety of kinematics problems, including analysis problems, such as the forward and inverse kinematics problems for robots, and synthesis problems that seek to design mechanisms that produce a desired motion. The current frontiers of work in the area include improving methods for working with real sets, which is of obvious importance in applications, and furthering initial progress that has been made towards algorithms to find exceptional sets. In algebraic kinematics, exceptional sets correspond to overconstrained mechanisms, an area of interest since the earliest days of formal kinematics but which to date has resisted efforts to develop effective general methods.

Acknowledgements We thank Jonathan Hauenstein for helpful comments.

Appendix: Study coordinates Study coordinates, also known as dual quaternions and called soma coordinates by E. Study (Bottema and Roth 1979), represent SE (3) as points on a quadratic surface, the Study quadric, in seven-dimensional projective space, P7 . Although presentations may be found in numerous places, including Study (1903), Husty, Karger, Sachs and Steinhilper (1997), McCarthy (2000), Selig (2005), Husty, Pfurner, Schr¨ ocker and Brunnthaler (2007), for completeness we give a brief description here. The representation may be developed as follows. First, we note that if e = e0 1 + e1 i + e2 j + e3 k is a quaternion and v = v1 i + v2 j + v3 k is a pure vector (a quaternion with real part equal to zero), then u = e ∗ v ∗ e /(e ∗ e )

(A.1)

is also a pure vector. Moreover, u and v have the same length u∗u = v∗v . In fact, casting u and v as three-vectors, one may see that equation (A.1) is equivalent to multiplication by a 3 × 3 rotation matrix R(e) ∈ SO(3). That is, u = R(e)v with   2 e0 + e21 − e22 − e23 2(−e0 e3 + e1 e2 ) 2(e0 e2 + e1 e3 ) 1  2(e0 e3 + e2 e1 ) e20 − e21 + e22 − e23 2(−e0 e1 + e2 e3 ) , R(e) = ∆(e) 2(−e e + e e ) 2(e e + e e ) e2 − e2 − e2 + e2 0 2

3 1

0 1

3 2

0

1

2

3

(A.2)

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where ∆(e) is the squared length of e: ∆(e) = e ∗ e = e20 + e21 + e22 + e23 .

(A.3)

Of course, we must restrict R(e) to the set of quaternions having non-zero length. In addition to a quaternion e for rotation, Study coordinates include a second quaternion g related to position. Consider the mapping from Study coordinates (e, g) to C4 × C3×3 defined by St : (e, g) → (g ∗ e /(e ∗ e ), R(e)),

e ∗ e = 0.

(A.4)

In this map, we would like g ∗ e to represent a translation vector, which is a pure vector. Hence, we require Q(e, g) = g0 e0 + g1 e1 + g2 e2 + g3 e3 = 0,

(A.5)

so that the real part of g ∗ e is zero. S62 = V(Q) is called the Study quadric. Since St, Q, and ∆ are all homogeneous, we may regard (e, g) as homogeneous coordinates on seven-dimensional projective space: [e, g] ∈ P7 . Restricting St to (S62 \ V(∆)) ⊂ P7 gives a map from a six-dimensional quasi-projective set to SE (3). The map St induces an isomorphism. That is, given (p, C) ∈ SE (3), C = R(e) has a unique inverse up to scale, and then p = g ∗ e /(e ∗ e ) may be inverted as g = p ∗ e. (Some authors use p = 2g ∗ e /(e ∗ e ) and g = (1/2)p ∗ e.) Letting cij be the i, jth element of C, the formulas for inverting C = R(e) are e = 1 + c11 + c22 + c33 : c32 − c23 : c13 − c31 : c21 − c12 = c32 − c23 : 1 + c11 − c22 − c33 : c21 + c12 : c13 + c31 = c13 − c31 : c21 + c12 : 1 − c11 + c22 − c33 : c32 + c23 = c21 − c12 : c13 + c31 : c32 + c23 : 1 − c11 − c22 + c33 .

(A.6)

For C ∈ SO(3), at least one of these four formulas is always non-zero.

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Variationally consistent discretization schemes and numerical algorithms for contact problems∗ Barbara Wohlmuth Technische Universit¨ at M¨ unchen Fakult¨ at f¨ ur Mathematik M2, Boltzmannstr. 3, 85748 Garching, Germany E-mail: [email protected] URL: www-m2.ma.tum.de We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

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Colour online available at journals.cambridge.org/anu.

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CONTENTS 1 Introduction 2 Problem setting for mechanical contact 3 Variationally consistent space discretization 4 Optimal a priori error estimates 5 Semi-smooth Newton solver in space 6 A posteriori error estimates and adaptivity 7 Energy-preserving time-integration scheme 8 Further applications from different fields References

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1. Introduction In many industrial applications or engineering problems, contact between deformable elastic bodies plays a crucial role. As examples we mention incremental forming processes, the simulation of rolling wheels, braking pads on tyres and roller bearings. Although early theoretical results go back to Hertz (1882), there are still many open problems, and the numerical simulation of dynamic contact problems remains challenging. These problems are discussed in several monographs on contact mechanics such as Fischer-Cripps (2000), Johnson (1985) and Kikuchi and Oden (1988). More recent theoretical results on existence and uniqueness can be found in Eck, Jaruˇsek and Krbec (2005) and Han and Sofonea (2000), and on mathematical models and numerical simulation techniques in Laursen (2002), Willner (2003), Wriggers (2006), Wriggers and Nackenhorst (2007) and the references therein. One of the main challenges relates to the fact that the actual contact zone is not known a priori and has to be identified by use of an iterative solver. Moreover, the transition between contact and non-contact is characterized by a change in the type of the boundary condition, and thus possibly results in a solution of reduced regularity. From the mathematical point of view, contact problems can be formulated as free boundary value problems and analysed within the abstract framework of variational inequalities (Facchinei and Pang 2003a, 2003b, Glowinski, Lions and Trémolières 1981, Harker and Pang 1990, Kinderlehrer and Stampacchia 2000). This work is an overview of theoretical and numerical results obtained in recent years. Most of the numerical examples are therefore taken from the original papers which are cited in the reference list. The numerical implementation is based on different software codes. In particular, DUMUX (Flemisch, Fritz, Helmig, Niessner and Wohlmuth 2007), DUNE (Bastian, Blatt, Dedner, Engwer, Klöfkorn, Kornhuber, Ohlberger and Sander 2008), NETGEN (Schöberl 1997), PARAVIEW (Ahrens, Geveci and Law 2005),

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PARDISO (Karypis and Kumar 1998, Schenk and Gärtner 2004, 2006), and UG (Bastian, Birken, Johannsen, Lang, Neuß, Rentz-Reichert and Wieners 1997) have been used. The structure of this paper is as follows. In Section 2, the governing equations and corresponding inequality constraints for frictional contact problems are stated. Different equivalent formulations are discussed and a weak saddle-point formulation is presented. Section 3 is devoted to space discretization. Special emphasis is placed on uniformly inf-sup stable pairings and a suitable approximation of the dual cone. Optimal a priori estimates for the discretization error in the displacement and in the surface traction are given for low-order finite elements in Section 4. Here, we restrict ourselves to very simple contact settings with given friction bounds and do not take non-matching contact zones into account, but allow for non-matching meshes. In Section 5, we survey different solver techniques for the non-linear inequality system. Of special interest are so-called semi-smooth Newton methods applied to an equivalent non-linear system of equations. We discuss in detail the structure of the systems to be solved after consistent linearization. In particular, in the case of no friction the Newton solver can easily be implemented as a standard primal–dual active set strategy, updating in each iteration step the type and the value of the boundary condition node-wise. Section 6 is devoted to different aspects of adaptive refinement. Bearing in mind that the mechanical role of the discrete Lagrange multiplier is that of a surface traction, different error indicators can easily be designed. However, the analysis is quite challenging and only a few theoretical results exist, taking into account possibly non-matching meshes and the inequality character of the formulation. Here, we provide upper and lower bounds for a simplified setting and comment on possible generalizations. Section 7 is devoted to aspects of time integration. For many applications structure-preserving time-integration schemes are of special interest. In this context, energy preservation is of crucial importance. Unfortunately most of the standard techniques result either in very high oscillations in the dual variable or in numerical dissipation. We apply a newly combined time and space integration scheme which is motivated by a reduction of the index of the differential algebraic system. Finally, in Section 8 we illustrate the flexibility of the proposed approach by considering applications from different areas. In particular, an example from finance shows that the Lagrange multiplier approach based on a (d − 1)-dimensional H 1/2 -duality pairing can also be applied to obstacletype inequalities reflecting a d-dimensional H 1 -duality pairing. Of special interest are examples where d- and (d − 1)-dimensional constraints are imposed, such as phase transition problems in heterogeneous porous media and elasto-plastic mechanical contact problems.

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2. Problem setting for mechanical contact In many applications involving several deformable bodies, frictional contact has to be considered in conjunction with inelastic material behaviour such as plasticity. A great deal of research has been done on both of these topics: see for example Boieri, Gastaldi and Kinderlehrer (1987), Eck et al. (2005), Johnson (1985), Laursen (2002), Willner (2003), Wriggers (2006) and the references therein for an overview of contact problems. Characteristically, this type of application leads to a constrained minimization problem or more generally to a variational inequality (Harker and Pang 1990, Haslinger, Hlavá˘cek, Ne˘cas and Lov´ı˘sek 1988, Kikuchi and Oden 1988, Kinderlehrer and Stampacchia 2000). Mathematical analyses of variational inequalities and constrained minimization problems can also be found in Facchinei and Pang (2003a, 2003b), Geiger and Kanzow (2002), Glowinski (1984), Glowinski, Lions and Trémolières (1981) and Haslinger, Hlavá˘cek and Ne˘cas (1996). We refer to the recent monographs by Han and Reddy (1999) and Han and Sofonea (2002) and the references therein for an overview of the mathematical theory and numerical analysis for inequality problems in continuum mechanics. Our formulation will be based on a primal–dual pair of variables. In addition to the displacement which represents the primal variable, the surface traction on the possible contact zone is introduced as dual variable: see, e.g., Christensen, Klarbring, Pang and Strömberg (1998). This new pair of variables has to be admissible, i.e., satisfy the inequality constraints arising from the non-penetration condition and the friction law. In this section, we provide the setting of a quasi-static frictional contact problem between elastic bodies. Figure 2.1 shows the stress components σxx and σxy for two different situations in the case of three elastic bodies in contact. Here, the contact of a deformable body with a rigid obstacle has been taken into account as well as the contact between deformable bodies. A fully symmetric situation is shown in Figure 2.1(a,b), and no Dirichlet boundary condition is imposed. The rigid body motions are fixed by the nonpenetration condition and a zero tangential displacement of the centre of the upper circle. In Figure 2.1(c,d), there is an additional rigid obstacle on the right of the three circles. Then all rigid body modes are automatically fixed by the contact conditions. To simplify the notation, we restrict our attention to two bodies, linear elasticity in the compressible range and a given constant Coulomb friction coefficient. However, most of our algorithmic results can easily be extended to more complex situations. We refer to the early papers by Laursen and Simo (1993a), Oden, Becker, Lin and Demkowicz (1985), Puso and Laursen (2004a, 2004b), Puso, Laursen and Solberg (2008), Yang and

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(d)

Figure 2.1. Stress component σxx and σxy for a symmetric (a,b) and a non-symmetric setting (c,d).

Laursen (2008a, 2008b) and Yang, Laursen and Meng (2005) for large deformation contact discretizations on non-matching meshes and to the recent contributions on solvers by Gitterle, Popp, Gee and Wall (2010), Popp, Gitterle, Gee and Wall (2010), Popp, Gee and Wall (2009) and Krause and Mohr (2011). Numerical examples also illustrate the performance of these approaches in more general formulations, e.g., for nearly incompressible materials and the inclusion of thermal effects with a temperature-dependent friction coefficient. The two bodies in the reference configuration are given by two open bounded domains Ωs and Ωm ⊂ Rd , d = 2, 3, with Lipschitz boundary ∂Ωs and ∂Ωm , respectively. The notation is adapted to the standard mortar framework, i.e., the upper index s stands for the slave side, and the index m refers to the master side. The contact conditions will be imposed weakly in terms of Lagrange multipliers defined on the slave side. Thus the displacement on the slave side has to follow the displacement of the master side in the event that the constraints are active. This observation motivates the terminology. The boundary ∂Ωk is partitioned into three open disjoint measurable parts ΓkD , ΓkN , and ΓkC , k ∈ {m, s}. Dirichlet conditions will be set on ΓkD and Neumann data on ΓkN . For simplicity of notation, we assume firstly that meas(ΓkD ) > 0, k ∈ {s, m}, secondly that ΓsD is compactly ems bedded in ∂Ωs \ ΓC , and thirdly that the actual contact zone Bn ⊂ ΓsC is compactly embedded in ΓsC . The first assumption on the Dirichlet boundary part means that Korn’s inequality holds on each body, and thus that we do not have to deal with extra rigid body motions. The second assumption guarantees that the trace space restricted to ΓsC does not see any boundary condition originating from ΓsD . As we will see later, the third assumption on the actual contact zone guarantees that the support of the surface traction on ΓsC is compactly embedded in ΓsC . Throughout this paper, we use the standard notation for the Sobolev space H s (ω), s ≥ 1, where ω is a suitable subdomain of Ωk , k ∈ {s, m}, and denote the associated norm with · s;ω . The broken H s -norm on Ω is given by v2s;Ω := v s 2s;Ωs +v m 2s;Ωm for v := (v s , v m ) ∈ H s (Ωs )×H s (Ωm ).

574

B. Wohlmuth

On (d − 1)-dimensional manifolds γ such as ΓsC , we use the Sobolev space H s (γ), s ≥ 0, and its dual space H −s (γ). We point out that in our notation 1/2 H −1/2 (γ) is not the dual space of H00 (γ) but of H 1/2 (γ). The dual norm is defined in the standard way by µ−s;γ :=

µ, vs;γ , v∈H s (γ) vs;γ sup

s ≥ 0,

(2.1)

where ·, ·s;γ denotes the duality pairing. We note that the second assumption on the Dirichlet boundary part allows one to work with H 1/2 (ΓsC ). 1/2 Otherwise, we would have to consider the more complex H00 (ΓsC ) space, as in the mortar framework with cross-points (Bernardi, Maday and Patera 1993, 1994). We refer to the recent monograph on the theory and implementation of mortar methods by Lacour and Ben Belgacem (2011) In what follows, we shall frequently use the generic constants 0 < c, C < ∞, which are independent of the mesh size but possibly depend on the regularity of the domain or the mesh. Vectorial quantities are written in bold, e.g., x, y, and for simplicity of notation xy stands for the scalar product between x and y. Tensorial quantities are represented by bold greek symbols. 2.1. Problem formulation in its strong form For the moment we restrict ourselves on each body to a homogeneous isotropic linearized Saint Venant–Kirchhoff material and also to the small strain assumption. Then, the strain-displacement relation is defined by ε(v) := 1/2(∇v + (∇v) ) and the constitutive equation for the stress tensor is given in terms of the fourth-order Hooke tensor C by σ(v) := λ tr((v))Id + 2µ(v) =: C(v).

(2.2)

Here tr denotes the trace operator and Id the identity in Rd×d . The positive coefficient λ and the shear modulus µ are the Lamé parameters, which are assumed to be constant in each subdomain Ωk , k ∈ {s, m}, but have possibly quite different values on the slave and master side. We note that the Lamé parameters can be easily calculated from the Poisson ratio and Young’s modulus. Then, the linearized elastic equilibrium condition for the displacement u := (um , us ) can be written as − div σ(u) = f in Ω, s u = uD on ΓD := Γm D ∪ ΓD , s σ(u)n = fN on ΓN := Γm N ∪ ΓN ,

(2.3)

where n stands for the outer unit normal vector, which is almost everywhere well-defined. Here, the volume force f , the Neumann data fN , and


575

the Dirichlet condition uD are assumed to be in (L2 (Ω))d , (L2 (ΓN ))d and 1/2 (Γm ))d , respectively. Moreover, we (C(ΓsD ) ∩ H 1/2 (ΓsD ))d × (C(Γm D) ∩ H D assume that Creg < ∞ exists such that Ωs fv dx + ΓsN fN v ds Ωs fv dx + ΓsN fN v ds sup ≤ Creg sup . v1;Ωs v1;Ωs v∈(H 1 (Ωs ))d v∈(H 1 (Ωs ))d v| s =0 Γ D

v| s =0 Γ ∪Γs D C

(2.4) These regularity assumptions on the data can be considerably weakened but for most examples these hold. In addition to (2.3), we have to satisfy the contact constraints on ΓsC : the linearized non-penetration condition in the normal direction and the friction law in the tangential direction. These constraints can be formulated by means of the displacement and the surface traction λ := −σ(us )ns . The linearized non-penetration condition reads as [un ] ≤ g,

λn ≥ 0,

λn ([un ] − g) = 0,

(2.5)

where g ∈ H 1/2 (ΓsC ) is the linearized gap function between the two deformable bodies. The linearized setting can be expressed in terms of the normal contributions with respect to the reference configuration. Here λn := λns is the normal component of the boundary stress, and [un ] := (us − um ◦ χ)ns is the jump of the mapped boundary displacements, where χ(·) denotes a suitable mapping from ΓsC onto Γm C. In addition to (2.5), we have to satisfy the quasi-static Coulomb law λt ≤ νλn ,

[u˙ t ]λt − νλn [u˙ t ] = 0,

(2.6)

where the tangential components are defined by λt := λ − λn and [ut ] := [u] − [un ]ns , [u] := us − um ◦ χ, ν ≥ 0 is the friction coefficient, and · stands for the Euclidean norm. We note that for the dynamic case, inertia terms have to be included, and the volume mass density of the two bodies has to be taken into account: see Section 7. For the moment we focus on the static Coulomb law, i.e., we replace the tangential velocity by the tangential displacement in (2.6). This problem type has then to be solved in each time step if an implicit time-integration scheme is used. Figure 2.2 illustrates the notation and the situation for finite deformations. In that case the Jacobian of the deformation mapping has to be taken into account and the non-penetration has to be formulated with respect to the actual configuration. We recall that in the reference configuration ΓsC and Γm C do not have to be matching, and thus the displacement from the master side has to be projected onto the slave side. Moreover, the contact surface tractions on the master and slave body have to be in equilibrium in the actual configuration. In the case of linear elasticity, reference and actual configuration can be ns

576

B. Wohlmuth ΓsD ΓsN

us (Xs )

Ωs Xs

ΓsC

Γm C X

Γm N

ns

m

e3

Γm D

um (Xm )

e2 Ωm e1 Figure 2.2. Illustration of the notation.

identified, and thus the approach is simplified considerably compared to the case of finite deformations. We note that the Signorini problem with Coulomb friction was introduced by Duvaut and Lions (1976); see also Fichera (1964) and Lions and Stampacchia (1967). Although it is widely studied, not all aspects are yet fully understood and some open questions remain. It is well known that frictional contact problems do not necessarily have a unique weak solution. Many of the early results are designed for the case of one elastic body in contact with a rigid foundation. First existence results can be found in Demkowicz and Oden (1982), Jaruˇsek (1983) and Ne˘cas, Jaruˇsek and Haslinger (1980). An alternative proof based on a penalization technique is given in Eck and Jaruˇsek (1998). Examples for non-uniqueness are given in Ballard (1999) and Hild (2003, 2004), and uniqueness criteria are studied, e.g., in Ballard and Basseville (2005), Hild and Renard (2006) and Renard (2006). We refer to the recent monograph by Eck et al. (2005) for an excellent overview of existence and uniqueness results for these contact problems. Roughly speaking, for ν ≥ 0 small enough, the existence and uniqueness of a weak solution is guaranteed. Figure 2.3 shows the influence of the friction coefficient ν on the normal and tangential component of the surface traction for the classical Hertz problem (Hertz 1882). In between the maximal and minimal value of the tangential contact traction, the two bodies stick together. On the rest of the actual contact zone, a relative tangential displacement occurs. As can be seen directly from (2.6) for ν = 0, we have λt = 0, which is also obtained by the numerical scheme. We observe that the size of the slippy contact zone is largely influenced by ν. The smaller ν, the larger is the slippy contact zone, whereas the contact radius is not very sensitive with respect to ν.


577

No friction 0.05 0.15 0.3 0.5

500 400 300 200 100 0 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Figure 2.3. Surface traction of a Hertz contact problem with Coulomb friction for different values of the friction coefficient ν ∈ {0, 0.05, 0.15, 0.3, 0.5}.

2.2. Formulation of the weak problem We start with a detailed discussion of different but equivalent formulations of the contact problem with no friction, i.e., ν = 0. Then the problem is equivalent to a standard variational inequality of the first kind or to an energy-minimization problem on a convex set. The frictionless contact problem with a linearized non-penetration condition can be stated as follows. Find u ∈ K such that J(u) = inf J(v),

(2.7)

v∈K

where the convex set K is given by all admissible solutions, and the energy is defined by J(v) := 12 a(v, v)−f (v). Here, the bilinear form a(·, ·) is given, for v, w ∈ V := Vm × Vs := (H 1 (Ωm ))d × (H 1 (Ωs ))d , by a(w, v) := am (wm , vm ) + as (ws , vs ), k k ak (w , v ) := σ(wk ) : (vk ) dx, k ∈ {m, s}, Ωk

and the linear form f (·) is defined for all v ∈ V in terms of s m k k fv dx + fN vk ds. f (v) := fs (v ) + fm (v ), fk (v ) := Ωk

ΓkN

In addition to the Hilbert space V, we introduce the subset K ⊂ V: K := {v ∈ V | v = uD on ΓD and [vn ] ≤ g on ΓsC }. By definition K is a closed convex non-empty set, f (·) is a continuous linear functional on V, and the bilinear form a(·, ·) is continuous on V × V and elliptic with respect to the Hilbert space V0 := {v ∈ V, v = 0 on ΓD }.

578

B. Wohlmuth

Thus the constrained minimization problem (2.7) has a unique solution: see, e.g., Glowinski (1984). Due to the symmetry of the bilinear form a(·, ·), (2.7) can be equivalently written as a variational inequality of the first kind, i.e., find u ∈ K such that a(u, v − u) ≥ f (v − u),

v ∈ K,

(2.8)

or as a variational inequality of the second kind, i.e., find u ∈ V such that a(u, v − u) + χK (v) − χK (u) ≥ f (v − u),

v ∈ V,

(2.9)

where χK is the indicator functional of K, i.e., χK (v) := ∞ if v ∈ K and zero otherwise. We refer to Brezis (1971), Duvaut and Lions (1976), Fichera (1964), Glowinski (1984), Glowinski et al. (1981), Kinderlehrer and Stampacchia (2000) and the references therein for an abstract mathematical framework on inequalities as well as for the so-called Signorini problem and its physical and mechanical interpretation. It is easy to see that in the special case of a variational inequality of the first kind with the convex set being a Hilbert space, existence and uniqueness of a solution follow directly from the Lax–Milgram theorem. Alternatively to the pure displacementbased formulation, the Signorini problem can be characterized in terms of the contact pressure as unknown variable; see, e.g., Demkowicz (1982). We note that the convex set K can be characterized in terms of a dual cone. To do so, we introduce the dual space M := (M )d := (W )d =: W of the trace space W := W d := (H 1/2 (ΓsC ))d and define the bilinear form b(·, ·) by b(µ, v) := µ, [v]ΓsC ,

v ∈ V, µ ∈ M,

where ·, ·ΓsC stands for the H 1/2 -duality pairing on ΓsC , and [v] := vs − vm ◦ χ. It is assumed that χ is smooth enough such that for v ∈ V we have [v] ∈ W. In terms of the bilinear form b(·, ·), the closed non-empty convex cone M+ is set to M+ :={µ ∈ M | µ, wΓsC ≥ 0, w ∈ Wn+ }, Wn+

:={w ∈ W | wn ∈ W }, +

W

+

(2.10a)

:= {w ∈ W | w ≥ 0}.

(2.10b)

Here, we assume that ns is smooth enough such that for w ∈ W and w ∈ W we also have wn ∈ W and wns ∈ W, respectively. Then the definitions of µn ∈ M for µ ∈ M and of µn ns ∈ M for µn ∈ M given by µn , wΓsC := µ, wns ΓsC , µn ns , wΓsC := µn , wn ΓsC ,

w ∈ W, w ∈ W,

respectively, are consistent in the sense that µn , wn ΓsC = µn ns , wn ns ΓsC for all µ ∈ M and w ∈ W. As a result the bilinear form b(·, ·) can be split


579

in a well-defined normal and tangential part, b(µ, v) = bn (µ, v) + bt (µ, v)

(2.11)

with bn (µ, v) := b(µn ns , v), bt (µ, v) := b(µt , v), µt := µ − µn ns . The situation that ns is only piecewise well-defined on ΓsC can be handled by decomposing the contact zone into non-overlapping subparts γj ⊂ ΓsC , defining all quantities with respect to γj and using product spaces and broken duality pairings. Now, observing that K can be written as K = {v ∈ V | v = uD on ΓD and b(µ, v) ≤ g(µ), µ ∈ M+ }, where g(µ) := µn , gΓsC , µ ∈ M, we obtain the saddle-point formulation of (2.8). Find (u, λ) ∈ VD × M+ such that a(u, v) + b(λ, v) = f (v),

v ∈ V0 ,

b(µ − λ, u)

µ ∈ M+ ,

≤ g(µ − λ),

(2.12)

with the convex set VD := {v ∈ V, v = uD on ΓD }. Lemma 2.1. The three inequality formulations (2.8), (2.9) and (2.12) are equivalent in the sense that if (u, λ) solves (2.12), then u is the solution of (2.8) and (2.9), and if u solves (2.8) or (2.9), then (u, λ), with λ ∈ M, λ, wΓsC := f (Hw) − a(u, Hw),

w∈W

(2.13)

satisfies (2.12). Here we have used Hw := (0, Hs w), where Hs is the harmonic extension onto V0s := {v ∈ Vs | v = 0 on ΓsD } with respect to the bilinear form as (·, ·). Proof. For convenience of the reader we recall some of the basic steps and refer to the monographs by Glowinski (1984) and Glowinski et al. (1981) for further details. In particular, we comment on the formula (2.13) for the Lagrange multiplier. The equivalence between (2.8) and (2.9) is standard. Let (u, λ) be a solution of (2.12); then for all µ ∈ M+ we have µ+λ ∈ M+ and thus u ∈ K, and moreover b(λ, u) = g(λ). For v ∈ K we find v−u ∈ V0 and thus a(u, v − u) = f (v − u) − b(λ, v − u) ≥ f (v − u) + g(λ) − g(λ). Let u be the solution of (2.8). Then, for all w ∈ Wn+ , we have v := u − Hw ∈ K. Now the definition (2.13) of λ yields λ, wΓsC = a(u, v − u) − f (v − u) ≥ 0 for all w ∈ Wn+ and thus λ ∈ M+ . Moreover, observing that v± := u ± H((g − [un ])ns ) is in K, we get a(u, H((g − [un ])ns )) = f (H((g − [un ])ns )), from which we conclude that 0 = λ, (g − [un ])ns ΓsC = λn , gΓsC − bn (λ, u) = g(λ) − b(λ, u). Then the second line of (2.12) holds by the definition of K. To see that (2.13) also satisfies the first line of (2.12), we set w := u ± (v − H[v]) ∈ K

580

B. Wohlmuth

for v ∈ V0 and use w as a test function in (2.8), resulting in 0 = a(u, v − H[v]) − f (v − H[v]) = a(u, v) − f (v) + λ, [v]ΓsC . Remark 2.2. We note that the saddle-point formulation (2.12) also has a unique solution. The uniqueness of the displacement is already established by Lemma 2.1. For the uniqueness of the surface traction a suitable infsup condition has to be satisfied. By definition, M is the dual space of W, which is the trace space of V0s := {v ∈ Vs | v = 0 on ΓsD }, and the extension theorem yields that inf sup

µ∈M v∈V0

b(µ, v) b(µ, v) ≥ inf sup µ− 1 ;Γs v1;Ω µ∈M v∈V0s µ− 1 ;Γs v1;Ωs 2

C

2

C

µ, wΓsC = Cinf . µ∈M w∈W µ− 1 ;Γs w 1 ;Γs

≥ Cinf inf sup

2

C

2

C

The case of Coulomb friction is more involved; we refer to Eck et al. (2005) for existence and regularity results and only mention that, for ν small enough, a unique solution exists. For contact problems in viscoelasticity we refer to Eck and Jaruˇsek (2003) and Han and Sofonea (2002). In particular, the admissible solution space depends on the solution itself and cannot be characterized without knowledge of the contact pressure. After these preliminary remarks, we can now easily extend our saddlepoint formulation (2.12) for ν = 0 to ν ≥ 0. We observe that M+ defined by (2.10a) can also be written as M+ = {µ ∈ M | µn ∈ M + , µt = 0}, with M + := {µ ∈ M | µ, wΓsC ≥ 0, w ∈ W + }. For ν > 0 the tangential part of the surface traction, in general, does not vanish, and thus one has to work with a vectorial Lagrange multiplier, which is not necessary for a frictionless contact problem. Replacing the convex cone M+ in (2.12) by M(λn ) := {µ ∈ M | µ, vΓsC ≤ νλn , vt ΓsC , v ∈ W with − vn ∈ W + }, (2.14) we obtain the weak saddle-point formulation of a static Coulomb problem between two linearly elastic bodies as follows. Find (u, λ) ∈ VD × M(λn ) such that a(u, v) + b(λ, v) = f (v), v ∈ V0 , (2.15) b(µ − λ, u) ≤ g(µ − λ), µ ∈ M(λn ). In the case of the quasi-static version, one has to replace b(µ − λ, u) in the ˙ Comparing (2.12) and second line of (2.15) by bn (µ − λ, u) + bt (µ − λ, u). (2.15), we find that the only, but essential, difference is the solution cone for the Lagrange multiplier λ. The key idea for the proof of existence is to


581

define a series of solutions (uk , λk ) ∈ VD × M(λk−1 n ) and apply Tikhonov’s fixed-point theorem: see, e.g., Eck et al. (2005). In the following we will frequently make use of the Karush–Kuhn–Tucker (KKT) conditions. Lemma 2.3. Let (u, λ) ∈ VD × M(λn ) be the solution of (2.15); then the non-penetration KKT condition λn ∈ M + ,

g − [un ] ∈ W + ,

bn (λ, u) = λn , gΓsC

(2.16)

holds. Moreover, under suitable regularity, the Coulomb law in its weak form (2.17) λ ∈ M(λn ), bt (λ, u) = νλn , [ut ]ΓsC is satisfied. Proof. We observe that the constraint on λn in (2.16) follows directly from the definition of M(λn ). Using the additive splitting (2.11) and setting as test function µ := λ ± λn ns in (2.15), we get the equality in (2.16). Observing that W + , defined in (2.10b), can also be characterized by W + = {w ∈ W | µ, wΓsC ≥ 0, µ ∈ M + }, it trivially holds that g − [un ] ∈ W + . For (2.17) we assume that there exists a χ such that χ ≤ 1, χ[ut ] = [ut ], and we have χv ∈ W for v ∈ W. Then, we get µ := λn ns + νλn χ ∈ M(λn ) and b(µ − λ, u) = b(νλn χ, u) − b(λt , u) = νλn , [ut ]ΓsC − bt (λ, u) ≤ 0, from which the equality in (2.17) follows from the definition (2.14). Remark 2.4. The special case of a contact problem between one elastic body and a rigid obstacle can be obtained from the two-body situation. A rigid body can be regarded as an infinitively stiff elastic body, and thus the limit case λm , µm → ∞ results in a one-body case where formally um = 0. To conclude this section, we briefly comment on numerical stability issues in elasticity and on the extension to the case of a solution-dependent friction coefficient. 2.3. Nearly incompressible materials In the nearly incompressible case, the Poisson ratio tends to 0.5 and thus the ratio between λ and µ tends to infinity. The definition of the bilinear form a(·, ·) in terms of the linearized stress (2.2) shows that the continuity constant depends on max(λ, µ), while the coercivity constant depends on µ. As a consequence, a priori estimates for standard low-order finite elements involve large constants, and volumetric locking can be observed numerically. To handle such a case appropriately, special discretization techniques are required. Methods associated with the enrichment or enhancement of the strain or stress field by the addition of carefully chosen basis functions have proved to be highly effective and popular. The key work dealing with

582 (a)

B. Wohlmuth (b)

(c)

(d)

Figure 2.4. Von Mises stress and deformed mesh of a contact between a soft nearly incompressible material (lower body) and two hard compressible ones (upper bodies): standard low-order (a,b) and Hu–Washizu-based (c,d) discretization.

enhanced assumed strain formulations is Simo and Rifai (1990). Figure 2.4 shows the setting of a contact problem between a nearly incompressible soft rubber-like material and a compressible hard one. In Figure 2.4(a,b) standard conforming low-order finite elements are applied, whereas in (c,d) special low-order Hu–Washizu-based elements are used. Here the starting point is the Hu–Washizu formulation (Hu 1955, Washizu 1955), in which the unknown variables are displacement, strain, and stress. This formulation (see also Felippa (2000) for some historical comments) can serve as the point of departure for the development of enhanced strain formulations; see also, e.g., Braess, Carstensen and Reddy (2004), Kasper and Taylor (2000a, 2000b), Simo and Armero (1992), Simo, Armero and Taylor (1993) and Simo and Rifai (1990). For both discretization schemes the numerically obtained von Mises stress and a zoom of the deformed meshes are depicted in Figure 2.4. As can be clearly observed in the case of the standard scheme, volumetric locking occurs, resulting in a very stiff response of the soft material; only a modified scheme can provide a good approximation. In the numerical experiment, we use a pure displacement-based formulation obtained from local static condensation of a three-field formulation: see Lamichhane, Reddy and Wohlmuth (2006) for details. To get a better feeling for the influence of Poisson’s ratio on the quality of the discretization, we consider the classical Hertz contact problem between a circle and a half-plane, which is approximated by a rectangle. In that situation the maximum of the contact pressure as well as the contact radius can be computed analytically in terms of the material parameters (Johnson 1985) 2 f r(1 − νPoisson ) 2f pmax = , , rcont = 2 πrcont Eπ

583

Numerical algorithms for variational inequalities Table 2.1. Comparison of standard scheme (Q1) with a displacement-based Hu–Washizu method (HW) for different values of the Poisson ratio. Poisson ratio νPoisson 0.1 0.45 0.49 0.499 0.4999 0.49999 0.499999

Contact pressure: pmax Q1 HW ‘Exact’ 16.7840 19.0973 19.9255 21.5690 31.0458 52.2592 58.4622

16.7708 19.0460 19.5690 19.7007 19.7144 19.7158 19.7159

17.1564 18.9238 19.3374 19.4394 19.4497 19.4508 19.4509

Contact radius: rcont Q1 HW ‘Exact’ 0.3437 0.3125 0.2812 0.2500 0.1875 0.1250 0.0938

0.3750 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125

0.3709 0.3363 0.3291 0.3274 0.3272 0.3272 0.3272

where r is the radius of the circle, f the applied point load, νPoisson the Poisson number and E Young’s modulus. A quantitative comparison of the two discretization schemes is given in Table 2.1. In the compressible range, both schemes provide quite good and accurate numerical approximations even for coarse meshes. However, the situation is drastically changed if Poisson’s ratio tends to 0.5. From 0.49 on, from row to row, λ is increased by a factor of 10. In the case of standard conforming low-order elements, the contact radius tends to zero, and thus the maximal contact pressure tends to infinity. In contrast, the analytical solution as well as the Hu–Washizu-based formulation yield convergence to a non-zero contact radius, and the maximum of the contact pressure remains finite. The limit of the standard scheme is an unphysical point contact, with the surface traction being a delta distribution. From now on, we assume that we are in the compressible range and that we do not have to face numerical problems due to the material parameters. 2.4. Thermo-mechanical contact problem Coupled contact problems where the coefficient of friction depends on the solution itself are quite difficult to analyse. Although non-trivial from the theoretical point of view, these generalized settings do fit perfectly well into the computational framework. Figure 2.5 shows the temperature distribution of a sliding body undergoing thermo-mechanical contact. The discretization in space is based on non-matching meshes and no re-meshing has to be done. For the time integration we apply a simple mid-point rule in combination with standard mass lumping techniques; see H¨ ueber and Wohlmuth (2009) for details and further numerical results for this example. In addition to the displacement, the temperature T is a primal variable, and a bi-directionally coupled thermo-mechanical system has to be con-

584

B. Wohlmuth

Figure 2.5. Temperature distribution at time tk−1/2 for k = 14, 26, 38, 50. 0.035

0.035 Friction coefficient Scaled max. temperature

Friction coefficient Scaled max. temperature

0.03

0.03

0.025

0.025

0.02

0.02

0.015

0.1

0.2

0.3

0.4

0.015

0.1

0.2

x 0.035

0.4

0.035 Friction coefficient Scaled max. temperature

Friction coefficient Scaled max. temperature

0.03

0.03

0.025

0.025

0.02

0.02

0.015

0.3 x

0.1

0.2

0.3 x

0.4

0.015

0.1

0.2

0.3

0.4

x

Figure 2.6. Friction coefficient and temperature at the contact nodes on the cutting line y = 0 for k = 14, 26, 38, 50.

sidered. More precisely, the relative temperature enters in terms of the thermal expansion coefficient in the definition of the mechanical stress. In addition, we have to consider the first and second law of thermodynamics. The heating from the Joule effect adds the source term div u˙ to the heat equation, and thus a fully coupled system is obtained: see, e.g., Fluegge (1972) and Willner (2003). From the theoretical point of view, thermomechanical contact problems have been analysed in Eck (2002) and Eck and Jaruˇsek (2001) Moreover a friction coefficient ν(T ) ≥ 0, which is monotone decreasing in T , modelling a thermal softening effect, has been applied. It tends to zero in the critical case that the temperature tends to the damage temperature: see, e.g., Laursen (2002). Figure 2.6 shows the evolution of the temperature-dependent friction coefficient and the temperature at the nodes in contact for different time steps. The dashed horizontal line marks the static coefficient of friction, and the two vertical lines indicate the actual contact zone. Due to the heating of the two bodies, the temperature increases over time and thus ν(T ) decreases.


585

3. Variationally consistent space discretization In the past, penalty methods and simple node-to-node coupling concepts have been widely used and are quite often integrated in commercial software codes. The starting point of penalty techniques is the observation that contact without friction can easily be formulated as a minimization problem on a constrained space. This approach is closely related to (2.9) when replacing the proper convex and lower semi-continuous indicator functional by a regularized one with finite values. Working on the larger unconstrained space, but incorporating the restriction as an additional finite energy contribution, a non-linear variational system of equations is obtained. Although relatively easy to implement, oscillations possibly occur, and the numerical results are very sensitive to the choice of the penalty parameter. A too-small penalty parameter gives considerable penetration and a poor approximation of the contact forces. A too-large penalty parameter yields a badly conditioned system which has to be solved by a suitable non-linear scheme. Figure 3.1 illustrates the influence of the penalty parameter on the deformed geometry. For a small penalty parameter (see Figure 3.1(a)), significant penetration of the rigid obstacle into the elastic material occurs, and the von Mises stress is highly underestimated. In Figure 3.1(b), a very large penalty parameter is applied which gives numerical results of high quality. However, the number of multigrid steps then required to solve the system is quite high compared to the case of a small penalty parameter. Figure 3.2 illustrates the difference between a penalty and a Lagrange multiplier-based approach. Figures 3.2(a) and 3.2(d) show, respectively, the influence of the penalty parameter on the penetration and the contact stress. As can be seen, for small penalty parameters there is significant penetration and the approximation of the contact stress is rather poor. In the limit → 0, the solution tends to the unconstrained one and the (a)

(b)

Figure 3.1. Comparison of the deformed geometry and the von Mises stress for a three-dimensional one-sided contact problem: small penalty parameter (a) and large penalty parameter (b).

586

B. Wohlmuth (b)

0 −0.2 Tool ε = 5e05 ε = 1e05 ε = 5e06 ε = 5e07

−0.4 −0.6 −4

(c)

Vertical displacement

Vertical displacement

(a)

−2

0 2 x−coordinate

(d)

−0.2 −0.3 −0.4 −2

0 2 x−coordinate

12000

8000 6000

ε = 5e07 Active set

4000 2000 0 −2

−1

0 1 x−coordinate

8000 6000 4000 2000 0 −4

2

−2

0 2 x−coordinate

Figure 3.2. Penalty versus Lagrange multiplier approach. Node-to-node

Node-to-node

Consistent 2

2

1

Node-to-node

Consistent 2

2

1

1

1

0

0

0

0

−1

−1

−1

−1

−2

−2

−2

−2

−3

−3

−3

−3

−4

−4

−4

Consistent

4

ε = 5e05 ε = 1e05 ε = 5e06 ε = 5e07

10000 Contact stress

10000 Contact stress

Tool ε = 5e07 Active set

−0.1

−0.5 −4

4

12000

0

Node-to-node

−4

Consistent

2

2

2

2

1

1

1

1

0

0

0

0

−1

−1

−1

−1

−2

−2

−2

−2

−3

−3

−3

−3

−4

−4

−4

−4

Figure 3.3. Node-to-node interpolation versus variationally consistent coupling.

4


1

(b)

Energy norm 1/2 O(h ) 2 L -norm O(h)

0.1

587

Energy error 1/2 O(h ) L2-error O(h)

0.001 0.0001

Error

1e-05 1e-06

0.01

1e-07 1e-08

0.001

1e-09 0.0001

1e-10 10

100 1000 10000 Degrees of freedom

100000

100

1000 10000 Degrees of freedom

100000

Figure 3.4. Non-optimal error decay for a node-to node coupling: scalar case (a) and linear elasticity (b).

contact stress vanishes. In the limit → ∞, the correct constrained solution is recovered but then the condition number of the system tends to be extremely large. Figures 3.2(b) and 3.2(c) compare the numerical results of a penalty approach with a large value for with a weakly consistent Lagrange multiplier-based formulation. Both results show the same good quality of approximation, but the latter approach has a much better condition number and thus is more suitable for fast iterative solvers such as multigrid or domain decomposition methods. Nowadays penalty techniques and simple node-to-node coupling strategies are increasingly replaced by variationally consistent methods which pass suitable patch tests in the case of non-matching meshes. The admissibility of the discrete solution is then formulated in a weak variational framework. Displacement and surface traction form a primal–dual pair of unknown variables and have to be discretized. Figure 3.3 illustrates the difference between a simple node-to-node coupling strategy and a variationally consistent approach. A constant force can only be mapped correctly from the slave to the master side if a weak coupling is applied. The simple node-to-node coupling yields poor numerical results if non-matching meshes are used, whereas a sliding of the mesh does not influence the approximation quality in the case of a variationally consistent scheme. Figure 3.4 shows the quantitative error decay for a node-to-node coupling in the case of non-matching meshes. As a patch test, we select a linear solution which can be represented exactly by standard low-order finite elements. In Figure 3.4(a), the scalar-valued Laplace operator is considered, whereas in Figure 3.4(b) the results for the vector-valued system of linear elasticity are presented; see Dohrmann, Key and Heinstein (2000) for the parameter specifications. In both cases, the exact solution cannot be reproduced, and the error decay is sub-optimal. A first-order error decay can be only observed for the L2 -norm but not for the H 1 -norm. However, in the

588

B. Wohlmuth

case of a variationally consistent weak formulation, the exact solution can be recovered and the error is equal to zero on all meshes. From the theoretical point of view, a node-to-node coupling is associated with a discrete Lagrange multiplier being represented by a linear combination of delta distributions, which is not compatible with the required H 1/2 -duality pairing. In the following, we restrict ourselves to discrete Lagrange multiplier spaces being L2 -conforming. To obtain a stable and well-posed discrete setting, a uniform inf-sup condition has to be satisfied. Roughly speaking this means that the trace space of the discrete displacement has to be well balanced with the finite-dimensional space for the surface traction, also called the Lagrange multiplier. A necessary condition is that the dimension of the Lagrange multiplier space is less than or equal to the dimension of the jump of the trace spaces. There exists a large variety of different construction principles, all leading to optimal a priori estimates in the case of standard variational equalities. Quite often such a condition is numerically verified by the Bathe–Chapelle inf-sup test (Chapelle and Bathe 1993). A mathematically rigorous analysis can be performed within the abstract framework of mortar settings on non-matching meshes; see, e.g., Ben Belgacem and Maday (1997) and Bernardi, Maday and Patera (1993, 1994). Early theoretical results on uniform stable discretization schemes for contact problems without friction can be found in Ben Belgacem (2000) and Ben Belgacem, Hild and Laborde (1997, 1999). When a vector-valued Lagrange multiplier is used, there is no algebraic difference between a contact problem with Coulomb friction and one without. Thus, quite often solvers and error estimators designed for contact problems without friction naturally apply to contact problems with Coulomb friction. However, we recall that from the theoretical point of view there is a possibly considerable difference, as for existence and uniqueness results; see, e.g., Eck et al. (2005) and Kikuchi and Oden (1988). In this section, we illustrate the fact that a weakly consistent discretization based on a biorthogonal set of displacement traces and surface tractions is well suited to the numerical simulation of contact problems. While simple node-to-node coupling strategies are known to show locking effects as well as unphysical oscillations when applied to non-matching meshes, variationally consistent formulations based on uniform inf-sup stable pairings do not exhibit this behaviour. Moreover, the biorthogonality of the basis functions of such a pairing yields a stable node-to-segment coupling concept where the simple interpolation is replaced by a quasi-projection. This is quite attractive because of the locality of the coupling constraints. The discretization of the system is based on the saddle-point formulation (2.15). Both cases, the frictionless case and that with Coulomb friction, can be handled within the same abstract framework. In the case ν = 0, we do not work with the primal variational inequality (2.8) but also use


589

the primal–dual variational inequality setting. A low-order pair of primal– dual variables for the displacement u and the surface traction λ on the contact zone will be applied. As usual in the mortar context, the Lagrange multiplier space is associated with the (d − 1)-dimensional surface mesh on ΓsC inherited from the volume mesh on the slave side. In addition, the degrees of freedom from the master side will not be required for the inf-sup condition to hold. Thus the inf-sup constant is independent of the ratio between the mesh sizes of the master and slave sides and also independent of the non-matching character of the meshes, which is quite attractive in dynamic situations when sliding geometries occur. In the linear saddle-point theory, it is well established (Brezzi and Fortin 1991, Nicolaides 1982) that a priori estimates in terms of the best approximation error of the primal and dual variables can be obtained if stability and continuity of the relevant bilinear forms are given. The norm for the displacement is the product H 1 -norm, and for the surface traction, the H −1/2 -norm defined as the dual norm of the H 1/2 -norm on ΓsC . Thus, to obtain first-order estimates for the best approximation error, the natural choice for the displacement is the lowest-order conforming finite element space on each of the two subdomains, whereas for the Lagrange multiplier several interesting choices exist. Basically all existing possibilities from the mortar literature can be used, e.g., piecewise constants associated with the dual mesh or low-order conforming finite elements. 3.1. A pairing, not uniformly stable Before going into the details of the discretization, we consider a counterexample. We note that element-wise constants for the Lagrange multiplier do not yield optimal estimates. Although the best approximation error of element-wise constants with respect to the L2 -norm is of order one, this combination of primal and dual variables is not uniformly inf-sup stable. A mesh-dependent inf-sup constant results in a reduced convergence rate. Figure 3.5 illustrates this non-uniformly stable pairing in a one-dimensional setting. Let the unit interval I := (0, 1) be decomposed into Nl := 2l , l ∈ N, sub-intervals Ii := (i − 1, i)/Nl , i = 1, . . . , Nl , of equal length and Wl := {v ∈ C(I); v|Ii ∈ P1 (Ii ), i = 1, . . . , Nl }, Ml := {v ∈ L2 (I); v|Ii ∈ P0 (Ii ), i = 1, . . . , Nl }. l Then each µl ∈ Ml can be written as µl = N i=1 αi ψi , with αi ∈ R, and where ψi stands for the characteristic function of the sub-interval Ii . Each l vl ∈ Wl has the form vl = N b φ , with b ∈ R, and where φi denotes the i i i i=0 standard hat function associated with the node xi := i/Nl , i = 0, . . . , Nl .

590

B. Wohlmuth

(a)

(b)

(c)

ψi φi xi

Figure 3.5. Nodal finite element φi and Lagrange multiplier basis function ψi (a), special choice of µl and vl for l = 4 and l = 5 (b,c).

Lemma 3.1. The pairing (Wl , Ml ) is not uniformly inf-sup stable with 1 respect to the H 2 -duality. Moreover there exists a cinf > 0 independent of the number of sub-intervals Nl such that 1 cinf 0 µl vl ds ≥ , (3.1) inf sup µl ∈Ml vl ∈Wl µl − 1 ;I vl 1 ;I Nl 2

2

and the estimate in (3.1) is sharp. Proof. To see that the non-uniform inf-sup condition given by (3.1) holds, 1 it is sufficient to define a Fortin operator Fl : H 2 (I) → Wl such that I µl Fl v ds = I µl v ds for all µl ∈ Ml and Fl v 1 ;I ≤ cNl v 1 ;I for all 1

2

2

v ∈ H 2 (I): see, e.g., Brezzi and Fortin (1991). Let v ∈ H 1/2 (I) be given. l In a first step, we set w1 := N i=0 γi φi with 2 Ii ∪Ii+1 v(2φi − φi−1 − φi+1 ) ds , i = 0, . . . Nl , I0 := INl +1 := ∅. γi := |Ii ∪ Ii+1 | In a second step, we define recursively ∆γi := 2Nl Ii (v − w1 ) ds − ∆γi−1 , l i = 1, . . . , Nl and ∆γ0 := 0, w2 := N i=0 ∆γi φi . We note from the definition 1 of w1 that it is H 2 -stable and has L2 -approximation properties. In terms of w1 and w2 , we set Fl v := w1 + w2 . Then by construction, Fl v and v have the same mean value on each sub-interval Ii , i = 1, . . . , Nl . Introducing the matrix Bl ∈ RNl ×Nl as follows, we get for its inverse     1 1 1 1 −1 1        1 1  1 −1 1       −1 Bl :=  = , B   . 1 1 l  −1 1 −1 1     1 −1 1 −1 1   1 1     .. .. .. .. . . . .


591

Moreover, a standard inverse estimate for finite elements and the fact that the Euclidean norm of Bl−1 is bounded in terms of Nl , i.e., Bl−1 ≤ CNl , gives Fl v 1 ;I ≤ w1 1 ;I + w2 1 ;I ≤ C(v 1 ;I + Nl w2 0;I ) 2 2 2 2 −1 ≤ C(v 1 ;I + Nl Bl Πl (v − w1 )0;I ) 2 ≤ C(v 1 ;I + Nl Nl v − w1 0;I ) ≤ CNl v 1 ;I , 2

2

L2 -projection

where Πl stands for the onto Ml . To show that the estimate (3.1) is sharp, we have to specify a µl ∈ Ml such that no better bound can be obtained. Let us consider the choice αi := (−1)i (i − 1)(Nl − i): see Figure 3.5(b,c). The definition of the dual √ √ l norm (2.1) yields µl − 1 ;I ≥ c N |αi |/(Nl Nl ) ≥ cNl Nl . Then, for i=1 2 l ≥ 2 a straightforward computation and a standard inverse estimate shows Nl µl vl ds = (−1)i (i − 1)(Nl − i)(bi−1 + bi )/(2Nl ) I

=

i=1 N l −1

(−1)i (2i − Nl )bi /(2Nl )

i=1

=

Nl /4

1 (Nl + 1 − 4i) (b2i−1 − b2i ) + (bNl −2i − bNl +1−2i ) 2Nl i=1

Nl /2−1 1 (bi − bNl /2+i ) 2Nl i=1 C vl 1 ;I µl − 1 ;I . ≤ C(|vl |1;I + vl 0;I ) ≤ C Nl vl 1 ;I ≤ 2 2 2 Nl

+

Remark 3.2. A possible remedy would be to use a coarser mesh for the Lagrange multiplier space. The pairing (Wl , Ml−1 ) is uniformly inf-sup stable. Alternatively, the space Wl can be enriched by locally supported bubble functions, as is done in Brezzi and Marini (2001) and Hauret and Le Tallec (2007). Here we do not follow these possibilities, but use only Lagrange multiplier spaces being defined on the same mesh as the trace space of the slave side and having the same nodal degrees of freedom. 3.2. Stable low-order discretization On each subdomain Ωk , k ∈ {m, s}, independent families of shape-regular triangulations Tlk , l ∈ N0 , will be used, and we set Tl := Tlm ∪ Tls and k Ωl := ∪T ∈T k T . The maximum element diameter of the triangulation Tl is l

592

B. Wohlmuth

(a)

(b)

(c)

ψp

ψp

ψp

φp

φp

φp

p

p

p

Figure 3.6. Discontinuous biorthogonal basis functions satisfying (3.2)–(3.5).

denoted by hl . The restriction of Tlk to ΓkC;l := ∪F ∈F k F , where Flk stands l for the set of all contact faces on the side k, defines a (d − 1)-dimensional −1 surface mesh. Moreover, the surface mesh on Γm C;l will be mapped by χl s onto ΓC;l , resulting in possibly non-matching meshes on the contact zone. For the displacement, we use standard low-order conforming finite elements and for the surface traction dual finite elements which reproduce constants Vl := Vlm × Vls , Ml :=

Msl ,

Vlk := (Vlk )d ,

Vlk := spanp∈P k {φp },

Mkl

Mlk

:=

(Mlk )d ,

l

:= spanp∈P k {ψp }, C;l

k

k is the where Plk stands for all vertices of Tlk not being on ΓD , and PC;l k

set of all vertices on ΓC , k ∈ {m, s}. Moreover φp denotes the standard conforming nodal basis function associated with the vertex p. The basis functions ψp ∈ Mlk are required to have the following properties. • Locality of the support: supp ψp = supp φp |Γk ,

k p ∈ PC;l ,

C;l

• Local biorthogonality relation: ψp φq ds = δpq φq ds ≥ 0, F

F

(3.2)

k p, q ∈ PC;l , F ∈ Flk ,

(3.3)

• Best approximation property: inf µ − µl − 1 ;Γk ≤ Cbest hl |µ| 1 ;Γk , 2

µl ∈Mlk

C;l

2

C;l

1

µ ∈ H 2 (ΓkC;l ),

(3.4)

where Cbest < ∞ does not depend on the mesh size. • Uniform inf-sup condition: ΓkC;l µl vl ds ≥ cinf µl − 1 ;Γk , sup 2 C;l vl ∈W k vl 1 ;Γk l

2

C;l

µl ∈ Mlk ,

(3.5)

593


(b)

(c)

ψp

ψp

ψp

φp

φp

φp

p

p

p

Figure 3.7. Continuous biorthogonal basis functions satisfying (3.2)–(3.5).

where cinf > 0 does not depend on the mesh size and the discrete trace space Wlk is given by Wlk := spanp∈P k {φp |Γk }. C;l

C;l

From the local relation (3.3) follows directly a global one, i.e., k ψp φq ds = δpq mp := δpq φq ds ≥ 0, p, q ∈ PC;l , ΓkC;l

(3.6)

ΓkC;l

with mp > 0. We note that there exists no set of non-negative basis functions satisfying (3.6). Moreover (3.3) in combination with (3.4) automatically yields p∈P k ψp = 1 and C;l k ψp ds = φp ds, p ∈ PC;l , F ∈ Flk . (3.7) F

F

The most popular choice of a dual Lagrange multiplier (Wohlmuth 2000, 2001) is obtained by an element-wise biorthogonalization process of the local nodal finite elements followed by a node-wise glueing step to reduce the number of degrees of freedom. As a result piecewise-linear but discontinuous basis functions are created for a one-dimensional contact zone. This technique works for the lowest-order finite elements on all types of surface meshes. In the case of higher-order elements, we have to use Gauss–Lobatto nodes. This approach is restricted to tensorial meshes where the element mapping is affine: see Lamichhane and Wohlmuth (2007). Alternatively, for low-order elements, as in the present discussion, we can apply piecewiseconstant or -quadratic basis functions on a sub-mesh: see Figure 3.6(b,c), respectively. In some applications it is of interest to work with continuous basis functions. Although the local construction of biorthogonal basis functions, nodewise defined, quite often results in a set of discontinuous basis functions, continuous ones do exist. Figure 3.7 shows basis functions which satisfy (3.2)–(3.5) and which in addition are continuous. In Figure 3.7(a), the basis function is cubic on each element. For the construction of the basis function shown in Figure 3.7(b,c), we use sub-elements and a piecewise quadratic and

594

B. Wohlmuth

linear approach, respectively. As will be seen, the cubic dual Lagrange multiplier goes hand in hand with the definition of H(div)-conforming mixed finite elements for linear elasticity and fits well into the construction of a posteriori error estimators based on element-wise lifting techniques: see Section 6. The following two remarks briefly comment on more general formulations. In particular, the construction of a dual Lagrange multiplier basis possibly depends on the geometry in the case of cylinder coordinates or of sliding meshes. Remark 3.3. If a three-dimensional situation is reduced to a two-dimensional setting by introducing cylinder coordinates and exploiting symmetry arguments, then duality has to be formulated with respect to a weighted scalar product. The distance to the symmetry axis enters as weight in the local biorthogonality relation (3.3). As example, we consider the piecewise affine case. Let r1 and r2 be the distances of the two face nodes to the symmetry axis. Then the dual Lagrange multiplier restricted to a face can be written as

2r1 + r2 ψ1 = 2 (r1 + 3r2 )φ1 − (r1 + r2 )φ2 , 2 r1 + 4r1 r2 + r2

r1 + 2r2 (3r , ψ2 = 2 + r )φ − (r + r )φ 1 2 2 1 2 1 r1 + 4r1 r2 + r22 where φ1 and φ2 are the two nodal basis function associated with the face. We note that ψ1 + ψ2 = 1, and for r1 = r2 we fall back to the piecewise affine dual Lagrange multiplier depicted in Figure 3.6(a). Remark 3.4. A typical benchmark problem for large deformation contact is a small cube sliding over a larger block. If the large block is defined as slave side, then the integral over the face F ∈ Fls in the local biorthogonality relation (3.3) has to be replaced by the integral over F ∩Γm C;l . Let us consider with s ∈ (0, 1). Then a the 2D reference case F = (0, 1) and (s, 1) ⊂ Γm C;l straightforward calculation shows 2(1 + s + s2 ) φ1 − (1 + 2s)φ2 , 1−s (1 + 2s)(1 + s) ψ2 = − φ1 + 2(1 + s)φ2 , 1−s ψ1 =

where φ1 is the nodal basis function associated with the endpoint p = 0 and φ2 is associated with p = 1. The weak problem formulation will be based on suitable subsets of Vl and Ml . For the displacement, we only impose the Dirichlet condition on the space and use no constraint related to the non-penetration condition.

595


The convex set Vl;D is given by m s × Vl;D , Vl;D := Vl;D

k Vl;D :=

uD (p)φp + Vlk ,

k p∈PD;l

k

k stands for the set of all vertices of the actual mesh T k on Γ . To where PD;l D l handle the contact conditions (2.5) and (2.6) in a weakly consistent form, we have to impose constraints on the Lagrange multiplier space. The nonpenetration condition restricts the normal part of Ml , and the Coulomb law requires a solution-dependent inequality bound for the tangential part. Let the discrete solution λl ∈ Ml be given by γ p ψp , γ p ∈ Rd , λl = s p∈PC;l

and denote the discrete normal component by γpn ψp , γpn := γ p nsp . λnl := s p∈PC;l

Here nsp stands for a discrete normal vector associated with the node p. In the case of a non-planar contact surface, it can be obtained as a weighted combination of the adjacent element centre normals. We observe that λnl ∈ Mls but, in general, it is not equal to λl ns . Based on the discrete normal surface traction λnl , we then define the convex set n d n t n Ml (λl ) := µl = β p ψp , β p ∈ R , βp ≥ 0, β p ≤ νγp , (3.8) s p∈PC;l

as an approximation for the solution-dependent cone M(λn ) defined in (2.14). Here βpn := β p nsp , β tp := β p − βpn nsp . We assume that ΓsC;l is large enough such that γ p = 0 for p ∈ ∂ΓsC;l . Remark 3.5. If ν = 0, the convex set Ml (λnl ) is solution-independent, and its definition reduces to + d n t β p ψp , β p ∈ R , βp ≥ 0, β p = 0 . Ml := µl = s p∈PC;l

In this case it is sufficient to work with a scalar-valued Lagrange multiplier space as is often done in the literature. Here, we use a vector-valued Lagrange multiplier to be in the same abstract framework for all ν ≥ 0. If the standard nodal Lagrange multiplier basis is used to define Ml , there are then two natural but different ways to discretize M + ; see also, e.g., Hild

596

B. Wohlmuth

(a)

(b)

1

a

−1/2

+ + Figure 3.8. Two elements of Ml;2 \ Ml;1 : local support (a) and global support (b).

+ and Renard (2010). The first yields Ml;1 ⊂ M + , whereas the second choice is based on the definition of M + as a dual cone: + s βp ψp , βp ≥ 0, p ∈ PC;l , Ml;1 := µl ∈ Ml | µl =

+ Ml;2

:=

s p∈PC;l

µl ∈ Ml |

ΓsC;l

µl φp ds ≥ 0, p ∈

s PC;l

.

+ We note that these two definitions yield two different spaces, with Ml;1 + + being a proper subspace of Ml;2 . Figure 3.8 shows elements of Ml;2 which + are clearly not in Ml;1 . In Figure 3.8(a), the element µl ∈ Ml depends on the parameter value a and, as a straightforward computation shows, + + µl ∈ Ml;2 \ Ml;1 if and only if a ∈ [−0.5; 0). The function in Figure 3.8(b) is + s , obviously not in Ml;1 . Testing it with all nodal basis functions φp , p ∈ PC;l + yields the non-negative values (marked with bullets) and thus µl ∈ Ml;2 . + Both choices of Ml;i , i = 1, 2, can be applied in the discrete setting. The first one yields a conforming approach, whereas in the second one the nonconformity of Ml+ has to be taken into account in the a priori estimates. The difference in the spaces stems from the fact that the matrix given by ψq φp ds is then the standard mass matrix, which is not an M-matrix. s Γ C;l

The situation is different if our Lagrange multiplier basis, satisfying (3.6), is applied. Then the mass matrix is diagonal and positive definite and both definitions yield the same space Ml+ . However, using a biorthogonal basis automatically results in a non-conforming approach, i.e., Ml+ ⊂ M + . To enforce conformity in that situation is not a good idea, since then the locality of the elements in Ml+ ∩ M + is lost; see also Figure 3.9. Having the conforming finite element space Vl ⊂ V0 and the non-conforming closed convex cone Ml (λnl ), we can formulate the discrete weak version s n of (2.15) as follows. Find ul := (um l , ul ) ∈ Vl;D , λl ∈ Ml (λl ) such that al (ul , vl ) + bl (λl , vl ) = fl (vl ), ≤ gl (µl − λl ), bl (µl − λl , ul )

v l ∈ Vl , µl ∈ Ml (λnl ).

(3.9)


597

1

1/2 1/4

Figure 3.9. Elements in Ml+ ∩ M + for different refinement levels.

The mesh-dependent bilinear and linear forms are obtained from the associated continuous ones in a natural way by replacing the volume and surface terms by the corresponding discrete analogue, i.e., summing over the volume and surface elements of the mesh. 3.3. Coupling in terms of the mortar projection Although the given variational setting is a two-body formulation with possibly non-matching meshes, we can reformulate the contact conditions in a way similar to a one-body system. To do so, we introduce the mortar projection Πl := (Πl )d onto Wls and the dual mortar projection Π∗l := (Π∗l )d onto Msl ; see, e.g., Bernardi, Maday and Patera (1993, 1994). We recall that due to the assumption on the Dirichlet boundary part, no modification at the endpoints is required. For w ∈ H 1/2 (ΓsC;l ) and µ ∈ H −1/2 (ΓsC;l ) we set Πl wµl ds := µl , wΓsC;l , µl ∈ Mls , (3.10a)

ΓsC;l

ΓsC;l

Π∗l µwl ds := µ, wl ΓsC;l ,

wl ∈ Wls .

(3.10b)

s We note that Πl and Π∗l restricted to Wls and M l is the identity, respectively. In terms of Πl , we can write Πl [ul ] = s αp φp . Moreover, p∈PC;l we define gp := Γs gl ψp ds/mp with a suitable approximation gl for the C;l

linearized gap, where mp is specified in (3.6). The following lemma shows that Lemma 2.3 has a node-wise discrete analogue. s Lemma 3.6. For each node p ∈ PC;l the following discrete node-wise KKT conditions hold for non-penetration:

0 ≤ γpn ,

αpn ≤ gp ,

γpn (αpn − gp ) = 0.

(3.11)

Moreover, a discrete static Coulomb law holds for each node: γ tp ≤ νγpn ,

αtp γ tp − νγpn αtp = 0.

(3.12)

598

B. Wohlmuth

Figure 3.10. Discrete approximations Ωl of the domain Ω.

Proof. We observe that the constraints on γpn in (3.11) and on γ tp in (3.12) follow directly from the definition of Ml (λnl ). The biorthogonality (3.6) plays an essential role in the proof. Using µl = λl ± γpn nsp ψp ∈ Ml (λnl ) as test function in (3.9), we find ±γpn b(nsp ψp , ul ) = ±γpn αpn mp ≤ gp γpn mp and thus the complementarity condition in (3.11). (3.12) obviously holds for αtp = 0. For αtp = 0, we set µl = λl − γ tp ψp + νγpn et ψp ∈ Ml (λnl ) with et := αtp /αtp as the test function in (3.9) and get (3.12). The discrete contact conditions (3.11) and (3.12) only involve quantities associated with the slave nodes and thus have the same structure as a onebody system. However, to compute αp , we do have to evaluate the mortar projection Πl applied on [ul ]. Its algebraic representation can be obtained from the entries of the mass matrix associated with bl (·, ·). For its implementation, we not only have to map the mesh elements on the possible contact zone but also the basis functions. In Figure 3.10, we show different possible matching and non-matching situations in the case of a non-planar contact surface. We note that in this situation standard triangulations do not resolve the domain exactly. In contrast to the continuous setting where ΓsC = Γm C , we find, in the discrete setting, that the possible contact zones on the master and the slave sides are not the same, i.e., ΓsC;l = Γm C;l . Using hierarchical tree structures or front tracking techniques, the assembly of the surface-based coupling matrices between master and slave side can be realized quite efficiently and is of lower complexity, whereas in 3D a naive approach results in a higher complexity compared to the assembling process of the volume contributions. For the integration in 3D we use quadrature formulas on surface sub-triangles. Figure 3.11 illustrates different steps of the projection and partitioning procedure. This algorithm goes back to Puso (2004) (see also Puso et al. (2008)) and, alternatively, the recent papers by Dickopf and Krause (2009a, 2009b). An analysis and a numerical study of the influence of curvilinear interfaces in the mortar situation can be found in Flemisch, Melenk and Wohlmuth (2005a). The abstract setting of blending elements (Gordon and Hall 1973a, 1973b) plays a key role in establishing optimal upper bounds


599

Figure 3.11. Element-wise mapping from the master to the slave side.

of the consistency error. From the theoretical point of view, the mapping between ΓsC;l and Γm C;l has to be globally smooth. From the computational point of view, an element-wise smooth but possibly discontinuous mapping is more attractive and works well in practice. Replacing the H 1/2 -norm on the contact interface by a weighted mesh-dependent L2 -norm, this simplification can also be theoretically justified. A 3D analysis, in the case of planar interfaces and mesh-dependent norms, can be found in Braess and Dahmen (1998). We note that for non-planar interfaces, a slave side associated with the coarser mesh and a vectorial partial differential equation, e.g., linear elasticity, possibly poor numerical results can be observed in the pre-asymptotic range, if the Lagrange multiplier with respect to its Cartesian coordinates is discretized by dual or piecewise constant basis functions: see, e.g., Flemisch, Puso and Wohlmuth (2005b). This effect can be explained by the observation that a constant normal surface and a zero tangential force cannot very well be approximated in terms of the Lagrange multiplier space. Firstly decomposing the Lagrange multiplier into its normal and tangential parts, and secondly discretizing yield much better results. However, in that case we have to be quite careful to handle rigid body motions correctly. These oscillations do not occur for the Laplace operator, with a finer slave mesh side, standard linear Lagrange multipliers, or quadratic dual Lagrange multipliers. For contact problems, non-penetration and the friction law are directly expressed in terms of the normal and tangential component of the surface traction. Thus a discretization of the traction in its locally rotated coordinate system seems to be quite attractive, in particular for finite deformations, and has already been used to define Ml (λnl ). For small deformations and a constant contact normal, both approaches give the same results.

4. Optimal a priori error estimates Abstract error estimates for variational inequalities can be found, e.g., in Brezzi, Hager and Raviart (1977), Falk (1974), Glowinski (1984) and Glowinski et al. (1981) and a priori bounds for the discretization error of unilateral contact problems are given, e.g., in Haslinger and Hlavá˘cek (1981) and Haslinger et al. (1996). It is well known that the finite element solution of a variational inequality may have a reduced convergence order, compared to that of the best approximation. This holds true for higher-order finite

600

B. Wohlmuth

elements but may also be true for low-order approaches. A√proof purely based on standard techniques will generally yield only O( hl ) a priori bounds. We refer to the monograph by Han and Reddy (1999) for an introduction into this area for applications in plasticity. Mortar techniques for contact problems without friction have been introduced in Ben Belgacem, Hild and Laborde (1998), Ben Belgacem (2000) and Lhalouani and Sassi (1999). We also refer to Ben Belgacem and Renard (2003), Coorevits, Hild, Lhalouani and Sassi (2001), Hild (2000) and Hild and Laborde (2002), where standard Lagrange multiplier spaces have been considered and analysed. In early papers on mortar, unilateral contact problems have quite often been considered, taking no friction into account, and using a scalar-valued Lagrange multiplier. The choice of the contact pressure as Lagrange multiplier is motivated by the fact that in that case the tangential component of the surface traction is zero. There is a series of papers on a priori estimates for two-body contact problems with no friction on non-matching meshes starting with order 1/4 bounds for the discretization error (Ben Belgacem et al. 1998). A priori error estimates for the displacements in the H 1 -norm and for the Lagrange multiplier in the H −1/2 -norm of order 3/4 have been established; see, e.g., Ben Belgacem, Hild and Laborde (1999), Ben Belgacem and Renard (2003), Coorevits et al. (2001) and Lhalouani and Sassi (1999), under an H 2 -regularity assumption. Using additional quite strong and restrictive regularity assumptions on the Lagrange multiplier, order one has been shown; see, e.g., Coorevits et al. (2001) and Hild (2000). These first a priori results have been considerably improved over the last decade. Under suitable assumptions on the actual 2 -regular solution quasi-optimal, i.e., h contact zone and a H | log hl | and l hl 4 | log hl | a priori estimates can be found in Ben Belgacem (2000) and Ben Belgacem and Renard (2003). Most of the theoretical results are obtained for standard Lagrange multipliers, no friction and in the two-dimensional setting. Here, we apply these techniques to vector-valued dual Lagrange multiplier spaces and provide a priori error estimates for the displacement in the H 1 -norm and for the surface traction in the H −1/2 -norm. In 3D, only sub-optimal bounds can be obtained for a problem with non-trivial friction. In 2D, we follow the lines of H¨ ueber, Matei and Wohlmuth (2005b) and H¨ ueber and Wohlmuth (2005a) and establish in a simplified problem setting optimal a priori bounds under some regularity assumption on the actual contact part and on the sticky zone. We assume that no variational crimes are committed, i.e., the discrete bilinear and linear forms are exact. In particular, this implies that no quadrature error occurs and that Ωkl = Ωk ; we refer to Ciarlet (1991, 1998) for a rigorous mathematical analysis of the influence of quadrature formulas. Moreover, we assume a zero gap, i.e., m g = 0, ΓC := ΓsC = ΓsC;l ⊂ Γm C = ΓC;l

601


(b)

(c)

5x5 4x4 3x3 2x2 O(h)

Error

10-1

-2

10

1

10

2

10

3

4

10 10 Degrees of freedom

5

10

Figure 4.1. Decomposition into 4 and 25 subdomains (a,b) and error decay in the H 1 -norm (c).

and a constant unit vector n := ns on the possible contact zone. The a priori analysis of a curvilinear interface for the classical linear- and scalar-valued mortar case can be found in Flemisch et al. (2005a). Figure 4.1 shows that the number of subdomains in the linear mortar setting does not influence the constants in the a priori bound. The unit square is decomposed into l2 , l = 2, 3, 4, 5 subdomains; the interface is given by a sinus wave function. Due to the curvilinear character of the interface, a mesh-dependent mapping between the discrete master and slave interface is required. In the case of a non-linear contact problem, the same approach can be applied. For simplicity of notation, we do not provide any technical detail here and restrict ourselves to simple geometrical settings such as, e.g., a square on a rectangle. Moreover, we will work with globally quasi-uniform meshes, such that there exists a regularity constant creg > 0 so that, for all nodes p on ΓC , we have Bp (creg hl ) ⊂ supp φp |ΓC , where Bp (creg hl ) is the (d − 1)-dimensional ball with centre p and radius creg hl . Most importantly, we replace for the rest of this section the Coulomb friction law (2.6) by the more simple Tresca law with a constant friction bound F, i.e., λt ≤ F, [u˙ t ]λt − F[u˙ t ] = 0. (4.1) In the following, we will frequently make use of the Tresca version of Lemmas 2.3 and 3.6. Following the lines of the proof, we find for the normal components the complementarity conditions bn (λ, u) = 0 = bn (λl , ul ),

(4.2)

in addition to λn ∈ M + , λnl ∈ Ml+ and −[un ], −Πl [unl ] ∈ W + . We note that due to the possibly non-matching meshes, −[unl ] is, in general, not in W + . The friction law (4.1) guarantees that the tangential components satisfy F[ut ] ds, bt (λl , ul ) = FΠl [utl ]l ds, (4.3) bt (λ, u) = ΓC

ΓC

602

B. Wohlmuth

(a)

(b)

(c)

vlt

vlt l

vlt

vlt

vlt − vlt l

Figure 4.2. Absolute value in 2D · (a), discrete absolute value · l (b) and the difference · − · l (c). (a)

(b)

Normal part

Tangential part

20

10

0

5

−20

0

F*gf −F*gf

−40

(−1)*gn

−60

λ*n 4 −10 *(us*n)

0

0.5

1

1.5 2 x−coordinate

λ*t 4 10 *(us*t)

−5

2.5

3

−10 0

0.5

1

1.5 2 x−coordinate

2.5

3

Figure 4.3. Numerical results for a contact problem with Tresca friction: normal displacement and contact pressure (a) and surface traction in normal and tangential direction (b).

where the mesh-dependent Euclidean norm · l is defined by Πl [utl ]l := αtp φp ∈ Wl+ . s p∈PC;l

Figure 4.2 illustrates the difference between · and · l . The dashed line shows an element in Wl if ΓC is a straight line. It is easy to verify that Πl [utl ] − Πl [utl ]l is equal to zero at all vertices of the slave-side mesh and that, in general, we do not have Πl [ut ] ∈ Wl . Moreover, we find that Πl [utl ] − Πl [utl ]l ≤ 0. Figure 4.3 shows a simple numerical example for a contact problem with a non-constant Tresca friction bound and a non-zero gap. Further, we observe that for such examples the discrete complementarity conditions hold true. Figure 4.3(a) shows that the coefficients γpn are non-zero only on the actual discrete contact zone and thus (4.2) is satisfied. In Figure 4.3(b), the tangential displacement vanishes if the tangential stress component is strictly below its given bound, and thus (4.3) holds. In addition to the best approximation properties of the discrete Lagrange multiplier space (3.4), our proof relies on the properties of Πl and Π∗l : see

603


Wohlmuth (2001). For 1 ≤ s ≤ 2, we have 3

µ − Π∗l µ− 1 ;ΓC ≤ Chs−1 |µ|s− 3 ;ΓC , l

µ ∈ H s− 2 (ΓC ),

|w|s− 1 ;ΓC , w − Πl w 1 ;ΓC ≤ Chs−1 l

w ∈ H s− 2 (ΓC ).

2

2

2

2

(4.4a)

1

(4.4b)

Remark 4.1. We note that our regularity assumptions on the data and geometry guarantee that for u ∈ (H s (Ω))d , 1 ≤ s ≤ 2, we automatically have λ = −σ(us )n ∈ (H s−3/2 (ΓC ))d , and moreover |λ|s−3/2;ΓC ≤ C|u|s;Ω . For s = 2 this is obvious; for s = 1 this does not hold for all f ∈ V0 and fN ∈ {w = v|ΓN , v ∈ V0 }. However, by increasing the regularity of the data, as we do here, this holds true with a constant depending on Creg (see (2.4)), and then a standard interpolation argument yields the result for all s ∈ [1, 2]. 4.1. Upper bound for the discretization error The starting point is the following abstract lemma. A similar lemma can be found in Hild and Laborde (2002) without friction and with quadratic finite elements associated with standard Lagrange multipliers. Here, we also have to consider the friction part and take into account the tangential component of the bilinear form b(·, ·). Introducing the error El := (u − ul , λ − λl ) and its associated norm El 2V×M := u−ul 21;Ω +λ−λl 2−1/2;ΓC , the standard saddle-point theory and the complementarity conditions (4.2) provide a first a priori result. Lemma 4.2. Let (u, λ) ∈ V×M(F) be the solution of (2.15) with M(λn ) replaced by M(F) and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9) with Ml (λnl ) replaced by Ml (F). Then, we have El V×M ≤ C inf u − vl 1;Ω + inf λ − µl − 1 ;ΓC vl ∈Vl

µl ∈Ml

2

1

1 + max bn (λl , u), 0 2 + max bn (λ, ul ), 0 2

1 + max bt (λl − λ, u), 0 2 + inf s [ut ] − wl 1 ;ΓC . wl ∈Wl

2

Proof. Introducing el := u−ul , we find for the error el in the energy norm, and for vl ∈ Vl a(el , el ) = a(el , u − vl ) − b(λ, vl − ul ) + b(λl , vl − ul ) = a(el , u − vl ) − b(λ − λl , vl − u) − b(λ − λl , el ). Then Korn’s inequality, which holds on both subdomains by assumption, and the continuity of the bilinear forms a(·, ·) and b(·, ·) yield an upper bound for the H 1 -error of the displacement:

el 21;Ω ≤ C (el 1;Ω + λ − λl − 1 ;ΓC ) u − vl 1;Ω − b(λ − λl , el ). (4.5) 2

604

B. Wohlmuth

Using standard techniques from the saddle-point framework and applying the discrete inf-sup condition (3.5), we get µl − λl − 1 ;ΓC ≤ C sup 2

wl ∈Vl

b(µl − λl , wl ) wl 1;Ω

b(µl − λ, wl ) + a(ul − u, wl ) wl 1;Ω wl ∈Vl

≤ C µl − λ− 1 ;ΓC + ul − u1;Ω . = C sup

2

Then the triangle inequality and Young’s inequality applied on λ−λl − 1 ;ΓC 2 and (4.5), respectively, give El 2V×M ≤ C inf u − vl 21;Ω + inf λ − µl 2− 1 ;Γ − b(λ − λl , el ) . vl ∈Vl

µl ∈Ml

2

C

We use the additive decomposition of b(·, ·) into bn (·, ·) + bt (·, ·) and recall (4.2) b(λl − λ, el ) = bn (λl , u) + bn (λ, ul ) + bt (λl − λ, u) − bt (λl − λ, ul ). Each of the first three terms on the right can be bounded by the maximum of zero and the term itself. To bound the last term, we use (4.3) and the H 1/2 -stability (see (4.4b)) of the mortar projection Πl defined by (3.10a). For all µl ∈ Ml and wl ∈ Wls , we have bt (λ − λl , ul ) = λt − λtl , [utl ] − Πl [utl ]ΓC + λt − λtl , Πl [utl ]ΓC ≤ λt − µl , [utl ] − Πl [utl ]ΓC + F, Πl [utl ] − Πl [utl ]l ΓC ≤ λt − µl , [utl ] − Πl [utl ]ΓC ≤ Cλ − µl − 1 ;ΓC (u − ul 1;Ω + [ut ] − wl 1 ;ΓC ). 2

2

Now Young’s inequality gives the required bound. The first two terms in the upper bound of Lemma 4.2 are the best approximation errors. They reflect the quality of the approximation of the spaces Vl and Ml . The third, fourth and fifth term are consistency errors of the approach. We remark that the term max(bn (λ, ul ), 0) takes into account the discrete penetration of the two bodies on the actual contact set. The term max(bn (λl , u), 0) can be greater than zero if the discrete Lagrange multiplier λnl is negative on a part of the actual contact set. We recall that Ml+ is not a subspace of M + , and thus λnl does not have to be non-negative. To some extent this term measures the non-conformity of λnl with respect to the physical requirement of a positive contact pressure. The fifth term satisfies max(bt (λl − λ, u), 0) ≤ max( ΓC (λtl − F)ut ds, 0). Using that the maximum of a nodal dual Lagrange multiplier is larger than one, γ tp ≤ F does not necessarily give λtl ≤ F, and thus this term is in general non-


605

zero and measures the violation of the friction law. Finally, the last term does not appear for contact problems without friction. A closer look into the proof reveals that the H 1/2 -norm estimate on ΓC is too pessimistic for Coulomb problems, and it would be sufficient to consider the H 1/2 -norm on the actual contact zone. Moreover, this term does not occur if we work with matching meshes. 4.2. Optimal a priori estimates To prove optimal a priori error estimates under the H s -regularity assumption for the displacements u with 1 ≤ s ≤ 2, we have to consider in more detail the three terms in the upper bound of Lemma 4.2 which involve the bilinear form b(·, ·). We now give three lemmas providing upper bounds for these consistency errors. Lemma 4.3. Let (u, λ) ∈ V×M(F) be the solution of (2.15) with M(λn ) replaced by M(F) and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9) with Ml (λnl ) replaced by Ml (F). Under the regularity assumption u ∈ (H s (Ω))d , 1 ≤ s ≤ 2, we then have the a priori error estimate

2(s−1) 2 (s−1) |u|s;Ω + hl |u|s;Ω u − ul 1;Ω . bn (λ, ul ) ≤ C hl Proof. For standard Lagrange multipliers, we refer to Hild and Laborde (2002). Although our dual basis functions of Ml are not positive, we can apply the same techniques. Using the discrete saddle-point formulation (3.9) and the definition of the mortar projection, we find, in terms of the approximation properties (4.4) and Remark 4.1, the upper bound bn (λ, ul ) = λn , [unl ] − Πl [unl ] + Πl [unl ]ΓC ≤ λn , [unl ] − Πl [unl ]ΓC ≤ λn − Π∗l λn , [unl ] − Πl [unl ]ΓC ≤ λn − Π∗l λn − 1 ;ΓC [unl ] − Πl [unl ] 1 ;ΓC 2 2

n [u ≤ Chs−1 |λ | ] − [u ] + [u ] − Π [u ] 3 1 1 n s− ;ΓC n n n l l ;Γ ;Γ l 2 2 C 2 C

s−1 2(s−1) 2 |u|s;Ω . ≤ C hl |u|s;Ω u − ul 1;Ω + hl Before we focus on the terms bn (λl , u) and bt (λl − λ, u), we consider a non-linear quasi-projection operator which preserves sign. This type of operator was originally introduced in Chen and Nochetto (2000). We also refer to Nochetto and Wahlbin (2002) for a negative result on the existence of higher-order sign-preserving operators and for a detailed discussion of the special role of extreme points. Let Sl : W −→ Wls be a Clément-type operator which is defined node-wise by 1 int Sl w(p) := w ds, p ∈ PC;l . |Bp (creg hl )| Bp (creg hl )

606

B. Wohlmuth

int := {p ∈ P s , p ∈ ∂Γ }, and for all nodes p ∈ ∂Γ , we use Here PC;l C C C;l a locally defined value depending only on the values of w restricted to Bp (creg hl ) ∩ ΓC such that Sl is L2 -stable and reproduces polynomials of degree one. Standard arguments show that for 1 ≤ s ≤ 2 we get

Sl w − w 1 ;ΓC ≤ Chs−1 |w|s− 1 ;ΓC . l 2

2

More importantly, Sl w preserves the sign of w ∈ W + in the mesh-dependent + that interior Γint s \P int supp φp ), i.e., we have for w ∈ W C;l := ΓC \ (∪p∈PC;l C;l Sl w(p) ≥ 0 for p ∈ P int . C;l

In terms of the linear operator Sl , we define the non-linear operator Sl : Sl w(p) supp φp ⊂ supp w or p ∈ ∂ΓC , s Sl w(p) := p ∈ PC;l , (4.6) 0 otherwise,

the definition of which guarantees that supp Sl w ∩ Γint C;l ⊂ supp w. Moreover, for w ∈ W + we have Sl w(p) ≥ 0 for p ∈ ∂ΓC . Assumption 4.4. Let us define Σnl := {x ∈ ΓC | dist (x, ∂Bn ) ≤ 2hl }, where Bn is the actual contact zone, i.e., Bn := supp λn . Then, we assume that Σnl and Bn are compactly embedded in ΓC and Γint C;l , respectively, and moreover that s− 12

[un ]0;Σnl ≤ Chl

|[un ]|s− 1 ;ΓC . 2

Let us briefly comment on different aspects of this assumption. We note that for hl small enough, Σnl and Bn are compactly embedded in ΓC and Γint C;l , respectively, due to the assumption that Bn is compactly embedded in ΓC . This assumption can be weakened, but then the notation would s−1/2 become more technical. Setting Bnc := ΓC \ Bn and defining H00 (Bnc ) := {w ∈ L2 (Bnc ) | w = v|Bnc for v ∈ H s−1/2 (ΓC ) and supp v ⊂ Bnc }, we get s−1/2

[un ] ∈ H00 (Bnc ) if u ∈ (H s (Ω))d . Now, if Bnc is regular enough, the assumption is followed by a Poincaré–Friedrichs-type argument, together with suitable interpolation and a scaling. We refer to Li, Melenk, Wohlmuth and Zou (2010), where similar estimates for interfaces have been used, and to Melenk and Wohlmuth (2011), where these types of estimates are used to obtain quasi-optimal a priori L2 -norm estimates for the Lagrange multiplier in a linear mortar setting. In particular, an order-hl estimate is given for H 1 -functions with vanishing trace. The assumption is naturally satisfied if Bn is regular enough. If the boundary ∂Bn is smooth enough, we can locally flatten ∂Bn , use the fact that Sobolev spaces are invariant under smooth changes of variables and apply the 1D Sobolev embedding result vL∞ (I) vH s (I) , s > 12 recursively, where I is a fixed interval and v ∈ H s (I) (see, e.g., Adams (1975)).

607


(b)

4hl

Bn

ΓC

∂Bn

Σnl

Figure 4.4. Actual contact zone Bn (a) and boundary strip Σnl (b).

Figure 4.4 illustrates the definition of the strip Σnl which has a diameter of 4hl perpendicular to ∂Bn . In terms of these preliminary considerations, we can show the following bound for the consistency error of λnl . Lemma 4.5. Let (u, λ) ∈ V×M(F) be the solution of (2.15) with M(λn ) replaced by M(F) and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9) with Ml (λnl ) replaced by Ml (F). Under Assumption 4.4 and the regularity assumption u ∈ (H s (Ω))d , 1 ≤ s ≤ 2, we then have the a priori error estimate |u|s;Ω λ − λl − 1 ;ΓC . bn (λl , u) ≤ Chs−1 l 2

Proof.

The operator Sl yields Sl [un ] − [un ] 1 ;ΓC ≤ Chs−1 |[un ]|s− 1 ;ΓC ≤ l 2

2

|u|s;Ω for 1 ≤ s ≤ 2. Due to the fact that λnl ∈ M + with γpn = 0 for Chs−1 l p ∈ ∂ΓC , we find λnl Sl [un ] ds = γpn Sl [un ](p)mp ≤ 0. ΓC

int p∈PC;l

Then, (4.2), the construction of Sl and the assumption Bn ⊂ Γint C;l yield λnl [un ] ds ≤ λnl ([un ] − Sl [un ]) ds bn (λl , u) = ΓC

ΓC

=

λnl

− λn , [un ] − Sl [un ]ΓC ≤ λnl − λn − 1 ;ΓC [un ] − Sl [un ] 1 ;ΓC

≤

λnl

− λn − 1 ;ΓC ([un ] − Sl [un ] 1 ;ΓC + Sl [un ] − Sl [un ] 1 ;ΓC ).

2

2

2

2

2

As already noted the linear operator Sl has best approximation properties and thus it is sufficient to consider the second term on the right in more detail. The properties of the non-linear operator Sl defined by (4.6) play a crucial role. We observe that Sl [un ] and Sl [un ] coincide in the two dark grey-shaded regions of Figure 4.4(a) and note that Sl [un ]− Sl [un ] = 0 on ΓC \Σnl . For the

608

B. Wohlmuth

(a)

(b)

[ut ] = α2 λt

[ut ] = α2 λt

−(λt + [ut ])

−(λt + [ut ]) 2F

2F

Figure 4.5. Friction cone for d = 2 (a) and d = 3 (b).

term Sl [un ] − Sl [un ] 1 ;ΓC , we can now apply a standard inverse inequality 2 and get C Sl [un ] − Sl [un ]20;ΓC hl C ≤C hd−2 (Sl [un ](p))2 ≤ [un ]20;Σn , l l hl

Sl [un ] − Sl [un ]21 ;Γ ≤ 2

C

p∈PΣn l

where PΣnl := {p ∈ such that supp φ ⊂ Σnl }. Now Assumption 4.4 can be applied and yields the required bound. s PC;l

Now we combine the previous results and formulate a first optimal a priori error estimate for a two-body contact problem with no friction. Theorem 4.6. Let (u, λ) ∈ V × M+ be the solution of (2.15) and let (ul , λl ) ∈ Vl × M+ l be the solution of the discrete formulation (3.9) with ν = 0. Under the Assumption 4.4 and the regularity assumption u ∈ (H s (Ω))d , 1 ≤ s ≤ 2, we then have the a priori error estimate u − ul 1;Ω + λ − λl − 1 ;ΓC ≤ Chs−1 |u|s;Ω . l 2

Proof. Using the well-known approximation property for the spaces Vl and Ml , the proof is a direct consequence of Lemmas 4.2–4.5 by applying Young’s inequality and noting that λt = λtl = 0. For a non-trivial given friction bound F, the situation is more complex, and moreover there is a substantial difference between the two- and three-dimensional setting. In 2D, the tangential stress component can be identified with a scalar-valued functional, and we can follow the proof of Lemma 4.5. Figure 4.5 illustrates the difference. In 2D, the tangential surface traction λt = λt t of a regular solution is either λt = F or λt = −F for sliding nodes. Moreover, the tangential displacement [ut ] = [ut ]t can be − + separated into two parts [ut ] = max(0, [ut ]) + min(0, [ut ]) =: u+ t − u t , ut ,


(b)

609

(c)

Tangential part F

200

−0.1

0.6

−F

150

λ*t 5000*[u]*t

100

0.3

−0.2

0

−0.3

50 0 −50

−0.3

−100

−0.4

−150 −200 0.5

−0.6 1 x −coordinate

1.5

−0.6

−0.3

0

0.3

0.6

−0.5

0.2

0.4

1

Figure 4.6. Tangential displacement for contact problems with friction: d = 2 (a) and d = 3 (b,c). + u− t ∈ W : see Figure 4.6(a). In 3D, the tangential part [ut ] is a vectorvalued function, for which we cannot apply the friction law component-wise. In Figure 4.6, we show typical numerical results for a contact problem with friction. We note that for these considerations there is no difference between a Tresca and a Coulomb problem. In Figure 4.6(a) a Hertz contact problem with a given but non-constant friction bound is simulated in 2D. In Figure 4.6(b), we show a visualization of the tangential displacement for d = 3 and Coulomb friction. The nodes in the centre marked with black bullets are sticky and do not carry a relative tangential displacement. For these nodes the tangential component of the stress satisfies the inequality of the Coulomb law strictly. The nodes situated in the outer ring do slide, and the sliding direction is possibly changing from node to node. As can be seen in the zoom in Figure 4.6(c), the sliding direction is, as required by the friction law, opposite to the tangential stress.

Assumption 4.7. (2D setting) Σ±;t l

Let us define

:= {x ∈ ΓC | dist (x, ∂Bt± ) ≤ 2hl },

where ∂Bt+ ∪ ∂Bt− is the boundary of the actual sticky zone. More precisely, ±;t and Bt± are we set ∂Bt± := ∂ supp u± t ∩ ΓC . Then, we assume that Σl compactly embedded in ΓC , and moreover that for p ∈ ∂ΓC ∩ supp u± t we have γpt = ±F, s− 12

u± t 0;Σ±;t ≤ Chl l

|[ut ]|s− 1 ;ΓC . 2

We note that for a Coulomb problem, we have Bt ⊂ Bn , and thus Bt is automatically compactly embedded ΓC if Bn is so. This is not necessarily the case for a Tresca friction problem. We point out that this assumption rules out the case s = 2 and λt = λt t with λt being a function which − jumps from plus to minus of the friction bound. In that case u+ t and ut 3/2 have a lower regularity than ut . To be more precise ut ∈ H (ΓC ), whereas

610

B. Wohlmuth

3/2− (Γ ) for all > 0, and thus only an order h u± can be expected C t ∈H l to hold true. Figure 4.7 shows the numerical solution for a Coulomb problem in 2D with two different friction coefficients. In Figure 4.7(a) the case ν = 0.8 is shown whereas in Figure 4.7(b,c) the case ν = 0.3 is presented. The close-up in Figure 4.7(c) reveals that ut is zero not only in the centre point but also in a non-trivial sub-interval, and thus Assumption 4.7 is satisfied for both cases. The following lemma is the counterpart of Lemma 4.5 for the tangential − component. Due to the partition ut = u+ t − ut it only holds true for d = 2. 3/2−

Lemma 4.8. Let (u, λ) ∈ V×M(F) be the solution of (2.15) with M(λn ) replaced by M(F) and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9) with Ml (λnl ) replaced by Ml (F). Under Assumption 4.7 and the regularity assumption u ∈ (H s (Ω))2 , 1 ≤ s ≤ 2, we then have the a priori error estimate for d = 2: bt (λl − λ, u) ≤ Chs−1 |u|s;Ω λ − λl − 1 ;ΓC . l 2

Proof. The proof follows the lines ofthe proof of Lemma 4.5. We start with − t the observation that bt (λl − λ, u) = ΓC (λtl − F)u+ t ds − ΓC (λl + F)ut ds, − where λtl := λtl t. Now we apply the operator Sl to u+ l and ul . Then, under Assumption 4.7 we get + − − s−1 u+ [un ]s− 1 ;ΓC . t − Sl ut 1 ;ΓC + ut − Sl ut 1 ;ΓC ≤ Chl 2

2

2

The construction of Sl yields that + − − t bt (λl − λ, u) = λtl − λt , u+ t − Sl ut + λl − λt , Sl ut − ut − t + λtl − λt , Sl u+ t − λl − λt , Sl ut

≤ Chs−1 [un ]s− 1 ;ΓC λtl − λt − 1 ;ΓC l 2

+

λtl

−

F, Sl u+ t

2

−

λtl

+ F, Sl u− t

= Chs−1 [un ]s− 1 ;ΓC λtl − λt − 1 ;ΓC l 2 2 + (γpt − F)Sl u+ t (p)mp s p∈PC;l

−

(γpt + F)Sl u− t (p)mp .

s p∈PC;l

Moreover, by definition of Ml (F), it is easy to see that γpt − F ≤ 0 and


(b)

611

(c)

Lagrange multiplier

Lagrange multiplier

Lagrange multiplier 6

15

15

λ

λ

λ

τ

2

150*uτ

10

150*uτ

5

4

n

λτ

n

10

± F*λ

± F*λn

n

0

5

0

−2

−5

0

−10

0.2

0.4

0.6

0.8

1

0

F*λ −|λ | n

τ

± F*λn

−5 −15 0

λτ 40000*uτ

−4

0.2

0.4

0.6

0.8

1

−6 0.495

0.5

0.505

Figure 4.7. Tangential displacement for different friction coefficients.

γpt + F ≥ 0, and thus we get − t (γpt − F)Sl u+ t (p) − (γp + F)Sl ut (p) mp ≤ 0. int p∈PC;l

s \P int , we have that (γ t −F)S u+ (p) = 0 = (γ t +F)S u− (p). We For p ∈ PC;l l t l t p p C;l note that these last arguments are typical for d = 2 but cannot be applied in 3D.

We now combine the previous results and formulate an optimal a priori error estimate for a two-body contact problem with Tresca friction in 2D. Theorem 4.9. Let (u, λ) ∈ V × M(F) be the solution of (2.15), with M(λn ) replaced by M(F), and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9), with Ml (λnl ) replaced by Ml (F). Under Assumptions 4.4, 4.7 and the regularity assumption u ∈ (H s (Ω))2 , 1 ≤ s ≤ 2, we then have the a priori error estimate for d = 2: u − ul 1;Ω + λ − λl − 1 ;ΓC ≤ Chs−1 |u|s;Ω . l 2

Proof. Using the well-known approximation property for the spaces Vl , Ml and Wls , the proof is a direct consequence of Lemmas 4.2–4.8 by applying Young’s inequality. Let us briefly comment on the three-dimensional case. Lemma 4.8 is the only one where we have explicitly used a 2D construction. All the other results hold true for d = 2 and d = 3. In 3D, we do still get a priori bounds for the discretization error, although the optimal order is lost if u ∈ (H s (Ω))d , 3/2 < s ≤ 2. We introduce the operators Z∗l : M → Wls and Zl : W → Wls by Z∗l := (Zl∗ )d and Zl := (Zl )d , Zl∗ : M → Wls and Zl : W → Wls : Zl∗ µ :=

µ, φp Γ C φp , mp s

p∈PC;l

Zl w :=

φp , wΓ C φp . mp s

p∈PC;l

(4.7)

612

B. Wohlmuth

Obviously, both Zl and Zl∗ are locally defined and reproduce constants. Moreover, Zl is L2 - and H 1 -stable. From the H 1/2 -stability we get the H −1/2 -stability of Zl∗ with the same stability constant Zl∗ µ− 1 ;ΓC = sup 2

w∈W

Zl∗ µ, wΓC µ, Zl wΓC = sup . w 1 ;ΓC w∈W w 1 ;ΓC 2

2

In terms of these operators, we obtain the following non-optimal a priori bound for a Tresca friction problem in 3D with non-trivial but constant friction coefficient. Theorem 4.10. Let (u, λ) ∈ V × M(F) be the solution of (2.15), with M(λn ) replaced by M(F), and let (ul , λl ) ∈ Vl × Ml (F) be the solution of the discrete formulation (3.9), with Ml (λnl ) replaced by Ml (F). If u ∈ (H s (Ω))d , 1 ≤ s ≤ 3/2, we then have the a priori error estimate u − ul 1;Ω + λ − λl − 1 ;ΓC ≤ Chs−1 |u|s;Ω . l 2

Proof. We have to re-examine the two terms bn (λl , u) and bt (λl − λ, u) in the upper bound of Lemma 4.2. To do so, we apply the operators Zl∗ and Z∗l and remark that Zl∗ λnl = p∈P s γpn φp ∈ W + and C;l γ tp φp − F ≤ γ tp φp − F ≤ Fφp − F = 0. Z∗l λtl − F = s s s p∈PC;l

p∈PC;l

p∈PC;l

These preliminary observations in combination with (3.7) yield for the normal part bn (λl , u) = λnl − Zl∗ λnl , [un ]ΓC + Zl∗ λnl , [un ]ΓC ≤ λnl − Zl∗ λnl , [un ]ΓC = λnl − Zl∗ λnl , [un ] − Π0;l [un ]ΓC s− 12

≤ Chl ≤

λnl − Zl∗ λnl 0;ΓC |[un ]|s− 1 ;ΓC

Chs−1 λnl l

2

−

Zl∗ λnl − 1 ;ΓC |u|s;Ω , 2

where Π0;l is the L2 -projection onto element-wise constants. Here we have also used additionally the inverse estimate for λnl − Zl∗ λnl 0;ΓC , which results from standard inverse estimates for finite elements and the fact that Zl∗ λnl ≥ 0. Keeping in mind that bt (λ, u) = ΓC F[u]t ds, the tangential part can be estimated in the same way: bt (λl − λ, u) = bt (λl − Z∗l λl , u) + bt (Z∗l λl − λ, u) λtl − Z∗l λtl − 1 ;ΓC |u|s;Ω . ≤ bt (λl − Z∗l λl , u) ≤ Chs−1 l 2

In a last step, we have to consider λl −

Z∗l λl − 1 ;ΓC 2

in more detail and


613

bound it. The stability of Z∗l in the H −1/2 -norm gives λl − Z∗l λl − 1 ;ΓC ≤ Cλl − λ− 1 ;ΓC + λ − Z∗l λ− 1 ;ΓC 2

2

2

≤ C(λl − λ− 1 ;ΓC + 2

hs−1 |u|s;Ω ). l

Now, Lemma 4.3 in combination with Young’s inequality yields the required a priori bound. Remark 4.11. We note that in contrast to Theorem 4.9 no additional assumptions on the actual contact zones are made in Theorem 4.10. The main advantage of the dual Lagrange multiplier space is the possibility of computing λl by a local post-process from the discrete displacement. Taking the local residual and using a simple scaling directly yield the coefficients. However, for the visualization in general Z∗l λl is plotted and not λl . However, both quantities have the same order of convergence. For 1 ≤ s ≤ 2 and u ∈ (H s (Ω))d we obtain in terms of the H −1/2 -stability of Z∗l and its approximation property λ − Z∗l λl − 1 ;ΓC ≤ λ − Z∗l λ− 1 ;ΓC + Z∗l (λ − λl )− 1 ;ΓC 2

2

≤ C(λ − λl − 1 ;ΓC + 2

2

hs−1 |u|s;Ω ). l

Remark 4.12. Quite often in the context of mortar methods (see, e.g., Braess and Dahmen (2002)), one prefers to work with a norm that is easier to handle than the H −1/2 -norm. Thus this is replaced by a weighted meshdependent L2 -norm, µ2Ml := hF µ20;F µ ∈ L2 (ΓsC ) or µ ∈ (L2 (ΓsC ))d . (4.8) s F ∈FC;l

Then all our theoretical results also cover λ − λl Ml if the regularity of the solution is good enough. The proof follows exactly the same lines as for the H −1/2 -norm and uses an inverse estimate. It is known from the linear mortar setting that in the case of weighted L2 -norms, a uniform inf-sup condition also holds. Moreover, one can replace, in the proof of the best approximation properties of the constrained space, the discrete harmonic extension by a discrete zero extension to the interior nodes. 4.3. Numerical results We note that in all our numerical results the mesh-dependent norm (4.8) has been used to measure the discretization error in the Lagrange multiplier and Z∗l λl has been used to make the plots. To illustrate the convergence rates of low-order finite elements numerically, we consider two simple twodimensional test settings. As a reference solution, we use the numerical solution on Tlmax +2 .

614

B. Wohlmuth

(a)

(b)

(c) Stress and jump (friction bound: 0.6)

m

Ω

φ

Ωs

Γ

0.6

λ

0.4

50*(us −um)

Displacement (friction bound: 0.6) 0.015

us

0.01

um h

h

h

h

h

0.005

0.2 0

0

−0.2

−0.005

−0.4

−0.01

−0.6 −0.5

0 Angle φ

0.5

−0.015

−0.5

0 Angle φ

0.5

Figure 4.8. Problem setting (a), tangential Lagrange multiplier (b) and tangential displacement (c) on ΓsC;l .

Our first example is a scalar-valued model problem for anti-plane friction; we refer to H¨ ueber et al. (2005b) for details regarding the problem formulation. The friction bound is set to F = 0.6. Figure 4.8 shows the geometry as well as the tangential Lagrange multiplier and the displacement on master and slave sides. In Figure 4.8(b), we can clearly observe the discrete complementarity (3.12) with νγpn being replaced by the given bound F. In Figure 4.8(c) the tangential displacements are shown. For γ tp strictly smaller than F, the two bodies have to stick together. Sliding is possible only on a part of the contact zone where γ tp = F. Table 4.1 shows the convergence rates for this simplified frictional contact problem. We show the L2 -norm and the H 1 -norm of the displacement error and the mesh-dependent norm of the error in the Lagrange multiplier. Our numerical results confirm the theoretical ones, and the computed rates are fairly close to the optimal order of convergence. Although we do not have a theoretical result for the L2 -norm, the numerical results also show a significantly better rate compared to the H 1 -norm. Standard duality techniques such as the Aubin–Nitsche approach are tricky to apply in the setting of variational inequalities since they depend on regularity assumptions of the dual problem. Our second test example is the classical Hertz problem (Hertz 1882, Johnson 1985, Kikuchi and Oden 1988) with Coulomb friction. Although our theoretical results do not cover the case of a Coulomb contact problem, Table 4.2 shows that also for this case we obtain almost optimal convergence rates. In that case, the convergence order for the error in the Lagrange multiplier in the weighted L2 -norm is close to the best approximation order of 3/2. Remark 4.13. This effect is also numerically well observed for mortar problems in the linear setting (Wohlmuth 2001). A theoretical analysis can be found in the recent contribution by Melenk and Wohlmuth (2011), where it is shown that quasi-optimal L2 -norm estimates for the Lagrange multiplier hold in the linear mortar setting under suitable regularity assumptions.

615

Numerical algorithms for variational inequalities Table 4.1. Convergence rates for a contact problem with given friction bound. Level 0 1 2 3 4 5 6

ul −uref 0;Ω uref 0;Ω

8.9343e−02 3.0960e−02 8.6519e−03 2.4272e−03 6.6657e−04 1.7918e−04 4.6522e−05

− 1.53 1.84 1.83 1.86 1.90 1.95


3.2317e−01 1.8933e−01 1.0343e−01 5.6431e−02 3.0414e−02 1.6132e−02 8.3113e−03

λl − λref ∗

− 0.77 0.87 0.87 0.89 0.91 0.96

7.7185e−02 2.2162e−02 8.1336e−03 3.7816e−03 1.3031e−03 3.5214e−04 1.6712e−04

− 1.80 1.45 1.10 1.54 1.89 1.08

Table 4.2. Convergence rates for a Hertz contact problem with Coulomb friction, ν = 0.5. Level 1 2 3 4 5 6


4.465867e−01 3.056095e−01 1.693210e−01 9.155008e−02 4.857727e−02 2.450933e−02

− 0.55 0.85 0.89 0.91 0.99

λl − λref ∗ 5.065628 2.381819e+01 1.037995e+01 3.909448e+00 1.493191e+00 0.523946e+00

− 1.09 1.20 1.41 1.39 1.51

Finally, we briefly comment on higher-order elements; see also Belhachmi and Ben Belgacem (2000) for an analysis and Fischer and Wriggers (2006) and Puso et al. (2008) for simulation results in applications, and the influence of the choice of the Lagrange multiplier space. Recently, hp-techniques have also been applied for contact problems in combination with boundary elements (Chernov, Maischak and Stephan 2008). The p-version in the boundary element method for contact is discussed in Gwinner (2009). From the algorithmic point of view, higher-order elements can be easily applied. As we will see, quadratic elements do yield higher-order a priori estimates, but optimal quadratic order cannot be achieved. This results from the fact that there exists no monotonicity-preserving operator of higher order; see also Nochetto and Wahlbin (2002). In addition, the solution of a contact problem is, in general, not in H 3 (Ω), and thus also, from the point of view of best approximation, no second-order error decay can be expected. Nevertheless, a higher-order a priori estimate can be obtained by replacing Vl by quadratic finite elements. Quadratic finite elements and linear dual

616

B. Wohlmuth

Table 4.3. Relative error for the displacement in the H 1 -norm for linear and quadratic finite elements. Level 0 1 2 3 4 5

(i, j) = (1, 1) 4.663632e−01 3.214737e−01 1.807130e−01 9.735853e−02 5.111965e−02 2.584391e−02

− 0.54 0.83 0.89 0.93 0.98

(i, j) = (2, 1) 3.159307e−01 1.592747e−01 6.777325e−02 2.992646e−02 1.340727e−02 −

− 0.99 1.23 1.18 1.16 −

(i, j) = (2, 2) 3.903263e−01 1.376072e−01 5.656398e−02 2.422295e−02 1.028243e−02 −

− 1.50 1.28 1.22 1.23 −

Table 4.4. Mesh-dependent L2 -error for the Lagrange multiplier in the linear and quadratic approach. Level 0 1 2 3 4 5

(i, j) = (1, 1) 5.845412e+01 4.999477e+01 2.121223e+01 8.378905e+00 3.269796e+00 1.168347e+00

− 0.23 1.24 1.34 1.36 1.48

(i, j) = (2, 1) 5.849757e+01 4.129640e+01 1.814467e+01 7.316218e+00 2.813967e+00 −

− 0.50 1.19 1.31 1.38 −

(i, j) = (2, 2) 1.323412e+02 3.621992e+01 1.389391e+01 5.230080e+00 2.015976e+00 −

− 1.87 1.38 1.41 1.38 −

Lagrange multipliers yield an order hs−1 , 1 ≤ s < 52 , upper bound for the discretization error if the solution is H s -regular. Replacing the linear Lagrange multiplier space by quadratic Lagrange multipliers does not give a higher order: see, e.g., Hild and Laborde (2002). Revising the proof of Theorem 4.9 shows that the crucial steps are Lemma 4.5 and Lemma 4.8. These parts do not yield estimates of order two even if the spaces used have higher-order best approximation properties. For a proof and more detailed numerical results, comparing quadratic finite elements with linear Lagrange multipliers and with quadratic Lagrange multipliers, we refer to H¨ ueber, Mair and Wohlmuth (2005a). As a test example, we choose the simple Hertz contact problem without friction. Table 4.3 shows the convergence rates for the relative H 1 -norm of the error in the displacement, whereas Table 4.4 refers to the error of the Lagrange multiplier in the mesh-dependent L2 -norm. Here, we illustrate the influence of linear and quadratic finite elements. The indices i = 1 and i = 2 stand for the use of standard conforming linear and quadratic finite elements for the displacement, respectively. The indices j = 1 and j = 2 indicate the


617

use of biorthogonal basis function of lowest and second order, respectively. We note that the pairing (i, j) = (1, 2) is not uniformly inf-sup stable with respect to the slave side and thus is not considered. Moreover j = 1 already gives a best approximation property of the Lagrange multiplier space of order 3/2, and thus we do not expect a qualitative increase in the case j = 2 compared to j = 1 for the Lagrange multiplier. Although the convergence order in the H 1 -norm for i = 2 is not equal to two, it is much higher than compared to i = 1. A more efficient strategy, however, is to combine higherorder elements in the interior with adaptive refinement techniques on the contact part. We refer to the recent hp-strategy for a simplified Tresca problem in 2D (Dörsek and Melenk 2010). The last test shows that the numerical solution is quite insensitive to the choice of the dual Lagrange multiplier basis. We test the discontinuous piecewise constant and linear one (see Figure 3.6(a,b)) and the continuous piecewise cubic one (see Figure 3.7(a)). Table 4.5 shows a comparison of the maximum contact pressure for different Lagrange multipliers. From the very first levels, the maximal value for all the three tested Lagrange multipliers is in very good agreement. Thus the choice of the Lagrange multiplier basis is not relevant as long as the conditions (3.2)–(3.5) are satisfied. Table 4.5. Maximum contact pressure for different low-order dual Lagrange multipliers. Level

Linear

Constant

Cubic

1 2 3 4 5 6 7 8

382.057 514.166 504.190 496.765 494.805 494.264 494.174 494.202

382.057 514.172 504.229 496.755 494.809 494.266 494.175 494.202

382.057 514.172 504.229 496.755 494.809 494.266 494.175 494.202

5. Semi-smooth Newton solver in space Early numerical approaches for two-body contact problems on non-matching meshes and for contact problems with Coulomb friction often tried to weaken the non-linearity by suitable fixed-point or decoupling strategies. Following the proof of existence, a Coulomb friction problem can be reduced to a sequence of simplified problems with given bound for the tangential traction,

618 (a)

B. Wohlmuth (b)

Error

Convergence rate 10

0

10

−2

10

−4

10

−6

10 10

fixed point full Newton

−8

−10

5

10 15 Iteration steps

20

Figure 5.1. Section view of the problem geometry (a) and convergence rate (b).

and thus fixed-point strategies naturally apply, where in each step a contact problem with given friction bound has to be solved numerically. Figure 5.1 shows the convergence rates of two different solvers for a Coulomb contact problem with a complex geometry in 3D. As expected, the simple fixedpoint approach has a linear convergence rate, whereas an alternative solver exhibits a super-linear rate. Using the concept of domain decomposition, a two-sided contact problem can be rewritten as a one-sided contact problem in addition to the equilibrium of contact forces. Thus Dirichlet–Neumann-type algorithms are suitable; see, e.g., Bayada, Sabil and Sassi (2002), Chernov, Geyn, Maischak and Stephan (2006), Chernov et al. (2008) and Krause and Wohlmuth (2002). More precisely, in each iteration step we solve on the master side a linear elasticity problem with given surface traction, and on the slave side we consider numerically a non-linear contact problem where the displacement of the master side acts as a rigid obstacle. The update of the interface data is realized globally after each cycle. From a theoretical point of view convergence can only be proved for a sufficiently small damping parameter (see Bayada, Sabil and Sassi (2008) and Eck and Wohlmuth (2003)), and in practice these methods require sophisticated damping strategies and are barely competitive. Figure 5.2 shows the undamped version applied to a long hexahedral bar between two cylinders. The length of the bar highly influences the convergence behaviour. If it is small, then the undamped algorithm shows fast convergence, whereas if the bar is long, no convergence at all is obtained unless a suitable damping parameter is applied. Although simple to apply, these coupling strategies give rise to inner and outer iteration schemes and are therefore quite expensive. Thus there is a strong need for efficient solvers which focus on all non-linearities at the same time and tackle the fully coupled system.


619

Figure 5.2. Oscillation of an undamped Dirichlet–Neumann contact solver.

Monotone multigrid methods have been shown to be very attractive due to guaranteed convergence, if the underlying system is equivalent to a constrained minimization problem. Early references on multigrid methods for variational inequalities or free boundary problems such as the obstacle problem are given by Brandt and Cryer (1983), Hackbusch and Mittelmann (1983), Hoppe (1987), Hoppe and Kornhuber (1994), Kornhuber (1994, 1996), Kornhuber and Krause (2001) and the monograph by Kornhuber (1997). Nowadays these techniques have been applied very successfully to more challenging contact problems including two bodies with non-matching meshes, finite deformations and complex geometries in 3D (Dickopf and Krause 2009a, Krause 2008, 2009, Krause and Mohr 2011, Wohlmuth and Krause 2003). Domain decomposition-based solvers such as FETI techniques are also widely applied. An excellent overview of these techniques applied to variational inequalities can be found in the recent monograph by Dostál (2009); see also the original research papers of Dostál, Friedlander and Santos (1998), Dost´ al, Gomes Neto and Santos (2000), Dost´ al and Horák (2003), Dost´ al, Hor´ ak, Kuˇcera, Vondrák, Haslinger, Dobiaˇs and Pták (2005), Dost´ al, Hor´ ak and Stefanica (2007, 2009) and Sch¨ oberl (1998). Different alternatives exist, e.g., interior point methods (Wright 1997), SQP algorithms (Pang and Gabriel 1993), the radial return mapping or the catching-up algorithm (Moreau 1977, Simo and Hughes 1998), as well as penalty or augmented Lagrangian approaches (Glowinski and Le Tallec 1989, Laursen 2002, Simo and Laursen 1992). We refer to the recent monograph by Ito and Kunisch (2008a) for an overview of Lagrange multiplier-based methods for variational problems. Here we choose an abstract and very flexible framework within which many different applications can be handled. The starting point is the observation that most inequality constraints can be equivalently stated in terms of a non-linear system. This holds true not only for contact problems but also

620

B. Wohlmuth

for other problems involving variational inequalities. The weak form of the underlying partial differential equation and the non-linear complementarity (NCP) function then form a coupled non-linear system on which a Newton scheme can be applied and easily combined with fast iterative solvers, such as multigrid (Hackbusch 1985) or domain decomposition techniques (Toselli and Widlund 2005), for the consistent linearized system. Due to the characteristic lack of classical differentiability of the NCP function, the assumptions for standard Newton methods (Deuflhard 2004) are not satisfied, but the so-called semi-smooth Newton methods (Facchinei and Pang 2003a, Ito and Kunisch 2003, Hinterm¨ uller, Kovtunenko and Kunisch 2004) can be applied; see also Pang (1990) and Pang and Qi (1993). Early applications of this type of method can be found in the engineering literature. The classical radial return mapping in plasticity (see Moreau (1977) for an early variant of it) can be handled within this abstract framework, but it has also been successfully applied to contact problems for roughly two decades (Alart and Curnier 1991, Christensen 2002a, 2002b, Christensen and Pang 1999, De Saxcé and Feng 1991, Simo and Laursen 1992). It is well established that the semi-smooth Newton method converges locally super-linearly: see, e.g., the monograph by Facchinei and Pang (2003b). Global convergence can be shown for some special cases, e.g., the Laplace operator-based obstacle problem: see Ito and Kunisch (2008a). A simplified Signorini problem has been analysed in Ito and Kunisch (2008b). For contact problems no global convergence holds, but the pre-asymptotic robustness can, in particular in 3D, be widely improved by a suitable local rescaling of an NCP function and a local node-wise regularization of the Jacobian. In each Newton step, the contact condition and its boundary type have to be updated locally, e.g., a Robin-type condition applies in the case of a sliding node. As a consequence, the semi-smooth Newton method can be implemented as a primal–dual active set strategy (Hinterm¨ uller, Ito and Kunisch 2002, H¨ ueber, Stadler and Wohlmuth 2008, Ito and Kunisch 2004). The use of active sets allows for local static condensation of either the dual variable or the corresponding primal degrees of freedom, such that only a system of the size of the displacement has to be solved in each Newton step. One of the attractive features of this class of algorithms is that it can be easily combined with other types of non-linearities, such as non-linear material laws, for example. No inner and outer iteration loop is required even in the presence of the different types of non-linearities. 5.1. Equivalent formulation as a non-linear equation system In a first step, we rewrite the inequality constraints associated with the discrete static Coulomb friction problem as a non-linear system. For simplicity of notation, we present the algebraic form only for homogeneous


621

Dirichlet boundary conditions. Thus, after discretization the weak formulaV M tion (3.9) has the following algebraic structure. Find ( ul , λl ) ∈ RNl ×RNl , NlV := dimVl , NlM := dimMl , such that l λl = l u l + B fl , A l (λl , u l ) = 0. C

(5.1)

Here we use the same symbol for λl ∈ Ml and its vector representation M l ∈ RNlV ×NlV , B l ∈ RNlV ×NlM and the rightλl ∈ RNl . The matrices A V hand side fl ∈ RNl result from the bilinear forms al (·, ·), bl (·, ·) and the linear form fl (·), respectively, and are assembled with respect to the nodal s basis functions φp , ψp . We note that d#PC;l =: Nls = NlM . The NCP l (·, ·) ∈ RNlM reflects the non-penetration condition (3.11) and function C the static Coulomb law (3.12). It has a node-wise form and can be written s . p (γ p , u l (λl , u l ))p = C l ) ∈ Rd , p ∈ PC;l as (C There exist many different choices for NCP functions in the literature. Quite often generalizations of the Fischer–Burmeister approach (Fischer 1992) are used. We refer to Chen, Chen and Kanzow (2000) for a penalized version and to Chen (2007), Hu, Huang and Chen (2009), Kanzow, Yamashita and Fukushima (1997) and Sun and Qi (1999) for the introduction and analysis of a family of NCP functions. An excellent overview can be found in the monograph by Facchinei and Pang (2003a). Here we use a different type of NCP function, which is based not on the root function but on the max function: see Alart and Curnier (1991). The main advantage of this type of NCP function is that the generalized derivatives are extremely easy to compute. This is quite important if not only the simple non-penetration law is considered but also more complex situations such as Coulomb friction or finite deformations. We refer to the series of recent papers by Gitterle et al. (2010) and Popp et al. (2009, 2010), where the concept of dual Lagrange multipliers and semi-smooth Newton schemes have been applied to finite deformation problems. Moreover, it can be implemented easily in terms of an active set strategy which switches off and on different types of non-linear boundary conditions on the possible contact zone. As is standard in the case of radial return mappings, we introduce trial test vectors. Here we need two of them, one in the normal and one in the tangential direction, that is, n γp;tr := γpn +

2µcn n (αp − gp ), mp

γ tp;tr := γ tp +

2µct t α , mp p

(5.2)

where cn and ct are two positive mesh-independent constants. Keeping s , are the coefficients with respect to the nodal in mind that αp , p ∈ PC;l basis functions on the slave side of Πl [ul ], we find that αp is a function

622

B. Wohlmuth

l . In terms of these trial vectors, which depend on γ p and u l , one can of u easily reformulate the inequality constraints (3.11) and (3.12) as equality conditions. Lemma 5.1. The inequality constraints (3.11) and (3.12) for each node s p (γ p , u l ) := Cp (γ p , αp ( p ∈ PC;l are equivalent to C ul )) = 0, where the n n n normal component Cp (γp , αp ) of Cp (γ p , αp ) is given by n Cpn (γpn , αpn ) := γpn − max(0, γp;tr ),

(5.3)

and the tangential component of Cp (γ p , αp ) is defined by Cpt (γ p , αp ) :=  t  γp Fp;tr = 0, s t

t (5.4) Fp;tr  max γ p;tr , Fp;tr γ p − min 1, t otherwise, γ tp;tr γ p;tr n ). where s ≥ 0 stands for a scaling parameter and Fp;tr := ν max(0, γp;tr

Proof. The proof is a rather straightforward calculation, but for convenience of the reader it is given. If Cpn (γpn , αpn ) = 0, then obviously γpn ≥ 0. If γpn = 0, we get αpn −gp ≤ 0, and for γpn > 0, we have αpn = gp , and thus (3.11) is satisfied. Let Cpn (γpn , αpn ) = 0 and Cpt (γ p , αp ) = 0; then Fp;tr = νγpn . In the case Fp;tr = 0, s (3.12) trivially holds. For Fp;tr > 0, the scaling factor

max γ tp;tr , Fp;tr is non-zero, and thus from Cpt (γ p , αp ) = 0 it follows that Fp;tr t γt . γ p = min 1, t γ p;tr p;tr If γ tp;tr ≤ νγpn , then γ tp − γ tp;tr = 0, and thus αtp = 0 and γ tp ≤ νγpn . For γ tp;tr > νγpn , we get γ tp = νγpn . Moreover, γ tp;tr = (1 + β)γ tp with some positive value for β. Using the definition (5.2) of γ tp;tr , we find that αtp points in the same direction as γ tp and thus (3.12) is satisfied. Let (3.11) be true; then a straightforward computation shows that we have Cpn (γpn , αpn ) = 0. The situation is more complex for the discrete Coulomb law. If (3.11) and (3.12) hold, then either αtp = 0 or it points in the same direction as γ tp or γpn = 0. In the first two cases, we can thus write γ tp;tr = (1 + β)γ tp with some non-negative β. For β = 0, we trivially find that γ tp;tr = γ tp and thus Cpt (γ p , αp ) = 0. For β > 0, we have γ tp = νγpn and moreover γ tp;tr > νγpn , yielding Fp;tr 1 1+β t t γ tp;tr = γ tp − γ tp;tr = γ tp − γ = 0. γ p − min 1, t γ p;tr 1+β 1+β p The case that γpn = 0 yields Fp;tr = 0 and γ tp = 0 and thus Cpt (γ p , αp ) = 0.

623


Remark 5.2. We note that (3.11) and (5.3) are equivalent but (3.12) and (5.4) are not. In particular, the trial vector in the normal direction enters into the definition of Cpt (γ p , αp ). 5.2. Basis transformation: from nodal to constrained As already mentioned, from the algebraic point of view each two-body problem can be rewritten formally as a one-body problem by introducing a new basis. The weak inequalities then result in node-wise inequalities for the slave side, even in the case of non-matching meshes. One of the main advantages of the choice of a dual Lagrange multiplier space is that the basis transformation is a local operator with a sparse matrix representation. This is not the case for standard Lagrange multipliers. p (γ p , u l ) only depends on the nodal coefficient of λl at p, Although C l . A more local it does not have this simple structure with respect to u and thus implementationally attractive representation can be obtained by a suitable basis transformation. This transformation was introduced in Wohlmuth and Krause (2001) to construct a multigrid scheme for mortar finite element discretizations. The V-cycle analysis of a level-independent convergence rate can be found in Wohlmuth (2005). Introducing the nodal block structure for the displacement l = ( m sl ) , u uil , u l ,u

i

il ∈ RNl , u

m

Nl m u , l ∈R

s

sl ∈ RNl , u

m and N i := N V − N m − N s (see also Figure 5.3), we where Nlm := d#PC;l l l l l l and B l : obtain the following structure for A     i,m A i,s i,i A A 0 l l l  l =  l = −M , m,i A m,m A 0 , B A l l l s,s s,i Dl 0 A A l l

where the entries of the coupling matrices Dl and Ml are defined by s m dpp := mp Idd×d , mpq := ψp φq ◦ χl ds Idd×d , p ∈ PC;l , q ∈ PC;l . ΓsC;l

The diagonal structure of Dl is a consequence of (3.6). In the case of standard Lagrange multipliers Dl has the band-structure of a (d−1)-dimensional mass matrix, and thus Dl−1 is dense. We recall that, with respect to the nodal basis functions, the coefficient αp in (3.11) and (3.12) does depend on the coefficients of the master and the slave nodes. The definition (3.10) of the mortar projection yields that s m sl − Dl−1 Ml u =: α = u (αp )p∈PC;l l .

To eliminate the dependence of the coefficient vector on the master side, we

624

B. Wohlmuth Slave side Dirichlet boundary

Inner nodes Master side

Master nodes Slave nodes

Dirichlet boundary

Figure 5.3. Partitioning of the nodes into the three blocks. m , will be use a basis transformation. The nodal basis function φp , p ∈ PC;l replaced by a constrained basis function φ˜p := φp + rpq φq , (5.5) s q∈PC;l

m . while all other basis functions remain unchanged, i.e., φ˜p := φp , p ∈ Pl \PC;l

Lemma 5.3. gives

s , p ∈ P m in (5.5) Setting rpq := ((Dl−1 Ml )qp )11 , q ∈ PC;l C;l

Πl [φ˜p ] = 0,

m p ∈ PC;l .

Proof. The definition of the mortar projection (3.10a) shows that Πl , restricted to Wls , is the identity and that m rpq φq , p ∈ PC;l Πl (φp ◦ χl ) = s q∈PC;l

and thus Πl [φ˜p ] =

rpq φq − Πl (φp ◦ χl ) = 0.

s q∈PC;l

Algebraically this basis transformation can be realized very efficiently in a local pre-process. The coefficients with respect to the constrained basis multiplied by the matrix   Id 0 0 Id 0 Ql =  0 (5.6) −1 0 Dl Ml Id ˜ φp , yield the coefficients with respect to the nodal basis. If vl = p∈Pl β p ˜ ˜ then vl = p∈Pl β p φp with (β p )p∈Pl = Ql (β p )p∈Pl . To obtain the stiffness matrices Al , Bl and the right-hand side fl with respect to the new l , B l and constrained basis, we have to apply Ql in a suitable way on A fl : Al = Q l Al Ql ,

Bl = Q l Bl ,

fl = Q l fl .


625

Due to the definition (5.6) of Ql , it is easy to verify that the block structure of the matrix Bl , with respect to this new basis, has the form Bl = (0, 0, Dl ) . The non-linear system to be solved is equivalent to (5.1), and can be written as A l u l + Bl λ l = f l , (5.7) s . Cp (γ p , up ) = 0, p ∈ PC;l s . Here we have used that, with respect to the new basis, αp = up , p ∈ PC;l We point out that the same notation for ul ∈ Vl is applied as for its coeffiV cient vector ul ∈ RNl with respect to the new constrained basis. Finally, we mention that, with respect to this new basis, the p-component of the NCP function only depends on the p-component of λl and ul . The coefficients in the new basis with respect to the slave nodes no longer specify the total nodal displacement but now describe the movement of the underlying finite element node relative to the master side. Having ul , the Lagrange multiplier can easily be obtained, from the node-wise residual, by a diagonal scaling i sm m ss s λl = Dl−1 (fls − Asi l ul − Al ul − Al ul ).

(5.8)

Remark 5.4. Such a basis transformation can be carried out for all types of Lagrange multiplier spaces as long as a discrete inf-sup condition holds. In most cases, it will result in a dense block corresponding to the possibly global character of the mortar projection. However, in our situation the basis transformation is a local operator for all space dimensions and thus inexpensive. Moreover, for uniformly stable pairs (Vl , Ml ), the basis transformation does not influence the order of the condition number of Al , which l . is then comparable to that of A 5.3. Semi-smooth Newton solver As can be easily seen from (5.3) and (5.4), the NCP function Cp (γ p , up ) is not globally but only piecewise smooth. Thus, to solve (5.7), a Newton-type solver can be implemented in terms of an active set strategy. The active and inactive sets are defined by the different cases of Cp (·, ·) and can be selected node-wise. , uk−1 ) be the previous iterate. We then obtain the new iterate Let (λk−1 l l (λkl , ukl ) = (λk−1 , uk−1 ) + (δλk−1 , δuk−1 ) l l l l of the semi-smooth Newton step by solving a linear system for the update s , the local system , δuk−1 ). For each node p ∈ PC;l (δλk−1 l l

k−1

k−1 k−1 δγ p , δuk−1 (5.9) DCp γ k−1 = −Cp γ k−1 p , up p p , up has to be satisfied. Here, DCp denotes the Jacobian of the local NCP function Cp (·, ·). We note that due to our basis transformation, DCp can

626

B. Wohlmuth

be regarded as a d × 2d matrix. Defining the trial vectors of the increment (δγ p , δup ), in a way similar to (5.2), by n δγp;tr := δγpn +

2µcn n δup , mp

δγ tp;tr := δγ tp +

2µct t δup , mp

n ): we introduce three sets in terms of Sn (γ p , up ) := Sn := max(0, γp;tr ! s : Sn = 0 , In := In (γ p , up ) := p ∈ PC;l ! s It := It (γ p , up ) := p ∈ PC;l : γ tp;tr < νSn , (5.10) ! s A := A(γ p , up ) := p ∈ PC;l : γ tp;tr ≥ νSn , Sn > 0 .

Now, we can easily define the generalized derivative DCp (·, ·) for each p ∈

s in the direction of the update δγ , δu . We observe that I , I and A PC;l p n t p s . On I the nodes are free and not form a non-overlapping partition of PC;l n in contact and no boundary forces apply. On It ∪ A the nodes are actually in contact and sliding on A. Before we discuss the general case, we focus on ν = 0. Here It = ∅, and the Newton algorithm simplifies considerably. Observing that the tangential component of Cp (·, ·) is then linear, we get p ∈ In , (δγpn , δγ tp ) DCp (γ p , up )(δγ p , δup ) = n n t p ∈ A. (δγp − δγp;tr , δγ p ) The Newton update (5.9) then gives (γ p )k = 0 for p ∈ Ink := In (γpk−1 , uk−1 p ) t k n k k k−1 k−1 and (γ p ) = 0, (up ) = gp for p ∈ A := A(γp , up ). For all possible contact nodes, we have a homogeneous Neumann boundary condition in the tangential direction, whereas in the normal direction we have a Dirichlet or Neumann condition. Thus, the implementation can be easily realized as a primal–dual active set strategy. The situation is more involved for ν > 0. For a Coulomb problem with a non-trivial friction coefficient, we obtain three different situations for Cp := Cp (γ p , up ) and DCp := DCp (γ p , up )(δγ p , δup ). Using the differential 1 x⊗x x = Id − , x = 0, ∂ x x x2 a straightforward computation shows the following. • If p ∈ In , then Cp = (γpn , γ tp ) and DCp =

δγpn . δγ tp

627

Numerical algorithms for variational inequalities n , (νγ n )s (γ t − γ t )) , • If p ∈ It , then Cp = (γpn − γp;tr p;tr p p;tr " # n − δγ n δγ p p;tr DCp = (νγ n )s δγ t − δγ t + s δγ n (γ t − γ t ) . p;tr p p;tr p;tr p p;tr γn p;tr

n , γ t s (γ t + α γ t )) , • If p ∈ A, then Cp = (γpn − γp;tr 2 p;tr p;tr p n δγpn − δγp;tr

DCp = , γ tp;tr s δγ tp + α1 γ tp + α2 δγ tp;tr + α3 γ tp;tr

where the factors α1 , α2 and α3 are given by n γ tp;tr δγ tp;tr νγp;tr , , α := − 2 γ tp;tr 2 γ tp;tr n n νγp;tr νδγp;tr α3 :=(1 − s) t 3 (δγ tp;tr γ tp;tr ) − t . γ p;tr γ p;tr

α1 :=s

For all cases, the normal component of the NCP function is linear; this is not the case for the tangential component. Only for p ∈ In do we have a linear tangential component. Here, we restrict ourselves to s ∈ [0, 1] and note that for s = 0 or s = 1 some of the terms cancel. Lemma 5.5. The semi-smooth Newton solver applied to (5.7) can be implemented as a primal–dual active set strategy, where in each Newton step s we have to update for each node p ∈ PC;l the type and the value of the boundary condition. Moreover, for ν > 0 we have the following. k−1 • Homogeneous Neumann conditions for p ∈ Ink := In (γ k−1 p , up ):

(γ p )k = 0

(5.11)

k−1 • Inhomogeneous Dirichlet conditions for p ∈ Itk := It (γ k−1 p , up ):

(unp )k = gp

and

(utp )k =

n )k−1 −s(δγp;tr t k−1 , n )k−1 (up ) (γp;tr

(5.12)

where the condition in the tangential direction also depends on the update of the normal surface traction. • Dirichlet conditions in the normal and Robin conditions in the tangenk−1 tial direction for p ∈ Ak := A(γ k−1 p , up ): (unp )k = gp

and

t k t k (Id + Lk−1 ˆt Lk−1 p;s )(γ p ) + c p;s (up ) −

(5.13a) ν(γpn )k (γ tp;tr )k−1 = gpk−1 , (5.13b) (γ tp;tr )k−1

t k−1 , the mesh-dependent scaling factor c where gpk−1 := Lk−1 ˆt is p;s (γ p;tr )

628

B. Wohlmuth

given by cˆt :=

2µct mp

and the matrix Lk−1 p;s by

+ (s − 1)α2k−1 Lk−1 sLk−1 k−1 k−1 1 2 := β Id + Lk−1 α , p;s p 2 (γ tp;tr )k−1 2 α2k−1 := −

n )k−1 ν(γp;tr , (γ tp;tr )k−1

βpk−1 := 1,

:= ωpk−1 (γ tp )k−1 ⊗ (γ tp;tr )k−1 , Lk−1 1

ωpk−1 := 1, Lk−1 := (γ tp;tr )k−1 ⊗ (γ tp;tr )k−1 . 2

Proof. The proof is based on (5.9) and the partitioning (5.10). For each node p ∈ Ink , we get n k−1 n k−1 (γp ) (δγp ) =− (δγ tp )k−1 (γ tp )k−1 and thus (5.11). From now on, to simplify the notation we suppress the upper index k − 1 of the Newton iteration. If p ∈ Itk , then # " n n − δγ n n δγ p p;tr γp − γp;tr =− . n (γ t − γ t ) (γ tp − γ tp;tr ) δγ tp − δγ tp;tr + γ ns δγp;tr p p;tr p;tr

Observing that n γpn − γp;tr =

2µcn (gp − unp ) mp

and

n δγpn − δγp;tr =−

2µcn n δup , mp

we get the normal part of (5.12). For the tangential part, we use γ tp − γ tp;tr = −

2µcn t u mp p

and

δγ tp − δγ tp;tr = −

2µcn t δup . mp

For p ∈ Ak , we find the same condition as in (5.12) for the normal part and thus (5.13a). Using the tangential component of the system (5.9), we find ∂γ tp + α1 γ tp + α2 δγ tp;tr + α3 γ tp;tr = −γ tp − α2 γ tp;tr . (5.14) Unfortunately, the new unknown updates are hidden in the coefficients α1 and α3 . Using the definition of the matrix Lp;s and applying the formula (x ⊗ y)z = (yz)x, we get α1 γ tp + α3 γ tp;tr = (Lp;s − α2 Id)δγ tp;tr − = (Lp;s −

α2 Id)δγ tp;tr

n νδγp;tr γt γ tp;tr p;tr

n )k ν(γp;tr γ t − α2 γ tp;tr . − γ tp;tr p;tr

n )k = (γ n )k , and inserting the Using (5.13a), which guarantees that (γp;tr p t t equality for α1 γ p + α3 γ p;tr in (5.14), we get (5.13b).


629

Remark 5.6. If a non-local friction law or a simple combination of Coulomb friction and Tresca friction is applied, possibly more cases have to be considered, we refer to Hager and Wohlmuth (2009b) for an abstract framework. If the friction bound depends not only on the contact pressure, then nodes which are not in contact are not automatically surface tractionfree nodes. Table 5.1 shows the convergence rate of the semi-smooth Newton scheme with s = 1 applied to the 2D contact problem with given friction bound depicted in Figure 4.3. Here we have used a discrete mesh-dependent norm for the Lagrange multiplier. The third column shows the ratio of the error in step k and k − 1. As expected, from the theoretical point of view, this ratio tends to zero. Table 5.1. Super-linear convergence for the semi-smooth Newton scheme applied to a contact problem with given friction bound. k

ek := λkl − λl

ek /ek−1

1 2 3 4 5 6 7 8 9

5.622e+02 2.553e+02 1.087e+02 3.515e+01 5.761e+00 2.362e−01 3.360e−03 1.760e−07 1.004e−12

− 4.541e−01 4.257e−01 3.233e−01 1.638e−01 4.100e−02 1.422e−02 5.239e−05 5.707e−06

For now, we do not comment on the solvability of the global system. However, we note that if convergence can be achieved, then the Newton update will converge to zero and thus (5.12) yields that the physical condition of a sticky node is satisfied, i.e., the relative tangential displacement is equal to zero. However, in contrast to the physical non-penetration condition, it is not directly imposed as one might expect from a primal–dual active set strategy. A value for s strictly smaller than one provides an additional damping. Moreover, (5.12) and (5.13b) show that the Newton solver couples the tangential and normal parts in the boundary conditions. Figure 5.4 shows the influence of the friction coefficient ν on the convergence rate of the semi-smooth Newton scheme applied to a 3D contact problem. A comparison between a fixed-point approach and the Newton solver is given. For all three cases, super-linear rates are obtained for the

630

B. Wohlmuth

(a)

(b)

(c)

Convergence rate (ν =0.2)


0

0

10

−2

10

−2

−2

10 Error

−4

10

−6

10

10

−4

Error

10 Error


0

10

10

−6

10

−8

−8

10

−10

10

−10

10

−10

10 2

4

6 8 10 12 14 Iteration steps

−6

10

−8

10

−4

10

10 2

4


2

4


Figure 5.4. (a) ν = 0.2, (b) ν = 0.4, (c) ν = 0.6.

semi-smooth Newton algorithm. The number of Newton steps required is quite insensitive to the friction coefficient, whereas for the simple fixed-point approach more iteration steps are required for a larger ν. 5.4. Influence of the scaling parameter It is well known, e.g., in plasticity, that the classical radial return algorithm, which is equivalent to setting s = 0, does not converge for large load steps. A similar observation can be made for contact problems with Coulomb or Tresca friction if s = 0. To get a better feeling for the scaling parameter s, we consider a simplified one-dimensional model h(r, u) := 5 s 3 max(|r + 100u|, 100) (r − 100(r + 100u)/ max(100, |r + 100u|)). Figure 5.5 shows the NCP function h(·, ·) for s = 0 and s = 1. A cut through u = const. shows that for s = 0 the function is almost constant and thus a Newton linearization involves a badly conditioned Jacobian. The situation is improved for s = 1. This observation motivates the use of the scaling parameter s. The value s = 1 is already introduced in H¨ ueber et al. (2008) and also successfully applied in Koziara and Bicanic (2008). (a)

(b)

1000

1

0

0

h

h

× 10

-1000 10

500

0

u

-10

0 -500

r

6

-1 10

500

0

u

-10

-500

Figure 5.5. NCP function for s = 0 (a) and s = 1 (b).

0

r

631


(b)

(c) 0

30

10

25

Rel. error

−5

10

−10

10

−15

10

20

s =0 s =0.25 s =0.5 s =0.75 s =1 s =1.25 s =1.5 s =1.75 s =2 5

15 10

t=1 t=2 t=3 t=4

5 10

15 20 No. of iterations

25

30

0.5

1

1.5

2

Figure 5.6. Geometry (a), error decay of the semi-smooth Newton iteration for different values of s (b) and total number of iterations for t1 to t4 with respect to s (c).

Figure 5.6(a) illustrates the geometry of a dynamic two-body contact problem in 3D with Coulomb friction. In Figure 5.6(b) we show the convergence of the semi-smooth Newton method for different scaling parameters. As can be clearly seen for small s, we do not have convergence. Numerically optimal convergence can be observed for s = 0.75. The convergence is also more robust for s = 1 than for smaller values of s. Figure 5.6(c) shows that for s ∈ [0.75, 1], the global number of iterations required is almost constant. For s > 1, we observe a linear increase in the number of iteration steps, whereas for s ≤ 0.5 we see divergence. From now on we will restrict ourselves to the case s ∈ [0, 1]. 5.5. Stabilization in the pre-asymptotic range Numerical experience shows that the use of the Robin boundary condition in the form of (5.13b) does not necessarily yield a robust and stable algorithm. Similar observations have been made in totally different fields of applications. We refer to Chan, Golub and Mulet (1999) and Hinterm¨ uller and Stadler (2006) for the use of primal–dual active set strategies in the field of image restoration and to the early work by Andersen, Christiansen, Conn and Overton (2000). To get a better feeling, we consider the definition of Lk−1 p;s in more detail. Let us assume that the Newton iterates converge towards a solution satisfying (3.11) and (3.12). Then, for a sliding node, we find that Lk−1 and 1 k−1 tends to the limit L∞ −α2k−1 Lk−1 tend to the same limit and thus L p;s p 2 independently of s: L∞ p :=

γ tp 1 t t γ Id. ⊗ γ − p γ tp;tr γ tp p γ tp;tr

Noting that a rank-one matrix of the form x ⊗ x has exactly one non-trivial eigenvalue, namely x2 , we find that Id+Lk−1 p;s tends to a symmetric matrix

632

B. Wohlmuth

with eigenvalues 1 and 1 − γ tp /γ tp;tr , which are strictly larger than zero for a physically correct sliding node. If no sliding occurs then γ tp = γ tp;tr , and the matrix is singular. In that situation, we face a standard Dirichlet boundary condition for utp . Recalling that γ tp stands for the negative surface traction in the tangential direction acting on node p, we find in the limit of a non-degenerate Robin boundary condition that k −(γ tp )k + Lp;R (utp )k = gR ,

Lp;R :=

1 2µct ∞ −1 ∞ 12 2 (−L∞ p ) (Id + Lp ) (−Lp ) . mp

k itself depends on the solution (γ n )k Here the Robin boundary data vector gR p and the previous iterate. The matrix Lp;R is symmetric and positive definite, and thus yields a well-posed Robin condition for elliptic systems, and unique solvability for the linearized system is established. Unfortunately, these observations do not necessarily hold true in the pre-asymptotic range. Then k−1 −1 k−1 it might occur that Id + Lk−1 p;s is non-singular but that −(Id + Lp;s ) Lp;s is not positive semi-definite. This may result in no convergence. To stabilize our approach, we introduce two modifications such that in each Newton step we obtain a well-defined system. This will be done by modifying ωpk−1 and βpk−1 , introduced in Lemma 5.5, so that

ωpk−1 := βpk−1

n )k−1 ν(γp;tr

,

n )k−1 , (γ t )k−1 max ν(γp;tr p 1 := min 1, , k−1 s(1 − χk−1 ) p ζp

(5.15) (5.16)

and ζpk−1 are defined by where χk−1 p χk−1 p

(γ tp )k−1 (γ tp;tr )k−1 := , (γ tp )k−1 (γ tp;tr )k−1

ζpk−1

(γ tp )k−1 := min n )k−1 , 1 . ν(γp;tr

Let us briefly comment on the definition of the two parameters ωpk−1 and βpk−1 . As we will see by our numerical results, both parameters play important roles in the pre-asymptotic range but do tend to 1 within the first few iterates if the algorithm converges. Thus asymptotically the exact Newton method is recovered. The damping parameter ωpk−1 can be regarded as a penalty term and only differs from 1 if the friction bound is violated. It is obvious that ζpk−1 ∈ [0, 1] and that χk−1 p , being the cosine of the angle between (γ tp )k−1 and (γ tp;tr )k−1 , is in [−1, 1]. It is now easy to see that βpk−1 is equal to 1 if s ∈ [0, 0.5] or if χk−1 ≥ 0. p The last condition is equivalent to the fact that the angle between the actual and the trial tangential stress is bounded by π/2. Figure 5.7 shows βpk−1 as a function of χk−1 ∈ [−1, 1] and ζpk−1 ∈ [0, 1] for s ∈ {0.5, 0.75, 1}. For p

633


(b)

2

(c) 1

1

1

0 1 0.5 ζpk−1

1 0 −1

0 χk−1 p

0.8

βpk−1

βpk−1

βpk−1

0.9 0.8 0.7

0.6 0.4 1

1 0.5 ζpk−1

1 0 −1

0 χk−1 p

0.5 ζpk−1

1 0 −1

0 χk−1 p

Figure 5.7. The damping term βpk−1 : (a) s = 0.5, (b) s = 0.75, (c) s = 1.

the physically correct solution, the angle between the actual and the trial tangential stress is zero. As our numerical results show, βpk−1 is mostly equal to one, reflecting the fact that within the first few iterations the orientation of the actual and trial tangential stress is adapted in the physically correct direction. Lemma 5.7. Let ωpk−1 and βpk−1 be defined by (5.15) and (5.16), respectively. Then the boundary condition (5.13) is well-posed. Proof. In the case that Id + Lk−1 p;s is singular, the Robin condition reduces to a Dirichlet condition for the component of (utp )k in the direction of the k−1 kernel K1 of Id + Lk−1 p;s . Moreover, if Lp;s is singular, then the Robin condition reduces to a Neumann condition for the component of (γ tp )k in the direction of the kernel K2 of Lk−1 p;s . Since K1 ∩ K2 = ∅, the Robin condition degenerates, in the direction of an element of the kernel K1 or K2 , to either a well-defined Dirichlet or Neumann condition. We have now to consider the non-degenerate case in more detail. Recalling k−1 (αk−1 Id + x ⊗ y), we find for a non-singular Id + Lk−1 and that Lk−1 p;s = βp p;s 2 a non-singular Lk−1 p;s , −1 k−1 k−1 −1 (Id + Lk−1 p;s ) Lp;s = Id − (Id + Lp;s ) βpk−1 x⊗y k−1 =: aZ. = α2 Id + 1 + βpk−1 α2k−1 1 + βpk−1 α2k−1 + βpk−1 xy

We note that 0 < βpk−1 ≤ 1 and α2k−1 ≥ −1 and thus the factor a in front of the matrix Z is positive. It is easy to verify that we obtain the two eigenvalues of the matrix Z by α2k−1 < 0,

(1 + βpk−1 α2k−1 )(α2k−1 + xy) 1 + βpk−1 (α2k−1 + xy)

.

634

B. Wohlmuth Size of ωk−1 p

k−1

Size of β p

1

Decay of the relative error −1

10

1

−3

0.8

10

0.9

−5

10 0.6

0.8

0.4

0.7

10

0.6

10

0.2

−7

10

−9 −11 −13

10

0.5 0 0


15

0

−15


15

10

0


15

Figure 5.8. Scaling factors ωpk−1 and βpk−1 at iteration step k and global convergence rate for s = 1.

If −1 < eZ := βpk−1 (α2k−1 + xy) < 0, then Z is negative definite. The definition of Lk−1 p;s in Lemma 5.5 shows that we can set x := sωpk−1 (γ p )k−1 + (s − 1)α2k−1 (γ p;tr )k−1 ,

y :=

(γ p;tr )k−1 . (γ p;tr )k−1 2

Then, the value eZ can be rewritten as k−1 k−1 k−1 k−1 k−1 k−1 (γ p ) eZ = β p α2 + sωp χp + (s − 1)α2 (γ p;tr )k−1 k−1 k−1 k−1 k−1 k−1 (γ p ) = sβp ωp χp + α2 (γ p;tr )k−1 k−1 k−1 k−1 k−1 k−1 (γ p ) k−1 = sβp α2 ) 1 − ωp χp = sβpk−1 α2k−1 (1 − χk−1 p ζp n )k−1 ν(γp;tr ≥ α2k−1 ≥ −1. The case eZ = −1 is ruled out by the assumption that Id + Lk−1 p;s is nonsingular, since eZ + 1 is an eigenvalue of this matrix. These considerations show that the Robin boundary condition, given by (5.13b), reduces to a Neumann or Dirichlet condition in the direction of k−1 the kernel of Lk−1 p;s or Id + Lp;s , respectively. For all other cases it forms a well-defined Robin condition. Remark 5.8. We note that the proposed modifications are just one possibility. An alternative choice is to symmetrize and rescale Lk−1 p;s . All these modifications work well as long as they correctly normalize and, in the limit, tend to the original version; see also H¨ ueber et al. (2008) and H¨ ueber (2008). To illustrate the effect of the modification, we consider different test examples in 3D. Figure 5.8 shows the case of a Tresca friction problem. We select a representative node and depict the value for ωpk−1 and βpk−1 . As can be seen, the modification actually applies only in the first few iteration steps.


(b)

635

(c)

1.1 0

0.8

0.9

0.6 0.4 0.2 0 0

non−nested nested 2

4

6 8 10 No. of iterations

12

14

0.8 0.7 0.6 0.5 0

nested with a nested no a no−nest. with a no−nest. no a

−5

Rel. error

1 Size of β k−1 p

Size of ω k−1 p

10 1

non−nested nested 2

4

6 8 10 No. of iterations

12

14

10

−10

10

−15

10

0

2

4

6

8 10 12 14 16 18 20 22 24 No. of iterations

Figure 5.9. Scaling factors ωpk−1 , βpk−1 at iteration step k (a,b) and convergence rates for different strategies (c).

In Figure 5.9, we apply the modifications to a Coulomb problem and consider the influence of the modification on the local and global convergence. To do so, we select the initial guess in two different ways. In the first case, it is chosen randomly on a fine mesh, whereas in the second situation it is interpolated from the solution of the previous coarser level and thus can be expected to be good. For this nested approach the modification does not affect the iteration scheme and thus the same rates of convergence can be observed. This is in good agreement with the observation that asymptotically ωpk−1 and βpk−1 tend to 1 in the case of convergence. The situation is considerably different if we apply a bad initial guess. Here, without the modification, no convergence at all can be observed, whereas the modified version still shows a reasonable rate. These two test examples show that the effect of the modification can be neglected if the start iteration is already good enough. However in the pre-asymptotic range, it is of great significance and highly enlarges the domain of convergence. Our last test combines the scaling s ∈ [0, 1] with the modification for the nodes p ∈ Ak . Here, we consider a dynamic Coulomb problem in 3D on non-matching meshes where the actual contact zone is not simply connected. Figure 5.10 shows the influence of the convergence rate as a function of s. As can be clearly seen in Figure 5.10(a), the number of Newton iterations required is quite insensitive provided that s is large enough. For small s, no convergence at all can be obtained. Therefore, at time t2 , only the results for s = 0.5, s = 0.75 and s = 1 are plotted. The choices s = 0 and s = 0.25 do not yield a convergent scheme. Finally Figure 5.11 shows that for a selected node both parameters ωpk−1 and βpk−1 tend to 1 very fast in the case of s > 0.5. Moreover, the cosine of the angle between the tangential stress and the trial tangential stress tends to 1. This indicates that our algorithm is able to adjust the correct sliding direction within the first few iterates. The situation is changed drastically

636

B. Wohlmuth

(a)

(b)

0

10

t=1 t=2

25

10

−5

−5

10

20

15

10 Rel. error

Rel. error

No. of iterations

(c)

0

30

−10

10

s =0 s =0.25 s =0.5 s =0.75 s =1

−15

10

10

−20

5 0.4

0.5

0.6

0.7 Size of s

0.8

0.9

10

1

2

4

−10

10

−15

10

s =0.5 s =0.75 s =1

−20

6


12

10

14

2

4

6


12

14

Figure 5.10. Number of required iterations (a) and convergence rates for two different time steps (b,c).

−0.5

−1

1

0.9

0.95 k−1

s =1

0

1

Size of ω p

0.5

k−1

s =0 s =0.25 s =0.5 s =0.75

Size of β p

Cosine of angle χ p

k−1

1

0.8

0.7 s =0 s =0.25 s =0.5 s =0.75 s =1

0.6

2

4

6 No. of iterations

8

10

0.5

2

4

6 No. of iterations

8

10

0.9

0.85 s =0 s =0.25 s =0.5 s =0.75 s =1

0.8

0.75

2

4

6 No. of iterations

8

10

Figure 5.11. Scaling factors χpk−1 , βpk−1 and ωpk−1 at iteration step k.

if s ≤ 0.5. Then the correct sliding direction cannot be identified and ωpk−1 and χk−1 do not tend to 1, reflecting the fact that no convergence is p obtained. 5.6. Mesh-dependent convergence rates Although the abstract framework of semi-smooth Newton methods is very flexible and quite attractive for a large class of problems, there is one bottleneck. The convergence rate is, in general, mesh-dependent. Numerical results show that the number of Newton steps increases linearly with the refinement level. Several strategies exist to overcome this problem, depending on the type of application. One of the most efficient ones is to embed the solver in a nested iteration. This is extremely easy to realize with time-dependent problems or with adaptive refinement techniques. In both situations the initial guess can be interpolated from the previous time step or mesh. This simple pre-processing is quite often sufficient to obtain a level-independent number of non-linear solver steps. Alternatively, or additionally, we can combine the Newton approach with an inexact solution strategy. Quite often the arising linear system is not solved by a fast direct solver but by preconditioned Krylov or subspace correction methods such as multigrid or FETI techniques. Then it is only natural to make the Newton update after a small number of steps of the linear solver. To avoid


637

over-solving, during the first non-linear iteration steps, the number of inner iterations should be set dynamically. We consider the same example as shown in Figure 4.3 to illustrate the effect of the mesh-dependent convergence rates. We use a hierarchy of uniformly refined meshes and compare three different strategies. The ‘exact’ one starts with a randomly chosen initial guess on each level and solves the resulting linearized system in each Newton step by a multigrid method with fixed small tolerance. By Kl , we denote the number of Newton steps required. As can be seen in Table 5.2, Kl depends linearly on l, and thus the total number NlMG of multigrid steps on level l increases linearly with l. The situation is different if we apply an inexact strategy, using the same bad initial guess, but do a Newton update after each multigrid step. This results in an ‘inexact’ strategy where the correct Jacobian of the NCP function is used but the system stiffness matrix is replaced by its multigrid approximation. Here, Ml stands for the number of iterations required to identify the correct active sets. As before, this number increases linearly, but the total number of multigrid steps is significantly reduced. Finally we combine this inexact strategy with a good initial guess obtained by interpolation from the solution on the previous level. We call this the ‘nested’ approach. Then the total number of multigrid steps is bounded independently of the refinement level and is comparable to the number required to solve one linearized problem. Table 5.2. Number of total multigrid steps for different strategies. Strategy Level l DOF 1 2 3 4 5

27 125 729 4913 35937

Exact Kl MG 3 3 4 6 7

46 62 72 86 106

Inexact Ml NlMG 3 3 4 6 9

12 16 17 17 19

Nested Ml NlMG 3 3 4 6 6

12 16 15 14 16

In more complex 3D situations, or for non-linear material laws, more than one multigrid step is required before an update can be performed. In a last test, we combine semi-smooth Newton techniques and inexact solvers for the linearized system with an overlapping two-scale domain decomposition method in 3D. We refer to Brunßen, Hager, Wohlmuth and Schmid (2008) and Brunßen and Wohlmuth (2009) for details of the model and the specification of the data. In addition to the contact formulation, plasticity effects are taken into account. The approach is motivated by possible

638 (a)

B. Wohlmuth (b)

(c) Inexact computation

||F|| ||F+DF X|| Start of Block GS

Block GS residual and Newton residual

Block GS residual and Newton residual

Exact computation

0

10

−5

10

−10

10

0

5

10

15 20 Block GS steps

25

30

||F|| ||F+DF X|| Start of Block GS 0

10

−5

10

−10

10

0

5

10

15 20 Block GS steps

25

30

Figure 5.12. Surface mesh of the two-scale domain decomposition approach (a), convergence rates for the non-linear Newton solver (b,c).

applications to incremental metal cold forming processes. Here, the forming zone is small, but very mobile, and the work tool is contacting almost every point of the workpiece at some time in the process. To avoid expensive re-meshing and to reduce the complexity of the elasto-plastic constitutive equations, an operator splitting technique in space can be introduced. A small local but mobile subdomain, with a fine mesh and the fully non-linear contact and plasticity model, will interact with the global coarse mesh associated with a simplified model. To solve the fully coupled non-linear system, semi-smooth Newton techniques in combination with a block Gauss–Seidel solver are quite efficient. This is in particular true if inexact strategies are applied. To avoid over-solving during the first non-linear iteration steps, the number of inner iteration should be set dynamically. Here, the stopping criterion is based on Dembo, Eisenstat and Steinhaug (1982) and Eisenstat and Walker (1996). Figure 5.12 shows the increase in efficiency of the non-linear solver if, within each Newton step, the linearized coupled domain decomposition system is not solved exactly but by a few block Gauss–Seidel steps. The number of inner iteration steps is set dynamically depending on the non-linear residual. As can be seen from Figure 5.12(c), during the first few Newton iterations, there is no need to solve the linearized system up to very high accuracy. To obtain the full efficiency of the approach, the accuracy of the linear solver has to be gradually improved during the non-linear solution process.

6. A posteriori error estimates and adaptivity Adaptive techniques based on a posteriori error estimators play an important role in enhancing the performance of the numerical simulation algorithm and are well established for finite element methods: see the monographs by Ainsworth and Oden (2000), Babuˇska and Strouboulis (2001), Han (2005), Repin (2008), Verf¨ urth (1996) and the references therein. For abstract variational inequalities we refer to Ainsworth, Oden and Lee (1993),


639

Bostan, Han and Reddy (2005), Erdmann, Frei, Hoppe, Kornhuber and Wiest (1993), Fuchs and Repin (2010), Liu and Yan (2000), Nochetto, von Petersdorff and Zhang (2010), Moon, Nochetto, von Petersdorff and Zhang (2007) and Suttmeier (2005), whereas obstacle-type problems are considered in Bildhauer, Fuchs and Repin (2008), Braess (2005), Chen and Nochetto (2000), French, Larsson and Nochetto (2001), Hoppe and Kornhuber (1994), Johnson (1992), Kornhuber and Zou (2011), Nochetto, Siebert and Veeser (2003, 2005) and Veeser (2001), and early approaches for contact problems can be found in Blum and Suttmeier (2000), Buscaglia, Duran, Fancello, Feijoo and Padra (2001), Carstensen, Scherf and Wriggers (1999), Lee and Oden (1994) and Wriggers and Scherf (1998). A residual-type error estimator is introduced and analysed in Hild and Nicaise (2005, 2007) and in Bostan and Han (2006) and Hild and Lleras (2009) for a one-sided contact problem without friction and with Coulomb friction, respectively. In addition to standard face and volume residual terms, extra terms reflecting the non-conformity of the approach are taken into account. For boundary element discretizations, we refer to Eck and Wendland (2003) and Maischak and Stephan (2005, 2007). Although the error estimator in Hild and Lleras (2009) provides a mesh-independent upper bound for the discretization error, not all terms result in optimal lower bounds. Thus the efficiency of the error estimator cannot be guaranteed from the theoretical point of view. Early results on hp-techniques for frictional contact problems can be found in Lee and Oden (1994), whereas in the recent contribution of D¨ orsek and Melenk (2010) a simplified Tresca problem with a given surface normal traction equal to zero has been studied. As it turns out, the saddle-point approach (2.15) and its discrete version (3.9) provide an excellent starting point for the construction of an error indicator. In Wohlmuth (2007) an estimator was introduced for the case of no friction and non-matching meshes. A theoretical analysis shows that a constant-free global upper bound and local lower bounds for the error can be established. However, in contrast to the standard linear conforming setting, the additional higher-order term is solution-dependent and cannot be controlled within the adaptive refinement process. These first results can be improved considerably by following a more general construction principle. Firstly, we use the equilibrium of the saddlepoint approach to construct our indicator. Secondly, we consider the influence of the discretization (3.8) of (2.14). For the special case of a one-sided contact problem without friction, it is shown in Weiss and Wohlmuth (2009) that an indicator, constructed in this way, provides upper and local lower bounds for the discretization error and is thus an efficient error estimator. Moreover, the higher-order terms are standard data oscillation terms and can be controlled within the adaptive refinement strategy. The results are shown for a flux-based approach but can easily be generalized to cover the

640

B. Wohlmuth

case of a residual-based error estimator: see H¨ ueber and Wohlmuth (2010) for numerical results. Here, we follow these lines and extend the approach and the analysis to contact problems with friction. In this section, we discuss different element-oriented error indicators, i.e., η 2 := T ∈Tl ηT2 , and analyse the influence of the variational inequality. We focus both on the theoretical bounds and on computational aspects. Due to the variational inequality character of the given problem, we have to include a term which measures the non-conformity of the Lagrange multiplier. This contribution can be decomposed into a contact and friction term. For ease of presentation and analysis, we restrict ourselves to the two-dimensional setting, polygonal domains, a zero gap and a constant contact normal. Moreover, we do not analyse the influence of non-matching meshes but do provide the construction of the error indicator. As is standard, we use a data oscillation term, h2 T ξi2 := (6.1) f − Πi f 20;T , i = 0, 1. 2µ T ∈Tl

To keep the notation simple, we further assume that we are working with simplicial meshes and that the given boundary data is compatible with the discretization, i.e., ul |ΓD = uD and fN is piecewise cubic and continuous on each straight segment γi of ΓN . Moreover, we assume that nj fN |γi (p) = ni fN |γj (p) with ∂γi ∩ ∂γj = p and that nfN |ΓN (p) = 0 where p = ∂ΓC ∩ ∂ΓN , and n is the outer unit normal on ΓC . Otherwise, as is standard, additional boundary face terms have to be included, measuring the weighted L2 -norms of the boundary error: see, e.g., Repin, Sauter and Smolianski (2003). The weight hf /(2µ) for the Neumann term is the inverse of the weight for the Dirichlet term, reflecting the H 1/2 -duality between displacement trace and surface tractions. Here hf stands for the diameter of the boundary face f . In the following, we use the piecewise cubic biorthogonal basis function shown in Figure 3.7(a). As a first preliminary step, we reformulate the coupled problem (3.9) by introducing a weakly consistent Neumann force in Mm l on the master ∗;m d side. To do so, we introduce Π∗;m := (Π ) as the dual mortar projection l l m with respect to the master side, Π∗;m : M → M l l Π∗;m vl ∈ Wlm . (6.2) l µvl ds = µ, vl ΓC , ΓC

Keeping in mind that Mlm and Wlm reproduce constants and have a locally defined basis, a straightforward Bramble–Hilbert argument implies the approximation properties 2 2 µ − Π∗;m µ ≤ hf µ − Π∗;m µ ∈ L2 (ΓC ), (6.3) 1 l l µ0;f , − ;Γ 2

C

f ∈Flm


641

where Flm stands for the set of all contact faces of the master subdomain. In terms of (6.2), we now define the discrete contact forces of the slave and master sides by on ΓsC , fCs := −λl (6.4) fC := m fCm := Π∗;m l λl on ΓC . Provided that the Lagrange multiplier λl on the slave side is known, we can rewrite the first line of (3.9). Recalling the definition of Π∗;m l , we obtain k a standard variational problem for ul on each subdomain Ωk , k ∈ {m, s}. k such that Find ukl ∈ Vl;D fCk vl ds, vl ∈ Vlk . (6.5) ak (ukl , vl ) = fk (vl ) + ΓkC

Then (6.5) shows that usl and um l are conforming finite element approximations of a linear elasticity problem on Ωs and Ωm , respectively. Here the unknown contact stresses on ΓsC and Γm C are replaced by the numerical approximation fC as defined by (6.4). Thus the contact zone ΓkC , k ∈ {m, s}, can be regarded as a Neumann boundary part where, additionally, the error in the Lagrange multiplier has to be taken into account. Unfortunately, in contrast to given Neumann data, this error cannot be estimated a priori and has to be controlled by the error indicator. There is a huge variety of different types of error estimators. One of the most simple approaches is based on the residual equation. A more recent and quite attractive alternative construction is related to local lifting techniques in combination with equilibrated fluxes. These element-wise conservative fluxes have a long tradition in structural mechanics and go back to the early papers by Brink and Stein (1998), Kelly (1984), Kelly and Isles (1989), Ladevèze and Leguillon (1983), Ladevèze and Maunder (1996), Ladevèze and Rougeot (1997), Prager and Synge (1947) and Stein and Ohnimus (1997, 1999). We refer to the monograph by Repin (2008) and to Luce and Wohlmuth (2004), where such techniques have been applied successfully and constant-free upper bounds have been established. Recently these ideas have been generalized to many situations and are widely applied: see, e.g., Braess, Hoppe and Schöberl (2008), Braess, Pillwein and Schöberl (2009b), Cheddadi, Fuˇc´ık, Prieto and Vohral´ık (2008, 2009), Ern and Vohral´ık (2009), Nicaise, Witowski and Wohlmuth (2008) and Vohral´ık (2008). 6.1. Construction of the equilibrated error indicator The construction of such type of indicators is done in two steps. Firstly, equilibrated fluxes on the faces are defined locally and secondly a local volume lifting is performed. For low-order finite elements the equilibrated fluxes g are defined on the set of all faces Fl and for each simplicial face f in P1 (f )d . A unit face

642

B. Wohlmuth

normal nf is fixed for each face f . If f is a boundary face of the domain Ωk , then we set nf to be the outer unit normal$on ∂Ωk . Then the global problem reads as follows. Find g := (gf )f ∈Fl ∈ f ∈Fl P1 (f )d such that, for all elements T ∈ Tl and all v ∈ (P1 (T ))d , σ(ul ) : (v) dx = fv dx + (nT nf )gf v ds, (6.6) T

T

f ⊂∂T

f

where nT stands for the outer unit normal on ∂T . Moreover, for all Neumann and contact faces, the fluxes have to satisfy gf v ds = fN v ds, f ∈ FlN , v ∈ (P1 (f ))d , (6.7a) f f gf v ds = fC v ds, f ∈ FlC , v ∈ (P1 (f ))d , (6.7b) f

f

where Flk is the set of faces on Γsk and on Γm k , k ∈ {N, C}. We note that formally (6.7a) and (6.7b) have the same structure but in contrast to fN , fC is not a priori known but depends on the dual variable λl given by (5.8). A simple counting argument shows that in (6.6) and (6.7), we have to satisfy (d + 1)d NT and dd (NfN + NfC ) equations, respectively. Here NT stands for the number of elements in Tl , and NfN and NfC denote the number of faces on the Neumann and contact boundary part, respectively. Remark 6.1. We note that from the point of view of approximation properties, there is no need to work with gf ∈ P1 (f )d . A face-wise constant approximation in combination with v ∈ (P0 (T ))d in (6.6) and v ∈ (P0 (f ))d in (6.7) would be good enough. However, then the system cannot be decoupled easily and a global system has to be solved. This is not very attractive from the computational point of view. It is well known (see the monograph by Ainsworth and Oden (2000)) that a possible solution can be constructed locally by introducing the moments of the fluxes gf . An abstract framework for the vertex-based patch-wise computation of the moments can be found in Ainsworth and Oden (1993). In particular, the size of the local system depends on the shape-regularity of the mesh but not on the mesh size. Depending on the type of the vertex, i.e., Neumann, or interior, Dirichlet, the system has a unique solution or the system matrix is singular but solvability is guaranteed. Then the solution is fixed by imposing an additional constraint resulting from a local minimization problem for the moments. Here, we briefly recall the main steps and provide the structure of the vertex-based patch system in 2D. Figure 6.1 illustrates the notation for an interior vertex patch.

Numerical algorithms for variational inequalities f1 Tnp fnp

643

f2 T1

T2 n f2

n f1 n fn p p

Figure 6.1. Enumeration of the elements and faces sharing a vertex in 2D.

The moments µpf ∈ Rd are given for each face f and vertex p by p gf φp ds µf := f

and uniquely define the fluxes gf by gf = p∈Pf µpf ϕp , where ϕp is the nor malized linear dual basis with respect to φp on the face f , i.e., f φp ϕq ds = δpq . Here Pf stands for all vertices of f . Here, we only work out the details for an interior vertex p in 2D and refer to Ainsworth and Oden (2000) for a discussion of the local system in the case of boundary vertices. Using φp |Ti ej , j = 1, 2, i = 1, . . . , np , where np stands for the number of elements sharing the vertex p, as a test function in (6.6), in 2D we get the linear system   p   p  µ1 r1 −Id Id p  p      −Id Id    µ 2   r2     ..   ..  .. .. (6.8)    .  =  .  =: rp , . .    p   p       µnp −1 rn p −Id Id rpnp −1 Id −Id µpnp where (rpi )j := Ti f φp ej dx − Ti σ(ul ) : (φp ej ) dx, j = 1, 2, i = 1, . . . , np , and µpi := µpfi , i = 1, . . . , np . It is easy to see that this system is singular, and the dimension of its kernel is independent of np and equal to two. The two eigenvectors v1 and v2 associated with the eigenvalue zero are given by v1 = (1, 0, 1, 0, . . . , 1, 0) and v2 = (0, 1, 0, 1, . . . , 0, 1) . Since vj rp = f (φp ej )−a(ul , φp ej ) = 0, j = 1, 2, the solvability of (6.8) is granted. We note that, in the linear setting, each solution of (6.8) will provide an upper bound for the discretization error but will, in general, not be suitable for getting lower bounds. Thus the solution of (6.8) has to be selected carefully. The flux gf will enter directly into the definition of the error estimator, and discrete norm equivalences show that the error estimator depends on gf − {σ(ul )nf }, where {·} stands for the average face contribution. This

644

B. Wohlmuth

observation motivates us to obtain µp := (µp1 , µp2 , . . . , µpnp ) as the unique solution of a local constrained minimization problem. We identify the index i = np + 1 with i = 1 and introduce the quadratic functional p

Jp (µ ) :=

np i+1 i=1 j=i

2µi

|Ti |

fj φp ds

p p 2 2 µi − τ i;j ,

(6.9)

where µi is the shear modulus of the element Ti , |Ti | stands for the element volume and τ pi;j := fj σ(ul )|Ti nfj φp ds. Based on Jp (·), we impose an additional constraint for each vertex p. Find µp such that µp is a solution of (6.8) and satisfies Jp (µp ) =

ηp

min

solves (6.8)

Jp (η p ).

We point out that in our setting the factor 2µi in the weight of (6.9) is constant on each vertex patch and thus can be removed without influencing the result. Moreover, for meshes with no anisotropy, the weight itself can be replaced by one. Remark 6.2. We note that the difference µpi − τ pi;j can also be used to define error estimators (see the monograph by Ainsworth and Oden (2000)), but then the upper bound is, in general, not constant-free. The second step, in the construction of our error indicator, is to map the surface fluxes in terms of local lifting techniques to volume H(div)conforming fluxes. In many cases, up to higher-order data oscillation terms, upper bounds for the discretization error with constant one can then be obtained. Thus these equilibration techniques form a flexible and an attractive class of error estimators and are of special interest if a reliable stopping criterion is required. For scalar elliptic equations it is quite easy to construct this type of estimator. Basically two types of approach exist. The first one works on a dual mesh and uses a sub-mesh for the recovery in terms of standard mixed finite elements, e.g., Raviart–Thomas (RT) or Brezzi–Douglas–Marini (BDM) elements: see Brezzi and Fortin (1991). Here the vertex patches are nonoverlapping, and the fluxes are simply given by the discrete finite element flux which is well-defined in the interior of each element: see, e.g., Luce and Wohlmuth (2004). Alternatively one can use the standard overlapping vertex patches and the face flux moments from the equilibrated approach as described above; see, e.g., Vohral´ık (2008). The situation is more involved in the case of linear elasticity. Firstly, the dual mesh approach cannot be applied in the linear elasticity setting due to the local rotations, which act as rigid body mode. Secondly, classical mixed finite elements for each row of the stress tensor cannot be used

645


because they violate the symmetry. Thus special mixed finite elements for symmetric tensor approximations have to be applied. This can be done by selecting Arnold–Winther-type mixed finite elements (Arnold, Falk and Winther 2006, Arnold and Winther 2002). Here, we only work with the twodimensional setting, but these types of elements do also exist in 3D (Arnold, Awanou and Winther 2008, Arnold and Winther 2003), and on hexahedral meshes (Arnold and Awanou 2005). Using the equilibrated fluxes and an H(div)-conforming lifting, as in Nicaise et al. (2008) for a linear elasticity problem, we define a globally H(div)-conforming approximation σ l of the stress σ(u). Then the error indicator is defined by 2 2 ηL2 := ηL;T , ηL;T := C −1/2 (σ l − σ(ul ))20;T . (6.10) k∈{m,s} T ∈Tlk 2 We note that once σ l is known, ηL;T can be easily evaluated by 1 λ 2 ηL;T = σ l − σ(ul )20;T − tr(σ l − σ(ul ))20;T . 2µ 2µ + dλ

Before we specify σ l , we recall the basic properties of the Arnold–Winther elements in 2D. The element space XT for a simplicial T ∈ Tl is given by ! XT := τ ∈ (P3 (T ))2×2 , τ 12 = τ 21 , div τ ∈ (P1 (T ))2 and has dimension 24. Based on this, we can define the global space by s Xl := Xm l × Xl , where Xkl := {τ l ∈ H(div; Ωk ) | τ l |T ∈ XT , T ∈ Tlk },

k ∈ {m, s}.

By definition Xkl is H(div)-conforming on each subdomain Ωk , and the degrees of freedom are given by (see Arnold and Winther (2002)): • the nodal values (3 dof) at each vertex p, • the zero- and first-order moments of τ l nf (4 dof) on each face f , • the mean value (3 dof) on each element T . We define our stress approximation σ l of σ(u) by setting σ l : (v) dx := σ(ul ) : (v) dx, v ∈ (P1 (T ))2 , T T (σ l nf )v ds := gf v ds, v ∈ (P1 (f ))2 , f

(6.11a) (6.11b)

f

σ l (p) :=

np 1 σ(ul )|T (p) + αp . i np

(6.11c)

i=1

In contrast to Nicaise et al. (2008), where only homogeneous Dirichlet boundary conditions have been considered, we have to include a suitable

646

B. Wohlmuth

αp in (6.11c). For each node p not on ΓN ∪ ΓC , we set αp := 0. Otherwise, it is a symmetric 2 × 2 matrix with minimal Euclidean norm under the constraint  np 1   f − σ(ul )|T (p)n p ∈ ΓN ,  N np i i=1 αp n := (6.12) n p   1  fC − n σ(ul )|T (p)n p ∈ ΓC , p i=1

i

where, for corner-points p, (6.12) has to be satisfied for both normal vectors. Our assumptions on the given data, the actual contact zone and our choice for Ml guarantee that αp is well-defined. Moreover, due to the symmetry of σ l , (6.11a) reduces to three independent conditions on each element. This definition has already been applied to contact problems in Wohlmuth (2007) and Weiss and Wohlmuth (2009). Lemma 6.3. The subdomain-wise H(div)-conforming Arnold–Winther element σ l is well-defined and satisfies − div σ l = Π1 f , fN σl n = fC

on Ω, on ΓN , on ΓC ,

where Πj stands for the L2 -projection on piecewise polynomials of degree at most j ∈ N0 . Proof. By definition of Xl , div σ l is element-wise in (P1 (T ))2 . Integration by parts and the symmetry of σ l shows that, for all v ∈ (P1 (T ))2 , div σ l v dx = − σ l : (v) dx + (σ l nT )v ds. T

T

∂T

Now we can use the definition (6.6) of the fluxes gf and the definitions (6.11a) and (6.11b) for the Arnold–Winther element σ l , and we obtain div σ l v dx = − σ(ul ) : (v) dx + (nf nT )gf v ds T

T

=−

f ⊂∂T

f

fv dx. T

The stress σ l is in Xl , and thus σ l n restricted to each face f is in (P3 (f ))2 . Let p1 and p2 be the two endpoints of f ; then (6.11c) and (6.12) show that we have σ l n(pi ) = fC (pi ) and σ l n(pi ) = fN (pi ) for f ⊂ ΓC and f ⊂ ΓN , i = 1, 2, respectively. Moreover, by assumption on the data fN and by choice of Ml , σ l n − fk , k ∈ {C, N}, is cubic with zero value at the endpoints. Then (6.7) in combination with (6.11c) shows that the zero- and first-order moments


647

of σ l n − fk vanish, from which we can conclude that σ l n − fk = 0 on each Neumann or contact boundary face. Remark 6.4. We note that Lemma 6.3 does not hold for other choices of biorthogonal Lagrange multiplier basis functions such as, e.g., the piecewise affine but discontinuous one. The error indicator ηL , defined by (6.10), is motivated by the observation that the discrete Lagrange multiplier λl acts as a Neumann boundary condition on the contact part. It takes into account neither the inequality constraints resulting from the non-penetration nor the friction law. To have the equality σ l n = fC on the contact zone, it is crucial that λl is mapped by ∗;m Π∗;m onto Mm l . In the case of matching meshes we have λl − Πl λl = 0, l but in a more general situation this difference is non-zero and has to be estimated and controlled within the adaptive refinement process. To do so, we introduce an extra term, which is restricted to the master side of the contact zone, hf 2 2 2 ηS := ηS;f , ηS;f := m λl − Π∗;m (6.14) l λl 0;f , 2µ m f ∈Fl

where µm is the Lamé parameter associated with the master body. We now provide a first preliminary result, which is the starting point for our a posteriori analysis. The error in the displacement will be estimated in the energy norm ||| · |||, which is defined by |||v|||2 := a(v, v), Lemma 6.5.

v ∈ V.

The upper bound for the error in the energy norm satisfies

|||u − ul ||| ≤ (ηL + CηS + Cξ1 )|||u − ul ||| + λl − λ, [u] − [ul ]ΓC . 2

Proof. We start with the definition of the energy norm and apply integration by parts. The assumptions on the Dirichlet and Neumann boundary conditions yield that u − ul = 0 on ΓD and that (σ l − σ(u))n = 0 on s ΓN . We then obtain in terms of Lemma 6.3 and (6.3) for el := (em l , el ) := s s (um − um l , u − ul ) |||el |||2 = (σ(u) − σ(ul )) : (el ) dx Ω ≤ ηL |||el ||| + (σ(u) − σ l ) : (el ) dx Ω = ηL |||el ||| + (f − Π1 f )el dx Ω s + (σ(u) − σ l )n el ds + (σ(u) − σ l )nm el ds ∂Ωs

∂Ωm

648

B. Wohlmuth m ≤ (ηL + Cξ1 )|||el ||| + λl − λ, esl ΓC − Π∗;m l λl − λ, el ΓC m = (ηL + Cξ1 )|||el ||| + λl − λ, [el ]ΓC − Π∗;m l λl − λl , el ΓC = (ηL + CηS + Cξ1 )|||el ||| + λl − λ, [el ]ΓC .

Remark 6.6. We note that ηS is equal to zero in the case of a one-sided contact problem or for matching meshes. We refer to Section 6.7 for more comments on non-matching meshes. In the following subsection, we introduce two extra terms which allow for the variational inequality. Although we restrict our analyses to very simple two-dimensional settings and to matching meshes, the definitions are given for the more general case, including non-matching meshes, a non-zero gap, and d = 3. As our numerical results will show, the error indicator can also be applied to such more general settings. 6.2. Influence of the contact constraints To bound the variational crime resulting from the non-penetration condition and the friction law, we introduce two additional terms ηC and ηF . Both terms are restricted to the slave side of the contact zone and are associated with the faces 2 ηj := ηj;f , j ∈ {C, F}. (6.15) f ∈Fls

Here Fls stands for the set of all faces f on ΓsC of the actual mesh. The term ηC measures the violation of the physical condition of a positive contact + n pressure and is associated with λC l := λl ∈ Ml . The term ηF;f measures the violation of the friction law and is associated with the scalar-valued tangential part λF l . For a Coulomb law, we set (νγpn − γ tp )ψp =: γpf ψp , (6.16a) λF l := s p∈PC;l

s p∈PC;l

and for a Tresca law, we set t := (F − γ )ψ =: γpf ψp . λF p p l s p∈PC;l

(6.16b)

s p∈PC;l

The local face contribution ηj;f is then defined in terms of a non-linear operator Plj and uses a correctly weighted L2 -norm, ηj;f :=

hf δ f λj − Plj λjl 20;f , 2 min(µs , µm ) l

j ∈ {C, F},

(6.17)

where µs and µm are the Lamé parameters associated with the slave and master body, respectively. Here hf stands for the face diameter and δf ∈

649


(b) −3

20

5

x 10

10

(d)

20

5

−3

x 10

10 0

0

0

0

−10 0

(c)

−10 0.1

0.2

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0.4

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0.1

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0

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C C C F F Figure 6.2. (a) Scaled λC 3 , u3 , (b) the product λ3 u3 , (c) scaled u3 , λ3 , F F (d) the product λ3 u3 .

{1, 1/hf }. The choice of δf and the definition of Plj will be specified in (6.20) and (6.19), respectively, and depend on the mesh and actual discrete contact zone. + F n Although both λC l and λl are, by definition of Ml (λl ) and Ml (F), in Ml , + they are not in M for non-trivial cases. As the a priori analysis has already shown, we have to face terms coupling the discrete Lagrange multiplier with the continuous relative displacement. We associate two scalar-valued F relative displacements with the discrete Lagrange multipliers λC l and λl n uC (αpn − gp )φp , (6.18a) l := Πl ([ul ] − g) = s p∈PC;l

t uF l := Πl [ul ]l :=

αtp φp .

(6.18b)

s p∈PC;l

C F F We recall that γpn (αpn − gp ) = 0 and αtp γpf = 0, but that λC l ul and λl ul C are, in general, non-zero. Figure 6.2 shows for a numerical example λ3 , uC 3 F C C F F and λF 3 , u3 as functions and the product λ3 u3 and λ3 u3 . For symmetry reasons, we plot only the left half of ΓsC . We note that both products are only non-zero on two faces of ΓsC . To get a better understanding, we illustrate the situation for d = 3 F in Figure 6.3. The possible support of λF l and ul is sketched in FigF ure 6.3(a,b). Then λF l ul does not vanish on the grey-shaded ring depicted F F in Figure 6.3(c). Starting with λF l , we construct a Pl λl such that the support of it is given as the grey-shaded region of Figure 6.3(d). Then it is F F F + obvious that PlF λF l ul = 0. Moreover, we will require that Pl λl ∈ M . F To measure the non-conformity of λC l and λl , we introduce mapped funcC C F F + tions Pl λl , Pl λl ∈ M . The construction is based on a decomposition of ΓsC into disjoint simply connected macro-faces F := ∪f ⊂F f , where f is an element of Fls . The set Fl;j , j ∈ {C, F}, of macro-faces forms a partition of ΓsC , i.e., ΓsC = ∪F ∈Fl;j F . Moreover, we require that the macrofaces satisfy hF := diamF ≤ C minf ⊂F hf for all F ∈ Fl;j and F = f if F ⊂ supp ujl \ supp λjl . The following assumption plays a crucial role in the proper scaling of the additional contact terms.

650 (a)

B. Wohlmuth (b)

(c)

(d)

F F F Figure 6.3. (a) supp uF l , (b) supp λl , (c) supp ul ∩ supp λl , F (d) supp PF λl and macro-faces F ∈ Fl;F .

Assumption 6.7. We assume that for j ∈ {C, F} there exists a macroface decomposition in the above sense such that, for F ∈ Fl;j with F = f there exists at least one f0 ⊂ ΓsC \ supp ujl with f0 ⊂ F . We note that the decomposition into macro-faces is not unique, and the macro-faces should be as small as possible. Moreover, each f ∈ Fls belongs to exactly one macro-face in Fl;j denoted by Ffj . In Figure 6.3(d), we show a possible macro-face decomposition. Remark 6.8. This assumption can easily be violated on coarse meshes but will hold asymptotically provided that the solution satisfies {[ut ] = 0} = O({[ut ] = 0}), where O(·) denotes the open set of its argument. In 2D this corresponds to the assumption that [ut ] cannot change its sign by passing through zero at a single point of ΓsC . Equivalently, we can require that in the case of a Coulomb law λt does not jump from +νλn to −νλn . In the case of a Tresca law, we assume that λt does not jump from +F to −F. For a more detailed discussion, we refer to Eck et al. (2005), Hild and Renard (2007) and Renard (2006). In 2D, it is automatically guaranteed that the support of λn does not contain isolated points, and thus we can always assume the existence of such a macro-face decomposition for j = C provided the mesh is fine enough. In terms of these preliminary settings, we define the operator Plj which is used in (6.17). It is face-wise given for the faces f ∈ Fls of the original mesh by:  0 f ⊂ supp ujl ,    j  j λl ds  F  f f ⊂ ΓsC \ supp ujl , Ffj ⊂ ΓsC \ supp ujl , (6.19) Plj λjl := |Ffj ∩(ΓsC \supp ujl )|  j j j j  s  f ⊂ Ff ⊂ ΓC \ supp ul , and λl ≥ 0, λ    l∗ j otherwise, Z l λl

651


(b)

(c)

10

10

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0

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C C C C C Figure 6.4. (a) λC 3 , (b) P3 λ3 , (c) λ3 − P3 λ3 .

(a)

(b)

(c)

10

10

10

0

0

0

−10

−10

−10

0

0.1

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0

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0

0.1

F F F F F Figure 6.5. (a) λF 3 , (b) P3 λ3 , (c) λ3 − P3 λ3 .

where | · | stands for the (d − 1)-dimensional area of its argument. Recalling the definition (4.7) of Zl∗ , it is easy to see that Zl∗ maps Ml+ onto Wl+ . We note that Plj λjl , j ∈ {C, F} is always well-defined. However, Assumption 6.7 guarantees that Plj λjl = 0 if λjl = 0. Figures 6.4 and 6.5 show λj3 , P3j λj3 and λj3 − P3j λj3 , j ∈ {C, F}, for the first example discussed in Section 6.8 and ν = 0.8. We note that λj3 − P3j λj3 = 0 only on four faces of ΓsC . For this example this also holds true for all refinement levels l ≥ 1. Our numerical results show that for all our examples Assumption 6.7 is asymptotically satisfied. Now, we specify our choice for δf in (6.17): 1 if f ⊂ Ffj such that F j Plj λjl ds = F j λjl ds, f f δf := (6.20) 1 otherwise. hf Lemma 6.9. Under Assumption 6.7, the following properties hold for Plj λjl , j ∈ {C, F}: (i) (ii) (iii) (iv)

Plj λjl ujl = 0, Plj λjl ≥ 0, j j j all F ∈ Fl;j , F Pl λl ds = F λl ds for j j j 2 λl − Pl λl − 1 ;Γ ≤ C f ∈F s hf λjl − Plj λjl 20;hf . 2

C

l

652

B. Wohlmuth

Proof. Property (i) follows directly from the first line in the definition (6.19) of Plj λjl . Recalling that λjl ∈ Ml+ and thus Zl∗ λjl ≥ 0 and f λjl ds ≥ 0 for all faces f , we find property (ii). For F ⊂ ΓsC \ supp ujl , the required equality follows from f λjl ds = f Zl∗ λjl ds. For F = f ⊂ supp ujl \ supp λjl , we have Plj λjl = λjl = 0. The only non-trivial case yields j j j j Pl λl ds = Pl λl ds = Plj λjl ds F

f ⊂F

=

|F ∩

f

f ⊂F \supp ujl

f

j F λl ds (ΓsC \ supp ujl )| f ⊂F \supp ujl

|f | = F

λjl ds.

Here we have used the fact that, by Assumption 6.7, the sum is not empty. To show property (iv), we use (iii). For v ∈ H 1/2 (ΓC ), we find j j j (λl − Pl λl )v ds = (λjl − Plj λjl )(v − Π0 v) ds ΓsC

F ∈Fl;j

≤C

F

hF λjl − Plj λjl 0;F v 1 ;F ,

F ∈Fl;j

2

where Π0 is the L2 -projection onto macro-elementwise constants. Property (iii) in Lemma 6.9 yields that δf = 1 for all faces. We note that the properties specified in Lemma 6.9 are also satisfied by more sophisticated operators. As in the case of dual basis functions, we can start from the existing choice and add suitable functions. By adding a quadratic function, we can make the result continuous. In Figure 6.6, we show two different alternatives for d = 2. Figure 6.6(a–c) illustrates a part of the contact zone C C and λC 3 and P3 λ3 . In (a,d), the operator defined by (6.19) is given. In (b,e) and (c,f), we depict an alternative definition using a piecewise affine and quadratic modification, respectively. By using polynomials of higher order in the definition (6.19), we can guarantee that λjl − Plj λjl is continuous, and that Plj λjl also satisfies the properties of Lemma 6.9. Although the piecewise quadratic modification results in a continuous C C λC 3 − P3 λ3 , the implementation of it in 3D is technically more involved and does not bring any qualitative benefit. Thus, from now on we only use the definition given by (6.19). Having introduced PjC and PjF , we can improve the upper bound for |||u − ul ||| for the special situation of a contact problem in 2D with matching meshes. However, an additional assumption is required. 1 (Γ ) Assumption 6.10. We assume that for d = 2, there exists a χ ∈ W∞ C t t such that |[ut ]| = χ[ut ] and moreover |[ul ]| = χ[ul ].


(b)

(c)

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653

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0.05

Figure 6.6. (a,d) Piecewise constant, (b,e) linear, (c,f) quadratic C C C C C modification; (a–c) λC 3 and P3 λ3 ; (d–f) λ3 − P3 λ3 .

Let us briefly comment on this assumption. The first part is closely related to Assumption 4.7 and is also reasonable to make in 3D. In both assumptions, the case that [ut ] changes its sign by passing through zero at a single point of ΓC is ruled out. The second part of the assumption is only reasonable in 2D. In fact, it follows from the first part and the a priori estimates for s > 1 for hl < H0 . But there is no possibility of estimating H0 . A direct consequence of Assumption 6.10 is the estimate |[ut ]| − |[utl ]| 1 ;ΓC = [ut − utl ]χ 1 ;ΓC ≤ C[ut ] − [utl ] 1 ;ΓC . 2

2

2

Lemma 6.11. Under Assumptions 6.10 and 6.7, we obtain the following upper bound in 2D, for the error in the energy norm for matching meshes, and a zero gap |||u − ul |||2 ≤(ηL + C(ηC + ηF + ξ1 ))|||u − ul ||| + Cνλnl − λn − 1 ;ΓC [unl ] − [un ] 1 ;ΓC , 2

2

where we formally set ν = 0 for a Tresca friction problem. Proof. We decompose the surface stress into a normal and a tangential part C n and recall that λn , [un ]ΓC = 0 = PlC λnl , uC l ΓC and that Pl λl , [un ]ΓC ≤ n 0. Furthermore, on matching meshes, (6.18a) states that uC l = [ul ] and n thus [ul ] ≤ 0. In terms of Lemma 6.9, the normal part in Lemma 6.5 can

654

B. Wohlmuth

then be bounded by λnl − λn , [un ] − [unl ]ΓC ≤ λnl , [un ] − [unl ]ΓC ≤ λnl − PlC λnl , [un ] − [unl ]ΓC ≤ CηC |||u − ul |||. For the tangential part, we have in 2D λtl − λt , [ut ] − Πl [utl ]ΓC = λtl − λt , [ut ] − Πl [utl ]ΓC . Assumption 6.10 and the fact that we are working with matching meshes guarantee that Πl [utl ]l = Πl [utl ] = |Πl [utl ]| = |[utl ]| and that uF l defined by (6.18b) is equal to χutl . In a first step, we consider the case of Tresca friction. We then find that, since λ ∈ M(F), λtl − λt , −Πl [utl ]ΓC ≤ −F, |[utl ]|ΓC + |λt |, |[utl ]|ΓC ≤ 0. Using Assumption 6.10 in combination with Lemma 6.9 and the definition of λF l given by (6.16b), we finally obtain λtl − λt ,[ut ]ΓC = λtl , χ|[ut ]|ΓC − F, |[ut ]|ΓC = −λF l , |[ut ]|ΓC F F F = PlF λF l − λl , |[ut ]|ΓC − Pl λl , |[ut ]|ΓC F ≤ PlF λF l − λl , |[ut ]|ΓC F t F F F t = PlF λF l − λl , |[ut ]| − |[ul ]|ΓC = Pl λl − λl , ([ut ] − [ul ])χΓC F t ≤ CPlF λF l − λl − 1 ;ΓC ([ut ] − [ul ])χ 1 ;ΓC 2

2

F t ≤ CPlF λF l − λl − 1 ;ΓC [ut ] − [ul ] 1 ;ΓC ≤ CηF |||u − ul |||. 2

2

In a second step, we consider the case of Coulomb friction. We observe that our assumptions guarantee λtl , [ut ]ΓC = λtl , χ2 [ut ]ΓC = νλnl − λF l , |[ut ]|ΓC . Using definition (6.16a) and applying the same techniques as before, we get n t λtl − λt ,[ut ] − [utl ]ΓC ≤ −λF l , |[ut ]|ΓC + νλl − λn , |[ut ]| − |[ul ]|ΓC

≤ C ηF |||u − ul ||| + νλnl − λn − 1 ;ΓC [ut ] − [utl ] 1 ;ΓC . 2

2

We note that Lemma 6.11 also holds true in 3D if we consider a contact problem on non-matching meshes and without friction. 6.3. Error bound for the Lagrange multiplier We now consider the error in the Lagrange multiplier. From the abstract theory of saddle-point problems, it is well known that the discretization error in the Lagrange multiplier can be bounded in terms of its best approximation error and the discretization error in the primal variable. Unfortunately, the

655


best approximation error is not computable and thus cannot be directly controlled within the adaptive refinement process. Therefore, we provide an estimate where the best approximation is replaced by the error indicator ηL . Of crucial importance for the stability of a saddle-point problem is the inf-sup constant. The bilinear form b(·, ·) reflects the H 1/2 -duality pairing. This observation motivates the use of a parameter-dependent norm for the Lagrange multiplier, which is equivalent to the H 1/2 -dual norm |||µ||| :=

sup

v=(0,vs ),vs ∈V0s

b(µ, v) . a(v, v)

(6.21)

We note that the norm of the Lagrange multiplier only depends on the Lamé parameters of the slave side but not on the master side. Alternatively, a different scaling can be used. Lemma 6.12.

The error in the Lagrange multiplier is bounded by |||λ − λl ||| ≤ |||u − ul ||| + ηL + Cξ1 ,

where ξ1 is the data oscillation term defined by (6.1). Proof. We start with the observation that b(·, v) restricted to v ∈ 0 × V0s is equal to ·, vs ΓC and recall that (u, λ) satisfies the equilibrium (2.15) and that ΓC = ΓsC . Using the definition (6.21), Lemma 6.3 and the symmetry of σ l , we find s s s s σ l nvs ds λ − λl , v ΓC = fs (v ) − as (u , v ) + ΓsC

=

(σ l − C(u )) : (v ) dx + s

Ωs

s

Ωs

(f − Π1 f )vs ds.

Here, we have used the fact that, by assumption, the Arnold–Winther space can exactly resolve the given Neumann data. Otherwise, an additional data oscillation term on the Neumann boundary has to be taken into account. Inserting the primal finite element approximation on the slave side, we get & λ − λl , vs ΓsC ≤ as (usl − us , usl − us ) + ηL + Cξ1 as (vs , vs ). Thus if ηL is an error estimator for the primal discretization error, it also provides an upper bound for the discretization error in the dual variable. 6.4. Upper bounds for the friction and contact terms In this subsection, we provide upper bounds for the terms ηC and ηF defined by (6.15) and (6.17). We recall that λjl is a numerical approximation of a non-negative functional, and λjl −Plj λjl measures the variational crime of the approximation λjl , j ∈ {C, F}. Due to our assumption that we work with

656

B. Wohlmuth

simplicial meshes, σ(ul )ns is constant on each face. In general, this does not hold for quadrilateral/hexahedral meshes. In that case, the estimates are technically more involved. Lemma 6.13. friction

Under Assumption 6.7, we have for Tresca and Coulomb ηC + ηF ≤ CηL .

n Proof. We recall λC l = λl and start with the normal contact term ηC . This term is naturally associated with the normal stress. For each face f ∈ Fls and its associated macro-face FfC , we then obtain, in terms of Lemma 6.3 and Lemma 6.9, C C 2 C C C 2 2 n 2 λC l − Pl λl 0;f ≤ λl − Pl λl 0;F C ≤ ChF C |λl |1;F C f f f 2 s 2 ≤C hf˜|n(σ l − σ(ul ))n|1;f˜ f˜⊂FfC

≤C

n(σ l − σ(usl ))n20;f˜

f˜⊂FfC

≤C

(σ l − σ(usl ))n20;f˜ ≤ C

f˜⊂FfC

1 σ l − σ(usl )20;˜ωf . hf

In the last step, we have used the properties of the macro-faces and a scaled inverse-type inequality for polynomials. Here ω ˜ f stands for the union of all s C elements T ∈ Tl such that ∂T ∩ Ff is non-trivial. Using the weighting of C C 2 λC of ηC;f , and (6.20) in combination l − Pl λl 0;f in the definition (6.17) 2 2 . Summing over all faces, with Assumption 6.7, we get ηC;f ≤ C T ⊂˜ωf ηL;T and noting that each element T is contained in at most a bounded number of ω ˜ f , we have ηC ≤ CηL . Now, we focus on the term ηF , which is associated with the friction law and thus involves the tangential stress component. We can follow the lines of the proof for ηC;f directly. Using the definition (6.16) for λF l , and formally setting ν = 0 in the case of a Tresca friction, both definitions (6.16a) and (6.16b) guarantee that

F F 2 2 ˜ t |2 λF h2f˜|λF h2f˜ ν|λnl |21;f˜ + |λ l − Pl λl 0;f ≤ C l |1;f˜ ≤ C l 1;f˜ f˜⊂FfF

f˜⊂FfF

t ˜ t := with λ s γp ψp . The first term on the right-hand side has already p∈PC;l l been discussed, and we only have to consider the second term in more detail.

657


(b)

(c)

10

10

10

5

5

5

0

0

0

−5

−5

−5

−10 0.06

0.07

0.08

0.09

0.1

−10 0.06

0.07

0.08

0.09

0.1

−10 0.06

0.07

0.08

0.09

0.1

C Figure 6.7. λC l − PC λl , l = 5 (a), l = 8 (b), l = 11 (c).

Using a discrete norm equivalence and noting that |γ tp −γ tq | ≤ γ tp −γ tq , we find that ˜ t | ˜ ≤ C|λt | ˜ ≤ C|λl | ˜ ≤ C|(σ l − σ(us ))n| ˜ |λ l 1;f l 1;f l 1;f 1;f and the same arguments as before apply. Remark 6.14. We note that if Assumption 6.7 is violated, the different scaling factor δf results in a mesh-dependent upper bound. Although we obtain ηC + ηF ≤ CηL theoretically, we observe for all our numerical tests that ηC +ηF decreases more rapidly than ηL . To get a better feeling for the role of ηC and ηF , we illustrate the decrease in the value of C λC l − PC λl for different refinement levels in Figure 6.7. It can be observed that only two neighbouring faces have a non-trivial contribution on the left part of the contact zone. Due to the scaling by hf of the L2 -norm of C λC l − PC λl in the definition of ηC , this contribution decreases rapidly within the adaptive refinement process. In particular, in this test example, ηC is C ∞ s equivalent to hmin λC l l − PC λl ∞ , where · ∞ is the L -norm on ΓC and min C C hl := minT ∈Tls hT . Due to the decrease of λl − PC λl ∞ with respect to the refinement level l, we expect that ηC can be asymptotically neglected compared to ηL . Theorem 6.15. Under Assumptions 6.7 and 6.10, for matching meshes and in 2D or in 3D with no friction, we obtain the upper bound |||u − ul ||| ≤ C(ηL + ξ1 ) for a Tresca or Coulomb friction problem provided that ν is small enough. Proof. The result follows by Lemmas 6.11 and 6.13, the application of Young’s inequality and Lemma 6.12 in the case of a Coulomb problem. We note that Assumption 6.10 is quite strong and cannot be verified within the adaptive refinement process. However, for all our numerical test

658

B. Wohlmuth

examples it is satisfied. We refer to Hild and Lleras (2009) for an alternative approach. However, additional terms then enter into the definition of the error estimator, which cannot be bounded with optimal lower constants. This is not the case for the approach we propose. 6.5. Lower bound for the discretization error In this subsection, we provide a local upper bound for our error estimator. For a one-sided contact problem without friction this result can be found in Weiss and Wohlmuth (2009); see Nicaise et al. (2008) for the linear elasticity setting. Here, we generalize it to Tresca and Coulomb friction problems in 2D and also to non-matching meshes. We restrict ourselves to d = 2, but the same type of argumentation can be applied to d = 3. We start with a preliminary result which bounds, in the discrete setting, the jump of the stress across a face by the jump of its surface traction. Lemma 6.16. For each face f ∈ Fl , we have [σ(ul )]0;f ≤ C[σ(ul )nf ]0;f . Proof. The proof is based on the observation that [∇ul tf ] = 0 on each face. Here tf is a normalized fixed orthogonal vector to nf . For each face f , the set {nf ⊗ nf , nf ⊗ tf , tf ⊗ nf , tf ⊗ tf } forms an orthonormal basis for the space of 2 × 2 constant tensors. Then, due to the symmetry of σ(ul ), we have (σ(ul )tf )nf = (σ(ul )nf )tf , and thus [σ(ul )]20;f = [σ(ul )nf ]20;f + [σ(ul )tf ]20;f = [σ(ul )nf ]20;f + [(σ(ul )tf )tf ]20;f + [(σ(ul )nf )tf ]20;f . Recalling (2.2) and that we have constant Lamé parameters on each body, we have [(σ(ul )tf )tf ] = 2µ[((ul )tf )tf ] + λ[tr (ul )] = 2µ[(∇ul tf )tf ] + λ[tr (ul )] = λ[tr (ul )] = λ[div ul ] = λ[(∇ul nf )nf ]. For the normal contribution [(σ(ul )nf )nf ], we can proceed in an analogous way. Using [((ul )nf )nf ] = [(∇ul nf )nf ], we get [(σ(ul )nf )nf ] = (2µ + λ)[(∇ul nf )nf ] and thus the jump of the discrete stress across a face is bounded by the jump of its surface traction, [σ(ul )]20;f = [σ(ul )nf ]20;f + [(σ(ul )nf )tf ]20;f 2 λ + [(σ(ul )nf )nf ]20;f ≤ 2[σ(ul )nf ]20;f . 2µ + λ In our proof for the lower bound, we start with an estimate for the contact term.

659


Lemma 6.17. There exists a constant independent of the mesh size such that for all contact faces f ∈ Flk , k ∈ {s, m}, we have # " hf Π1 f 0;ωf + [σ(ul )nF ]0;F , (σ l − σ(ul ))nf 0;f ≤ C F ∈Ffi

where Ffi stands for the set of all interior and Neumann faces in Ωk and on ΓkN such that F ∈ Ffi shares a vertex with f ∈ Flk and ω f := ∪T ∈T k T , with Tfk being the set of all elements in Tlk sharing a vertex with f .

f

Proof. To bound (σ l − σ(ul ))nf 0;f for f ∈ Flk , k ∈ {m, s}, we insert the with Π∗;s := Π∗l (see the definitions (3.10b) dual mortar projection Π∗;k l l and (6.2)), and get ∗;k (σ l − σ(ul ))n0;f ≤ (Π∗;k l − Id)σ(ul )n0;f + σ l n − Πl σ(ul )n0;f . (6.22) The first term in (6.22) can be bounded by using the properties of the dual mortar projection. We recall that Π∗;k reproduces constants and that l ∗;k Πl σ(ul )n restricted to f depends only on the values of σ(ul )n on f and its two adjacent faces. Let Ffk be the set of all faces in Flk such that its elements share at least one endpoint with f ; see Figure 6.8(a). Here the two elements in Ffk not equal to f are marked with a dashed line.

(a)

(b)

γf

γf

Figure 6.8. Definition of γf (a) and

Ffi

(b).

Setting γ f := ∪F ∈F k F , we obtain, by using the local L2 -stability of Π∗;k l , f

∗;k (Π∗;k l − Id)σ(ul )n0;f = (Πl − Id)(σ(ul )n − (σ(ul )n)|f )0;f ≤ Cσ(ul )n − (σ(ul )n)|f 0;γf .

Due to the fact that σ(ul ) restricted to each element is constant and that γf contains at most three faces, we find (σ(ul )|T (pF ) − σ(ul )|T (pF ))2 . σ(ul )n − (σ(ul )n)|f 20;γf ≤ Chf F ∈Ffk ,F =f

f

F

Here pF , F ∈ Ffk , F = f is the vertex shared between f and F , and TF ,

660

B. Wohlmuth

F ∈ Ffk , stands for the element in Tlk such that F ⊂ ∂T . Keeping in mind that all involved quantities are element-wise defined and constant on each element, the jump at the vertices can be bounded by the jump across the faces. Lemma 6.16 then gives (σ(ul )|T (pF ) − σ(ul )|T (pF )) ≤ [σ(ul )]|fˆ ≤ [σ(ul )nfˆ]|fˆ. F

f

fˆ∈Fpk

fˆ∈Fpk

F

F

Here FpkF stands for the set of all interior faces in Ωk sharing the node pF . Moreover, we introduce Ffi as the union of sets FpkF where pF is an endpoint of f ; see Figure 6.8(b). The two endpoints of f are marked with a bullet. To bound the second term on the right-hand side of (6.22), we use the fact that the local norm of an element in Mkl can be bounded by testing it with an element of Wlk = tr Vlk |ΓC with local support. Keeping in mind that σ l n− Π∗;k σ(ul )n ∈ Mkl , we can then write it as α1 ψpf + α2 ψpf on f , where 1

2

ˆ 1 φp f + α ˆ 2 φp f , pfi , i = 1, 2, are the two endpoints of f . We now set vh = α 1 2 ˆ i = 1 and α ˆ i αi = αi , i = 1, 2. Using the biorthogonality of φp where α and ψq , a simple calculation shows C ∗;k k k σ l n − Πl σ(ul )n0;f ≤ (σ l nk − Π∗;k l σ(ul )n )vl ds hf ΓkC C (fCk − σ(ukl )nk )vl ds. = k hf ΓC We note that the definition of vl yields that its support is in ω f and that its L2 -norm on Ωk is bounded by Chf and its L2 -norm on F ∈ Ffi is bounded by hf . We now apply Green’s formula on each element and find, in terms of (6.5), k k k k (fC − σ(ul )n )vl ds = ak (ul , vl ) − fk (vl ) − σ(ukl )nk vl ds ΓkC

= Ωk

"

fvl dx −

ΓkC

F ∈Ffi

≤ C hf Π1 f 0;ωf

[σ(ul )nF ]vl ds F

# + hf [σ(ul )nF ]0;F . F ∈Ffi

To obtain an upper bound for ηL;T , a discrete norm equivalence for Arnold–Winther elements is of crucial importance. For τ ∈ XT , we have cτ 20;T ≤ m0;T (τ ) + m1;T (τ ) + m2;T (τ ) ≤ Cτ 20;T .

(6.23)


661

Here m0;T (·), m1;T (·) and m2;T (·) are given by τ (p)2 , m0;T (τ ) := |T | p∈PT

2 2 f f m1;T (τ ) := τ nf ds + τ nf (φ1 − φ2 ) ds , f

f ∈FT

f

2 1 τ dx , m2;T (τ ) := |T | T

in terms of the degrees of freedom. The set PT stands for all vertices of T , FT is the set of all faces of T , and φfi , i = 1, 2, are the two nodal Lagrange basis functions associated with the two endpoints of f . Basically the proof is reduced to a scaling argument, the use of the matrix valued Piola transformation and the fact that in finite-dimensional spaces all norms are equivalent; see Arnold and Winther (2002) for details. Observing that σ(ul )|T ∈ XT , we can use (6.23) to bound the local contribution ηL;T . We do so by considering the three parts separately. Using (6.11), we get m2;T (σ l − σ(ul )) = 0, m1;T (σ l − σ(ul )) ≤ C

hf gf (nT nf ) − σ(ul )nT 20;f ,

f ∈FT

m0;T (σ l − σ(ul )) ≤ C

hf [σ(ul )]20;f ,

p∈PT f ∈Fp

where Fp is the set of all faces sharing the vertex p. For interior faces, [σ(ul )nf ] is the jump across the face, for Dirichlet boundary faces we define [σ(ul )nf ] := 0, for Neumann faces we set [σ(ul )nf ] := σ(ul )nT − fN , and for contact faces [σ(ul )nf ] := σ(ul )nT − fC . We note that the constant in the bound of m0;T (σ l −σ(ul )) depends on the maximum number of elements sharing a vertex but not on the mesh size. Let us briefly comment on the given bounds for mi;T (σ l −σ(ul )), i = 0, 1. Recalling that the equilibrated fluxes are consistent for each face, we find the upper bound 1 gf (nT nf ) − σ(ul )nT 0;f ≤ gf − {σ(ul )nf }0;f + [σ(ul )nf ]0;f , 2 and thus the bound for m1;T (σ l − σ(ul )) has the same structure as in the linear setting. The terms on the right are known from residual and equilibrated error estimators. In the linear setting (see Ainsworth and Oden (2000), Babuˇska and Strouboulis (2001) and Verf¨ urth (1994)), they can be bounded by the element and face residuals, and thus by the local discretization error and by

662

B. Wohlmuth

local higher-order data oscillation terms. We recall that the proof involves cubic element and quadratic edge bubbles. More precisely, it is based firstly on the observation that on a finite-dimensional polynomial space the L2 1/2 norm · 0;ω and the weighted L2 -norm bω · 0;ω are equivalent, where 1/2 bω is a suitably scaled bubble function. Secondly, integration by parts can be applied. Due to the local patch-wise construction of the error estimator, there is no difference compared to the linear setting, and as long as no contact face is involved the same techniques can be used. These preliminary observations and Lemma 6.16 can be used to bound m0;T (σ l − σ(ul )) by the jump of the surface traction: m0;T (σ l − σ(ul )) ≤ C hf [σ(ul )nf ]20;f . p∈PT f ∈Fp

Using the norm equivalence (6.23) and the bounds for the terms mi;T (σ l − σ(ul )), 0 ≤ i ≤ 2, we now obtain

2 ηL;T ≤C hf gf − {σ(ul )nf }20;f + [σ(ul )nf ]20;f f ∈FTi

+

hf (σ l − σ(ul ))nf 20;f .

(6.24)

f ∈FTb

Here FTi stands for the set of all faces not on ΓC ∪ ΓN and sharing a vertex with the element T , and FTb is the set of all faces on ΓC ∪ ΓN and sharing a vertex with the element T . Figure 6.9 illustrates for a given T the two sets: the elements in FTb are marked with dashed lines and the elements in FTi with bold solid lines.

T Figure 6.9. Definition of FTi and FTb .

Combining (6.24) and Lemma 6.17, we get the following lower bound for the discretization error. Theorem 6.18. The element contribution ηL;T of the error estimator can be bounded by the error on a local neighbourhood and some local oscillation terms 2 ηL;T |||u − ul |||2Tˆ + ξT2ˆ , ≤C Tˆ∈TT


663

where TT is the set of all elements sharing a vertex with T . We note that the number of elements in TT does not depend on the mesh size but only on the shape-regularity of the triangulation. Remark 6.19. As the construction of ηL;T is not restricted to 2D, and since all the proofs in this subsection also work out in 3D, Theorem 6.18 can also be shown to hold true in 3D. Moreover, in contrast to many other results, for the proof it does not make any difference if friction or no friction is applied. And as the proof shows, the results are also valid in the case of non-matching meshes. 6.6. Residual-type error estimator Lemma 6.13 and Theorem 6.18 justify that it is sufficient to take a standard estimator for linear elasticity problems and use λl as the Neumann condition on the possible contact boundary in the case of matching meshes or a one-sided contact problem. These considerations show that ηL can be replaced by any other error estimator suitable for the Lamé equation. Of special interest is a residual-type indicator ηR . Following the definition of the classical residual-based error estimator for the Laplace operator, we set hf h2 1 hf 2 ηR;T := T f + div σ(ul )20;T + [σ(ul )nf ]20;f + rf2 , 2µ 2 2µ 2µ ext int f ⊂ΓT

f ⊂ΓT

(6.25) where hT and hf stand for the element and face diameter, respectively, nf ext int denotes a unit face normal, and Γint T := {f ⊂ ∂T, f ⊂ Ω}, ΓT := ∂T \ ΓT . 2 The term rf depends on the type of the boundary part,  0 f ⊂ ΓD ,     σ(u )n − f 2 f ⊂ ΓN , N 0;f l rf2 := s s 2  f ⊂ ΓsC , σ(ul )n + λl 0;f    ∗;m m 2 f ⊂ Γm σ(um C. l )n − Πl λl 0;f The case f ⊂ ΓD ∪ ΓN is standard. For f ∈ ΓC , we apply the interpretation of λl as Neumann boundary data. It is possible to show that both error indicators ηR defined by (6.25) and ηL given in (6.10) are up to higher-order data oscillations locally equivalent. To do so, it is sufficient to consider the case of linear elasticity and given λl . Using the discrete norm equivalence (6.23) for Arnold–Winther-type elements, it is easy to see that ηL;T is equivalent to the sum of patch-wise contributions of ηR;T˜ and the face contributions of the difference between the equilibrated and the discrete fluxes. This difference satisfies a local system with a system matrix independent of the mesh size (see also (6.8)),

664

B. Wohlmuth

and the right-hand side is defined in terms of local face and element residual contributions. Then an algebraic argument and a correct scaling yield the equivalence. Remark 6.20. We point out that a numerical study shows that the ratio between ηC , ηF and ηL tends asymptotically to zero. As a result the terms ηC and ηF do not contribute significantly to the total estimated error and can thus often be neglected in the stopping criteria. The situation may be different if we only consider the local influence of ηC;f and ηF;f on the adaptive refinement process. Then these terms help the estimator to resolve, already on quite coarse meshes, the transfer between contact and no contact and the sliding and sticky part. 6.7. Non-matching meshes Let us briefly comment on the more general case of non-matching meshes. Then Lemma 6.5 still holds and shows that we also have to consider ηS defined by (6.14). Following the lines of a posteriori error estimates for a linear mortar setting (see, e.g., Belhachmi (2003, 2004), Bergam, Bernardi, Hecht and Mghazli (2003), Bernardi and Hecht (2002), Wheeler and Yotov (2005), Pousin and Sassi (2005) and Wohlmuth (1999a, 1999b)), suitable upper bounds can be shown for ηS . We note that in the case of non-matching meshes, mesh-independent upper and lower bounds always rely on some assumptions on the ratio of the coefficients and mesh sizes from master and slave side. In the case of globally constant Lamé parameters, we can bound ηS by ηL if the local ratio between the mesh size on the master and the slave side is bounded, i.e., hf m max < C. f m ∈Flm minf s ∈F s ;f m ∩f s =∅ hf s l The proof follows the lines of Section 6.4. However, Lemma 6.11 no longer holds. A more detailed look into the proof reveals, that then the term [ul ] − Πl [ul ] enters into the estimate. This term is zero if the mesh on the slave side of the contact is a refinement of the one on the master side, but on general non-matching meshes this term does not vanish. To obtain an upper bound for the error in the energy norm, an additional term of the form 2µs 2 ηˆD := [ul ] − Πl [ul ]20;f h f s f ∈Fl

is a possibility. But then, the lower bound is tricky and will not work out. This can be explained by the difference in the structure of the exact solution. In the linear mortar setting, we can exploit the fact that the jump of the exact solution vanishes across the interfaces and Πl [ul ] = 0. This no longer holds for contact problems. The jump in the normal direction is only zero


665

on the actual contact zone and in the tangential direction only on the sticky part but, in general, not on all of ΓC . A more careful analysis shows that n n one has to bound PlC λC l − λn , Πl [ul ] − [ul ]ΓC for the normal contribution. We refer to Coorevits, Hild and Pelle (2000) and Wohlmuth (2007) for some results on contact problems with non-matching meshes. However, we note that none of those is fully satisfying from the theoretical point of view. In particular, certain ‘higher-order’ terms depend on the unknown solution and are not accessible during the refinement process. Alternatively one can include in the indicator terms depending on 2µs 2 ηD := max(0, [unl ])20;f , (6.26) h f s f ∈Fl

to take into account the possible discrete penetration. But then the ratio between upper and lower bound will be not independent of the mesh size. We refer to Bernardi and Hecht (2002), Pousin and Sassi (2005) and Wohlmuth (1999a, 1999b) for error indicators and estimators in the case of non-matching meshes. One of the problems with non-matching meshes is that standard inverse estimates for finite elements do not necessarily apply for [vl ] on ΓC , vl ∈ Vl . A priori error estimates use the best approximation property of the spaces and the stability of mortar projections, while a posteriori error estimates work with duality and the residual. In the linear elasticity setting of a glueing problem, the jump [ul ] across the interfaces characterizes the nonconformity of the approach. The natural norm to associate with is the H 1/2 -norm. Unfortunately, in the case of non-matching meshes no inverse inequality holds. To get a better understanding of the influence of nonmatching meshes, we consider a simplified setting. Let I := (−1, 1); then we introduce two different globally quasi-uniform partitions given by the nodes p1 := −1, p2 := 0 and p3 := 1 and q1 := −1, q2 := t ∈ [−1/2; 1/2] and q3 := 1. Associated with these nodes are two finite element functions vp := 3i=1 αi φpi and vq := 3i=1 βi φqi , where φpi and φqi are the standard hat functions associated with the nodes pi , and qi , 1 ≤ i ≤ 3, respectively. Then the standard inverse estimates applied to this very special situation gives ws;I ≤ Cw0;I for s ∈ [0, 1] and w = vp or w = vq . Here the constant does not depend on s, or on the coefficient, or on t ∈ [−1/2; 1/2]. The situation is drastically different if we consider w = vp − vq . Figure 6.10 shows φp2 − φq2 for different values of t ∈ {−1/2, −2/3, −1/6, 0}. For t ∈ [−1/2; 1/2] and t = 0, we obtain  p  − φq2 0;I s ∈ [0; 12 ),   Cφ 2 p q C − log |t|φ2 − φ2 0;I s = 12 , φp2 − φq2 s;I ≤  p q C  s ∈ ( 12 , 1].  s− 1 φ2 − φ2 0;I

Lemma 6.21.

|t|

2

666

B. Wohlmuth

Figure 6.10. Influence of the node q2 on the basis function φq2 and φp2 − φq2 .

Proof. Due to symmetry arguments, it is sufficient to consider the case t ∈ [−1/2, 0). A straightforward computation then shows that ∆φ2 := φp2 − φq2 is given by  t  −1 ≤ x ≤ t,  t+1 (x + 1) 1 ∆φ2 (x) = x + 1−t (x − t) t < x < 0,   t 0 ≤ x ≤ 1, 1−t (x − 1) and that ∆φ2 20;I = O(t2 ) whereas ∆φ2 21;I = O(|t|). For 0 < s < 1, we use the standard definition of the Aronstein–Slobodeckij norm in 1D; see, e.g., Adams (1975). Introducing I1 := (−1, t), I2 := (t, 0) and I3 := (0, 1), we get 3 (∆φ2 (x) − ∆φ2 (y))2 2 |∆φ2 |s;I = dx dy. (x − y)2s+1 Ii Ij i,j=1

Using that ∆φ2 is piecewise affine, we find by a simple interpolation argument that 3 (∆φ2 (x) − ∆φ2 (y))2 dx dy = O(|t|2 + |t|3−2s ). (x − y)2s+1 Ii Ii i=1

Integration of the remaining terms yields for s ∈ (0, 1) 3 O(|t|3−2s ) (∆φ2 (x) − ∆φ2 (y))2 dx dy = (x − y)2s+1 O(−|t|2 log(|t|)) Ii Ij i,j=1,j =i

s = 12 , s = 12 .

Figure 6.11 illustrates the inverse inequality for different parameters s ∈ {0.1, 0.25, 0.5, 0.51, 0.75, 0.95}. The straight line is φp2 − φq2 s;I evaluated analytically, and the markers indicate qualitatively the upper bound. For s = 0.1 and s = 0.25, the upper bound in the inverse inequality is bounded independently of t. For s = 0.5 logarithmic growth can be observed, and for s > 0.5 the singularity is the more dominant the closer s is to 1.

Numerical algorithms for variational inequalities 1.2

s= 0.1

1.6

s= 0.25

0.6

667 s= 0.5

10

1.5 1.15 0.4

1.4

10

1.1 1.3 0.2

1.05 −0.5

0

0.5

−0.5

0

0.5

3

0.7

10

s= 0.51

10 −0.5

0

0.5

5

10

s= 0.75

10

s= 0.95

2

10 0.5

10

1

10

0.3

10 −0.5

0

0

0.5

10 −0.5

0

0

0.5

10 −0.5

0

0.5

Figure 6.11. Influence of the parameter s on the inverse inequality for t ∈ [−0.5, 0.5], t = 0.

This lemma shows that on non-matching meshes we cannot simply apply inverse estimates to go from one norm to another. As a possible remedy, one first has to apply approximation properties and secondly inverse estimates for the faces on the interface between master and slave side. But then the ratio between the mesh sizes on master and slave side must be considered. Although we do not give a rigorous mathematical analysis of the terms associated with the influence of non-matching meshes, we provide some numerical results. We consider the same numerical example as illustrated in the paragraph on different materials of Section 6.8 and set i = 3. The normal contact pressure has a mild singularity at the left endpoint of the contact interface. Figure 6.12 illustrates the normal displacement and the normal contact stress at the contact zone of the slave and the master body for Levels 3, 5 and 7. As can be seen from Figure 6.12, there is almost no penetration, although we do work on non-matching meshes. Thus we can expect ηD defined by (6.26) to be very small compared to ηL . The situation is different for the contact stress. Due to the choice of a biorthogonal set of basis functions, the visualization of λnl as a function shows oscillations. Figure 6.12(d– n f) illustrates how the difference λnl − Π∗;m l λl decays with respect to the refinement level. Figure 6.13(a) shows the error decay of the different contributions. From the very beginning the term ηD is much smaller than ηL and can thus be neglected. We note, however, that it has the same convergence order as ηL . For the term ηS defined by (6.14), we observe that it is quantitatively

668

B. Wohlmuth

(a)

(b) Level 3

−6

2

x 10

u

x 10

S M

1.5

uS−uM

x 10

M

1.5

uS−uM 1

0.5

0.5

0.5

0

0

0

−0.5

−0.5

−0.5

−1

0.002

0.004

0.006

0.008

0.01

(d)

−1.5 0

u

M

uS−uM

1

−1

u

S

u

1

−1.5 0

Level 7

−6

2

u

S

u

1.5

(c) Level 5

−6

2

−1

0.002

0.004

0.006

0.008

0.01

(e)

−1.5 0

0.002

Level 5 λM

400

700

λS λM

500

λ −λ S

0.008

λS λM

600

λ −λ

M

S

400

300

0.01

Level 7

600

λS

0.006

(f)

Level 3 500

0.004

M

λ −λ S

500

300

400

200

300

100

200

0

100

M

200 100 0 −100

−100

−200 0

0.002

0.004

0.006

0.008

0.01

−200 0

0

0.002

0.004

0.006

0.008

0.01

−100 0

0.002

0.004

0.006

0.008

0.01

Figure 6.12. Normal displacement (a–c) and normal contact stress (d–f) for different refinement levels. (a)

(b)

(c)

Error decay for same materials (i=3)

Level 3

Level 5

400

−3

10

500

λ

S

λM λS−λM

300

−4

10

250

Error

λ

S

350

λM

400

λS−λM

300

200

−5

10

200 150

−6

10

η

100

η

50

100

L C

η

0

S

−7

10

0

ηD 1

10

2

10

3

10

No. of elements

4

10

5

10

−50 0

0.002

0.004

0.006

0.008

0.01

−100 0

0.002

0.004

0.006

0.008

0.01

Figure 6.13. Normal contact stress for different refinement levels.

and qualitatively of the same order as ηC . This might be surprising since only a few faces contribute to ηC , whereas all contact faces contribute to ηS . We note, however, that closer to the singularities the face diameter hf , which enters as weight into the definition of ηS , is much smaller than at the end of the actual zone of contact. Figures 6.13(b) and 6.13(c) show Zl∗ λnl , ∗;m ∗;m n n ∗ n Zl∗;m Π∗;m Πl λl . Here, the operator Zl∗;m l λl and the difference Zl λl −Zl is defined similarly to (4.7) but with respect to the master side. As can be clearly observed, much smaller values for the difference are obtained and the plotted functions exhibit fewer oscillations. Thus, using these values to define ηS would result in a smaller value, but then the equilibrium (6.5) is no longer satisfied, and our proof does not apply.

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Remark 6.22. The term ηS measures how well the mesh on the master side can resolve the discrete surface traction provided by the Lagrange multiplier on the slave side. The weight corresponds to the case of a Neumann boundary. Correspondingly, the term ηˆD reflects the consistency error of a linear mortar approach and quantifies the difference in the discrete solution on master and slave sides. It has the standard weight of a Dirichlet boundary term. We recall that for contact problems, the principle of equilibrium of forces holds and thus ηS is also appropriate. This is not the case for ηˆD . Here one should use the modified definition ηD given in (6.26), which takes into account the contact constraints. 6.8. Numerical results for adaptive mesh refinement We consider a series of different test examples. A detailed discussion and the specific problem settings can be found in H¨ ueber and Wohlmuth (2010), Weiss and Wohlmuth (2009) and Wohlmuth (2007). We start with one-sided contact problems where ηL is a mathematically sound error estimator. One-sided two-dimensional Coulomb problem In a first test, we consider a one-sided Coulomb friction problem in 2D with the friction coefficient given by ν = 0, ν = 0.3 and ν = 0.8. Using ηL and a mean value strategy to define the adaptively refined meshes, we compute ηR , ηL , ηC and ηF on each refinement level and show the decay with respect to the number of elements in Figure 6.14. The normal and tangential stress for ν = 0.3 and ν = 0.8 is given in Figure 4.7. For ν = 0.0, the term ηF is equal to zero. For all three settings, we observe that we recover an optimal decay of the residual and equilibrated error estimator. As expected, the two additional terms ηC and ηF can be neglected asymptotically compared to ηL and ηR . These numerical results show that ηL yields a reliable stopping criteria. For ν = 0.8, we observe, from the very beginning, that both terms ηC and (a)

(b)

(c)

Error decay for ηL

Error decay for ηL

0

0

10

−1

10

−1

10

−1

−2

10

ηR

10 ηR

−2

ηL

10

η −3

O(h) ηC

−3

10

L

ηF

2

O(h ) −4

1

10

O(h) ηC

−3

10

ηF

O(h2)

−4

10

ηR

−2

10

η

O(h) ηC

L

10

Error

10

Error

Error

Error decay for ηL

0

10

2

10

3

4

10

10

No. of elements

5

10

10

O(h2)

−4

1

10

2

10

3

10

No. of elements

4

10

5

10

10

1

10

2

3

10

10

No. of elements

Figure 6.14. Error decay for adaptive refinement using ηL for ν = 0.0 (a), ν = 0.3 (b), ν = 0.8 (c).

4

10

5

10

670 (a)

B. Wohlmuth (b)

(c)

Figure 6.15. Distorted meshes after 8 adaptive refinement steps for different values of the coefficient of friction ν = 0.0 (a), ν = 0.3 (b), ν = 0.8 (c).

ηF tend with h2l to zero where we set hl := (#Tl )1/d . The situation is different for ν = 0.3. Here we observe, during the first refinement steps, a reduced decay order of ηF compared to ηC . This effect can be explained by Figure 4.7(b). For ν = 0.3, we have a very small sticky zone, and thus on coarse meshes Assumption 6.7 is violated, and we have only one vertex p with |γpt | − νγpn = 0. Then PlF λF l = 0 and δf = 1/hf on the two boundary faces sharing the vertex p. As soon as the sticky part is resolved, the friction term ηF drops down. The meshes obtained from the error indicator ηL after 8 refinement steps are shown in Figure 6.15. We remark that the interior of the contact boundary ΓsC will be refined considerably more at the sticky part of the boundary. This effect arises from the high gradient of the tangential component of the Lagrange multiplier. Furthermore, the boundary region of the contact boundary actually in contact is detected and thus refined by the error indicator. In the next test series, we illustrate the influence of the choice of error indicator on the adaptive refinement process for ν = 0.3. We compare ηL , ηL + ηC + ηF and ηR . Figure 6.16(a) shows a zoom of the very small sticky zone with the normal and the tangential components of the Lagrange multiplier as well as the friction bound νλn and the difference νλn −λt . The last expression is positive at all sticky nodes; for the sliding nodes it vanishes. Figure 6.16(b) shows that there is no significant difference in ηL and that the adaptive refinement process is not sensitive to the selected error indicator. In particular, a standard residual-based error indicator provides very good results, and no additional terms resulting from the contact situation with Coulomb friction are required. The only difference can be observed in ηF . In this example, the sticky zone is very small, and thus it cannot be well resolved on lower refinement levels. Using ηL + ηC + ηF as the indicator for the adaptive marking gives quite large element contributions for elements having both sticky and sliding vertices. Thus, these elements are within the pre-asymptotic range and all elements with δf = 1 are selected to be refined in the next step. As a consequence, the sticky contact zone can be resolved

671


(b)

(c) Error decay of η

Lagrange multiplier

Error decay of η

L

6

F

0

0

10

10

4 −1

−4

10

Error

Error

0

−2

−1

10

2

−2

10

ηL

40000*uτ

ηL+ηC+ηF

τ −3

10

η +η +η L

3/4

O(h) 0.505

10

F

O(h) O(h

−4

0.5

C

ηR

−3

R

± F*λn −6 0.495

η

10

η

F*λn−|λτ|

−2

10

L

λ

)

−4

1

10

2

10

3

10

No. of elements

4

10

5

10

10

1

10

2

10

3

10

4

10

5

10

No. of elements

Figure 6.16. Zoom of Lagrange multiplier (a), estimated error ηL (b) and ηF (c) for different refinement strategies for ν = 0.3. (a)

(b)

(c)

(d)

Figure 6.17. Distorted meshes at contact boundary after 6 adaptive refinement steps for ηL (a,b) and ηL + ηC + ηF (c,d) refinement for the coefficients of friction ν = 0.3 (a,c) and ν = 0.8 (b,d).

for a smaller refinement level compared to the case where the refinement is only controlled by ηL or ηR . However, this influence is quite small on the global estimated error, as can be seen in Figure 6.16(b). For all three series of adaptively refined meshes, we observe qualitatively and quantitatively the same results. To show the effect of ηF in more detail, we consider in Figure 6.17 a zoom of the meshes at the contact boundary for ηL and ηL + ηC + ηF used as the marking indicator for the two cases of ν = 0.3 and ν = 0.8. In both cases, we observe that the intersection between the sticky and the sliding zone as well as the intersection between contact and no contact is well resolved by the error indicator. The first one is resolved more accurately when the term ηF is used in the refinement strategy. Influence of the regularity of the solution To test the influence of the regularity of the solution on the adaptive refinement, we consider a parameter-dependent one-sided contact problem with no friction. The unit square is pushed onto a triangle with different opening angles α at the contact vertex. In our tests, we use α = 2/3π, α = π/2 and α = π/3. Due to the decreasing regularity of the solution for decreasing α, we observe that for α = π/3, the mesh is much more locally refined compared to α = 2π/3.

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B. Wohlmuth

(a)

(b)

(c)

Mesh at level 12

Mesh at level 12

Mesh at level 12

Figure 6.18. Square on triangle: mesh on level 12 for α = 2π/3 (a), α = π/2 (b), α = π/3 (c).

(a)

(b)

(c)

10

10

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10

−1

−1

10

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1

2

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3

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2

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3

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0

η O(h) η

0

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2

1

η O(h) η

−2

2

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3

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4

10

1

10

η O(h) ηC

−2

ηcontact C

O(h2)

10

−1

10

10

C ηcontact C

−3

O(h ) 10

−1

10

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C contact ηC

10

10

Est. error

Est. error

−1

4

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No. of elements

Error reduction

0

10

3

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(f)

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10

2

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Error reduction

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O(h0.3) adaptive O(h) 1

10

(e) Error reduction

−2

uniform −1

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4

10

No. of elements

(d)

Est. error

Error reduction uniform O(h0.5) adaptive O(h)

0

Est. error

uniform 0.7 O(h ) adaptive O(h)

0

Est. error

Error reduction

Est. error

Error reduction

O(h2)

−3

10 2

10

3

10

No. of elements

4

10

1

10

2

10

3

10

No. of elements

4

10

Figure 6.19. Square on triangle α = 2π/3, π/2, π/3: Estimated error ηL for adaptive and uniform refinement (a–c) and contact (d–f). comparison of ηL , ηC and ηC

The regularity of the solution is known and our numerical convergence rates for uniform refinement are in good agreement with the theory: see Figure 6.19. The slope in the estimated error decay is approximatively 0.7 for α = 2π/3, 0.5 for α = π/2 and 0.3 for α = π/3. Thus, with respect to the total degrees of freedom, we have only a sub-optimal convergence. The situation is drastically improved if adaptive mesh refinement techniques are applied. We then observe O(hl ) behaviour for all three cases. As in the first example, we observe for all α that the error contribution ηC decreases much faster compared to ηL . However, for low regularity, we

673


do not observe that ηC is of order O(h2l ). To get a better understanding of the situation, we define #1/2 " contact 2 := ηC;f ≤ ηC , ηC f ∈Fls;b

where Fls;b ⊂ Fls is the set of all faces being separated from supp(Πl [unl ] − gl ) by at most m faces, where m ∈ N is a small and fixed number. If contact = the singularity in λn is weak enough, we find asymptotically ηC ηC . However, for strong singularities, there are non-trivial contributions of C C C C PlC λC l − λl , whereas (Pl λl − λl )([un ] − g) = 0 in a neighbourhood of the singularity: see, e.g., the case α = π/3 in Figure 6.19(f). Two-sided contact problem with a corner singularity We use a geometry such that there is a zero gap between the two bodies and set ν = 0. Figure 6.20(c) shows how the error decays for uniform and adaptive refinement. Asymptotically we find better convergence rates for the adaptive setting. This results from the presence of the singularities which can be found at the two endpoints of the contact boundary. Due to the singularities at (−0.5, 0) and (0.5, 0), we observe strong local refinement at these points. (a)

(b)

(c) Punch problem adaptive uniform

0

Est. rel. error

10

−1

10

−2

10

1

10

2

10

3

4

10 10 No. of nodes

5

10

Figure 6.20. Deformed meshes after 3 and 6 adaptive refinement steps (a,b) and error decay (c).

Two-sided contact problem with different materials In this test setting we consider the influence of the material parameters on the adaptive refinement process. Two unit squares in contact are considered. Our initial mesh on the upper body consists of 9 uniform quadrilaterals, and on the lower body we have 4 uniform quadrilateral elements. Thus, we have non-conforming meshes at the contact boundary. In this example, we study the influence of the material parameters on the error indicator. We consider five different situations (i = 1, . . . , 5) for the material parameters. For the

674

B. Wohlmuth

upper subdomain, we select Young’s modulus as E1up = E2up = E3up = 2 × 105 ,

E4up = 2 × 106 ,

E5up = 2 × 109 ,

and for the lower body we define E1low = 2 × 109 ,

E2low = 2 × 106 ,

E3low = E4low = E5low = 2 × 105 .

We remark that for i = 3, both subdomains have the same material parameters, whereas for i = 1, 2 the upper subdomain is softer and for i = 4, 5 the lower subdomain is softer. For i = 1, 2, 3 the upper subdomain plays the role of the slave side, and for i = 4, 5 the lower subdomain is the slave side. Figure 6.21 shows the adaptively refined meshes after 8 refinement steps using ηL as error indicator. As expected, the adaptive refinement strongly depends on the material parameters. The softer the domain, the more it is refined. Having the same material parameters on both sides, we get the same level of mesh refinement on both sides; see Figure 6.21(c). In addition, we compare the estimated error decay between uniform and adaptive refinement using ηL as the error indicator. The decay of ηL and ηC for both approaches are shown in Figure 6.22 for i = 1, 3, 5. In all three situations, we observe that the error decay for the adaptive refinement shows the expected order. The different orders in the error decay of uniform and adaptive refinement can also be observed in the contact term ηC . (a)

(b)

(c)

(d)

(e)

Figure 6.21. Influence of the material parameter on the adaptive refinement, 1 ≤ i ≤ 5 (left to right). Error decay for upper softer (i=1)

Error decay for same materials (i=3)

−3

−3

10

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10

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−4

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ηL (uniform) ηL (adaptive)

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O(h)

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O(h)

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ηC (adaptive) 10

Error

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Error

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Error

Error decay for lower softer (i=5)

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1

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O(h)

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O(h2) 2

10

3

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No. of elements

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O(h2) 1

10

2

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3

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No. of elements

Figure 6.22. Error decay for uniform and adaptive refinement.

4

10

5

10


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In the rest of this section, we show that the error indicator can also be applied to more general situations such as non-matching meshes in 3D and large deformation. Dynamical contact problem We now apply our error indicator to a dynamic contact problem with Coulomb friction and use a refinement and coarsening strategy. We use a modified mid-point rule and a stabilized active set strategy as the timeintegration scheme and non-linear solver. In the case of a mesh which is constant in time, the total energy including contact work is preserved within each time step. As the initial condition, we have zero displacements and a constant velocity. The resulting adaptively refined meshes at different time steps are depicted in Figure 6.23. Here, a co-rotational formulation for the contact has been used (see Hauret, Salomon, Weiss and Wohlmuth (2008) and Salomon, Weiss and Wohlmuth (2008)), and the displacement is decomposed in each time step into a rotation and a small displacement which can be handled within the theory of linearized elasticity. (a)

(b)

(c)

(d)

(e)

Figure 6.23. Adaptive grid at time step tk : k = 0 (a), k = 20 (b), k = 40 (c), k = 80 (d), k = 120 (e).

Three-dimensional contact problem We consider the situation of a torus between two rectangular plates. The plates are considered as the slave sides defining the mesh for the Lagrange multiplier λ. We apply Coulomb’s law with ν = 0.8 and assume the plates to be softer. Figures 6.24(a) and 6.24(b) show the refined meshes with the effective von Mises stress. As can be observed, a local adaptive refinement occurs at the contact zone, resulting in highly non-matching meshes. Contact problem with large deformation In our last numerical example, we consider a contact problem without friction but with finite deformations in the two-dimensional setting. Instead of the linearized stress tensor σ (see (2.2)), we use a well-known neo-Hookean material law. In the definition of the error indicator, we replace σ by the first Piola–Kirchhoff stress tensor given by

λ 2 P= J − 1 F− + µ F − F− , 2

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B. Wohlmuth (b)


R η

R

O(h) 1

Error

10

3

10

4

5

10

10

No. of elements

Figure 6.24. Adaptively generated mesh (a,b) and error decay (c).

with the deformation gradient F = Id + ∇u and its determinant J := det(F). Due to this additional nonlinearity in the material, the first line in the discretized algebraic problem formulation (5.1) is no longer linear and the semi-smooth Newton scheme automatically takes into account the non-linear material law. Using the semi-smooth Newton method to treat nonlinear contact conditions has the main advantage that both nonlinearities, the contact conditions and the material nonlinearity, can be handled within one iteration loop. We press a half-ring onto a bar by applying suitable Dirichlet boundary conditions. The half-ring, being the lower half-part of a full ring, is assumed to be the slave side Ωs with inner radius ri = 80 and outer radius ro = 100 having its mid-point at the origin. The numerical results are presented in Figure 6.25. We adapt ηR to the non-linear material law and obtain an error indicator which can be easily evaluated. Figure 6.25(c) shows the estimated error decay. We also observe order-hl convergence in that situation. (b)


R ηR

1.9

10

O(h)

1.7

Error

(a)

10

1.5

10

1.3

10

2

10

3

10

No. of elements

Figure 6.25. Distorted meshes after 2 and 4 adaptive refinement steps (a,b) and error decay (c).


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6.9. AFEM strategies Finally, we point out that AFEM refinement strategies can also be designed for variational inequalities such as contact- or obstacle-type problems. These refinement strategies were originally designed for conforming finite elements applied to the Laplace operator. We refer to the original work by D¨ orfler (1996) and the surveys of Morin, Nochetto and Siebert (2002) and Nochetto, Siebert and Veeser (2009). Nowadays these techniques have been widely generalized and successfully applied to other types of equations and elements. Special refinement rules, in combination with a control over the data oscillation terms, lead to a guaranteed error decay. Moreover, optimal convergence results, under mild regularity assumptions, have been stated in Binev, Dahmen and DeVore (2004) and Stevenson (2005, 2007); see also Cascon, Kreuzer, Nochetto and Siebert (2008). For variational inequalities, the first theoretical results can be found in Braess, Carstensen and Hoppe (2007, 2009a) for obstacle problems. For one-sided contact problems and no friction, a guaranteed decay in the energy can be achieved: see Weiss and Wohlmuth (2009). Here we only show some numerical results. We point out that the proof relies on the fact that the discrete solution ul satisfies the non-penetration condition [unl ] ≤ 0 strongly. Thus the analysis can be based on the constrained minimization problem (2.7) for contact without friction. The discrete convex cone Kl is then a subset of K, and so we have J(u) ≤ J(ul ). This does not hold for two-body contact problems on non-matching meshes, and there is no straightforward way to generalize the result to non-matching meshes. The main difficulty in the application of AFEM results to variational inequalities is the loss of the Galerkin orthogonality compared to a standard conforming finite element discretization for linear problems. It is shown in Braess, Carstensen and Hoppe (2007) that a possible remedy is to consider the error in the energy δl := J(ul ) − J(u) and not the error in the energy norm. For the most simple case of ν = 0 and no gap, we get δl − δl+1 = J(ul ) − J(ul+1 ) 1 = |||ul − ul+1 |||2 + a(ul − ul+1 , ul+1 ) − f (ul − ul+1 ) 2 1 = |||ul − ul+1 |||2 − b(λl+1 , ul − ul+1 ) 2 1 ≥ |||ul − ul+1 |||2 . 2 Figure 6.26 shows the energy reduction for the second example in Section 6.8. The mean value of the energy decay per refinement step is between 0.7 and 0.8 for all three settings.

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B. Wohlmuth

(a)

(b)

(c)

Energy reduction

Energy reduction

Energy reduction 0

0

δ 2

O(h ) −1

10

−2

10

−3

10

δ

Appr. energy difference δ

10



0

2

O(h ) −1

10

−2

10

2

10

3

10

No. of elements

4

10

O(h2) −1

10

−2

10

10

10 1

10

δ

−3

−3

10

10

1

10

2

10

3

10

4

10

1

10

No. of elements

2

10

3

10

4

10

No. of elements

Figure 6.26. Energy reduction for α = 2π/3 (a), α = π/2 (b), α = π/3 (c).

6.10. Conclusion Based on the variational formulation (6.5), we have introduced an error indicator ηL , which is shown to be an error estimator for some simplified situations. For contact problems without friction this is true for matching meshes or one-sided contact problems with a zero gap and constant normal on the contact zone. Assumption 6.7 depends only on the discrete solution and can be checked easily for each pair (ul , λl ). The situation is more challenging for friction problems with Tresca or Coulomb friction. In that case, we have to restrict ourselves to 2D and the additional Assumption 6.10 is required. Our theoretical and numerical results show that there is no need to add terms related to the variational inequality, such as ηC and ηF , to the estimator. These observations provide an interesting and attractive general construction principle of a posteriori error estimators for variational inequalities. In a first step, the weak variational inequality for the primal variable has to be reformulated by means of a locally defined Lagrange multiplier as a variational equality. We recall that the Lagrange multiplier acts as an additional external source term, volume or surface, on the system. Then, in a second step, we apply any type of well-known a posteriori error estimator. If the pairing between discrete Lagrange multiplier and discrete finite element solution is suitable, we can then recover upper and lower bounds for the discretization error in the primal but also the dual variable. This is a very strong result and also applies to obstacle-type problems. As a by-product, we find that for a linear setting with inhomogeneous Neumann data, the boundary terms in the residual error estimator can be removed, and only the data oscillation enters the bounds.

7. Energy-preserving time-integration scheme In the previous sections, an abstract framework was provided to solve numerically a stationary contact problem efficiently in terms of Lagrange multipliers. The discretization is realized as weakly consistent and uniformly stable saddle-point formulation, and the Lagrange multiplier plays an essential role in the definition of the non-linear solver as well as in the design


679

of the error estimator. However, it is well known that these multipliers often show oscillations and numerical instabilities in dynamic situations: see, e.g., Borri, Bottasso and Trainelli (2001), Hauret and Le Tallec (2006), Ballard, Léger and Pratt (2006), Martins, Barbarin, Raous and Pinto da Costa (1999) and Raous, Barbarin and Vola (2002). Figure 7.1 illustrates this effect if a classical Newmark scheme is applied on a saddle-point formulation for a dynamical Hertz contact problem. Normal Lagrange multiplier 10 8 6 4 2 0 −2 0

0.2

0.4

0.6 Time

0.8

1 −3

x 10

Figure 7.1. Oscillations of the Lagrange multiplier in normal direction.

Thus there is huge demand for more robust numerical schemes. Recently, different techniques have been introduced for coping with these instabilities. The most promising approaches are based on a mass redistribution and go back to the early work by Khenous, Laborde and Renard (2006a, 2006b, 2008), and alternatively on a predictor–corrector scheme (see Deuflhard, Krause and Ertel (2008), Klapproth, Deuflhard and Schiela (2009), Klapproth, Schiela and Deuflhard (2010), Kornhuber, Krause, Sander, Deuflhard and Ertel (2007), Krause and Walloth (2009)), which is motivated by wellestablished two-stage schemes in plasticity; see the overview by Simo (1998). Although quite different, from the initial perspective, the proposed modifications in Khenous, Laborde and Renard (2006b) and Kornhuber et al. (2007) both require an additional global L2 -type projection step. The algorithm in the latter work involves per time step a global projection which is equivalent to solving a uniformly well-conditioned constrained minimization problem, whereas the mass redistribution can be worked out in a global pre-process. Stability is obviously of crucial importance, but in many engineering applications energy conservation is also essential. We refer to the early contributions of Armero and Petöcz (1998, 1999), Demkowicz and Bajer (2001), Laursen and Chawla (1997) and Pandolfi, Kane, Marsden and Ortiz (2002), and to the more recent work by Betsch and Hesch (2007), Gonzales, Schmidt and Ortiz (2010) and Hesch and Betsch (2009, 2010). Special emphasis on

680

B. Wohlmuth

(a)

(b)

t0

t50

t75

t100

t125

t150

t200

Figure 7.2. Deformation: no friction (a) and stick condition (b).

(a)

(b)

40 30 10 0 0.0

30

total kinetic strain

20

0.5

1.0

1.5

40 20

2.0

2.5

10 0 0.0

0.5

1.0

1.5

2.0

2.5

Figure 7.3. Energy: no friction (a) and stick condition (b).

the DAE aspect of mechanical systems with constraints can be found in Betsch and Steinmann (2002a, 2002b), Lunk and Simeon (2006) and Simeon (2006), and we refer to Gonzalez (2000), Hilber, Hughes and Taylor (1977) and Simo and Tarnow (1992) for time-integration schemes in non-linear elasto-dynamics. Figure 7.2 (see Hartmann, Brunßen, Ramm and Wohlmuth (2007) for details) illustrates the application of an energy-preserving method which is a combination of a velocity update motivated by the persistency condition of Laursen and Love (2002), and the generalized energy-momentum method proposed in Kuhl and Ramm (1999). The energy is shown in Figure 7.3(a,b). As can be seen from the two pictures, the total energy as the sum of the kinetic and strain energy is constant with respect to time. Although these approaches are energy-conserving, no reliable numerical results for the contact stresses can be obtained without additional postprocessing and stabilization. Here, we combine different techniques, a mass redistribution for its stabilization effect and the persistency condition for its role in the energy evolution. To start with, we extend our simple quasi-static model (2.3)–(2.6) to the dynamic case and include the density of the body, and we refer to Eck et al. (2005), H¨ ueber, Matei and Wohlmuth (2007), H¨ ueber and Wohlmuth (2005b) and Kikuchi and Oden (1988). The problem under consideration


681

can be written in its weak form as follows. Find u ∈ L∞ ((0, T ), VD ) and λ ∈ L2 ((0, T ), M(λn )) such that u˙ ∈ H 1/2 ((0, T ), V),

¨ ∈ L2 ((0, T ), V0 ) u

and v ∈ V0 t ∈ (0, T ], µ ∈ M(λn ), t ∈ (0, T ], v ∈ V0 , v ∈ V, where the zero-order bilinear form m(·, ·) is given by m(u, v) := Ω uv dx, and we assume that ρ is constant on each subdomain. Using the notation of Section 5 and the basis transformation of Section 5.2, we then obtain the following semi-discrete problem: m(¨ u, v) + a(u, v) + b(λ, v) = f (v), ˙ ≤ g(µ − λ), bn (µ − λ, u) + bt (µ − λ, u) (u(0, ·), v) = (u0 , v), ˙ (u(0, ·), v) = (v0 , v),

¨ l + Al ul + Bl λl = fl , Ml u s , Cpn (γpn , unp ) = 0, p ∈ PC;l s . Cpt (γ p , u˙ p ) = 0, p ∈ PC;l

(7.1)

Comparing (5.7) with (7.1), we find that both systems have a similar structure. The main difference is that in the dynamic case we have to use the already introduced splitting of the NCP function into its normal and tangential part. Formally, the semi-discrete system can be classified as a differential-algebraic equation with index three: see Brenan, Campbell and Petzold (1989) and Hairer and Wanner (1991). For this type of problem, standard time-integration schemes can result in strong oscillations; see also Figure 7.1. We do not follow the original approach of Khenous et al. (2006a, 2006b) but apply a locally defined mass modification, which can be directly assembled within the standard framework of finite element technology and does not require a global projection. Introducing a combined space–time integration, we have to replace the mass matrix Ml in (7.1) by a modified one, Mlmod . The presentation here follows the lines of Hager and Wohlmuth (2009a) and Hager, H¨ ueber and Wohlmuth (2008); see also the more recent contributions of Doyen and Ern (2009), Hager (2010) and Renard (2010). mod s Using an Mlmod such that mmod pq = mqp = 0 for all p ∈ PC;l reduces the index of the DAE system (7.1) from three to one and has a stabilization and regularization effect on the modified solution. Thus such an approach seems to be very attractive provided the computational cost is of low complexity and the order of the discretization is not reduced. To recover the motion r of a rigid body, we have to make sure that r Mlmod r = r Ml r. Sufficient conditions are formulated in Khenous et al. (2006a) as preservation of the

682

B. Wohlmuth

total mass Ω dx, of the centre of gravity Ω x dx and of the moments of inertia Ω xx dx: (M0) 1 Mlmod 1 = 1 Ml 1, (M1) 1 Mlmod xi = 1 Ml xi , 1 ≤ i ≤ d, mod x = x M x , 1 ≤ i, j ≤ d. (M2) x j l j i Ml i

Here we use the notation 1 = (1, . . . , 1) ∈ RNl , Nl := dimVl , and xi = Nl ((x p ei )ei )p∈Pl ∈ R , with xp being the coordinate vector of the vertex p and ei ∈ Rd the ith unit vector. 7.1. Local construction of Mlmod As standard in the finite element context, we assume that the elements of Ml are obtained by an element-wise assembling process and by the use of quadrature formulas on each element t ∈ Tl . Our definition of the modified mass matrix is based on a second triangulation which groups elements on the slave side near the contact zone into macro-elements. As a preliminary step, we introduce the strip Sl by s supp φp S l := ∪p∈PC;l

and

ΩSl := Ω \ Sl .

In the following, we assume that a fixed macro-triangulation TH is associated with Tl . By this we understand that there exists a second triangulation, possibly with hanging nodes, such that each element of TH can be written as the union of elements in Tl . Moreover TH has the following properties. • If T ∈ TH with T ⊂ ΩSl then T ∈ Tl . • If T ∈ TH \ Tl , then there exists exactly one element tT ∈ Tl with tT ⊂ ΩSl ∩ T and at most M elements t ∈ Tl with t ⊂ Sl ∩ T , where M is a fixed small number and not depending on l. Furthermore, all sub-elements of T can be accessed starting from tT by crossing only faces of sub-elements of T . We note that for a given Tl , there exists more than one macro-triangulation. Figure 7.4 illustrates different possibilities of TH for a given Tl . The elements of the original mesh are marked with dashed lines, whereas the elements of the macro-triangulation are given by bold lines. The shaded subdomains show the different types of elements in TH \ Tl and the strips Sl , Dl := ˜ l , which is defined as the union of all elements t in Tl such ∪T ∈TH \Tl T and D that ∂t ∩ Dl = ∅. Remark 7.1. If Tl is obtained from Tl−1 by uniform refinement based on a decomposition of each element into 2d sub-elements, then a natural construction for TH is straightforward: see Figure 7.4(b).


(b)

683

(c)

Figure 7.4. Different macro-triangulations TH for ˜ l (c). a given Tl (a,b) and the strips Sl , Dl , D

In the following, we restrict ourselves to simplicial triangulations Tl but note that these techniques can also be applied to more general meshes. We refer to Hager et al. (2008) for a discussion in the case of quadrilateral meshes. The mass matrix Mlmod is associated with the modified bilinear form mH (·, ·), which is defined in terms of a suitable quadrature formula applied to the elements of the macro-triangulation, i.e., mH (vl , wl ) :=

NT

wiT vl (qiT )wl (qiT ),

T ∈TH i=1

where NT is the number of quadrature nodes and qiT and wiT are the quadrature nodes and weights, respectively. From now on we omit the upper index T if it is clear from the context. Oneach element t ∈ TH ∩ Tl , we use a standard quadrature formula such that t ρφp φq dx is exactly evaluated by it. For each T ∈ TH \ Tl , we select our quadrature formula in a special way. The construction of the macrotriangulation guarantees that for T ∈ TH there exists a unique tT ∈ Tl such s that tT ⊂ T and ∂tT ∩ ΓC = ∅. On tT , we use the second-order Lagrange interpolation nodes qi , 1 ≤ i ≤ NT , as quadrature points. For a simplicial element tT , we have NT := 6 for d = 2 and NT := 10 for d = 3. For a quadrilateral/hexahedral element, we have NT := 9 for d = 2 and NT := 27 for d = 3. The weights are computed as wi := φ2qi dx, 1 ≤ i ≤ NT , T

where 1 ≤ i ≤ NT , is the second-order nodal Lagrange basis function on tT extended as polynomial to T . φ2qi ,

Lemma 7.2. The choice of the quadrature formula yields the following properties for the modified mass matrix Mlmod . (1) (M0)–(M2) hold. (2) (Mlmod )pq = (Mlmod )qp = 0 if p ∈ Pls .

684

B. Wohlmuth

(3) The local mass matrices MT associated with each macro-element are positive semi-definite and have rank rT . For a simplicial element tT , we have rT = d + 1, and for a quadrilateral/hexahedral element, rT is 2d . Proof. The definition of the macro-elements and the construction of the nodes and weights directly yield that globally quadratic functions are integrated exactly and thus (M0)–(M2) hold. The definition of tT and of the quadrature nodes gives that no quadrature node is placed in the interior of Sl or on ΓsC . Each basis function φp associated with a vertex p ∈ Pls is zero on ΩSl and thus φp (qiT ) = 0, for all quadrature nodes qiT , 1 ≤ i ≤ NT , and all T ∈ TH . If T ∈ TH ∩ Tl , it is obvious that MT is positive definite. For T ∈ TH \ Tl , MT is an nT × nT matrix, with nT ≥ rT . The kernel of MT has dimension at least nT − rT , and MT has a positive definite sub-matrix MtT ∈ RrT ×rT . In Figure 7.5, we present for d = 2 a suitable quadrature formula for two different macro-elements T . Here, we have selected the case when the macro-elements are associated with a coarser simplicial mesh from which Tl is obtained by uniform refinement. We note that in this special situation all sub-elements of a macro-element have the same volume, and the weights do not depend on the shape of tT . q3

(a)

q6

q3

(b)

q5

q1

q2

q6 q1

q5 q2 q4

q4 Figure 7.5. Quadrature rules for two different macro-elements.

In the situation in Figure 7.5(a) the weights are given by w1 = w2 = 1/3|tT |, w3 = 2/3|tT |, w4 = 2|tT |, and w5 = w6 = −2/3|tT |, and in the situation in Figure 7.5(b) the weights are w1 = w2 = 8/3|tT |, w3 = 4|tT |, w4 = 16/3|tT |, and w5 = w6 = −16/3|tT |. The local mass matrix MT associated with the macro-element T in Figures 7.5(a) and 7.5(b) reads as   8 4 −4 0 0 0   3 4 − 12 0 2 8 −4 0 0 0 2   3 1   | |tT |  |t −4 −4 4 0 0 0 2 − 0 T    , 2 2 MT = , MT =  0 0 0 0 0 0 3 − 12 − 12 1 0 3     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Numerical algorithms for variational inequalities Case I s s q7 q9 q8 s s q6 q4 s q5 s s q3 q1 s q2 s

Case II s s q5 q6 s s q3 q4 s s q1 q2

q7 q5 q3 q1

Case s s s s

III s s s s

q8 q6 q4 q2

685

Case IV s q6 s s q4 q5 s s q3 q1 s q2

Figure 7.6. Different quadrature rules for a quadrilateral macro-element.

respectively. Both local matrices are obviously singular but have rank three and are positive semi-definite. We briefly comment on the properties specified in Lemma 7.2 and recall that the first one guarantees that the rigid body motions are not affected by the modification. The second one reduces the index from three to one, and the third is essential to guarantee stability. For quadrilateral finite elements in 2D, we can also select different quadrature rules. The quadrature formula has to satisfy the properties (1) and (2) of Lemma 7.2. In addition, all elements in Q2 (tT ) extended as polynomials onto T have to be integrated , where FtT is the element mapping exactly. Here Qj (tT ) := Qj (Tˆ) ◦ Ft−1 T ˆ from the reference quadrilateral T onto tT , and Qj (Tˆ) is the space of all bi-linear elements for j = 1 and of all bi-quadratic elements for j = 2. This condition is very natural and results from the fact that Q1 (t), t ∈ Tl , is the local low-order finite element space. We refer to Ciarlet (1998) for a rigorous mathematical analysis of the influence of quadrature errors on the quality of the finite element approach. Then property (3) of Lemma 7.2 is automatically satisfied. Figure 7.6 shows a typical macro-element T for a quadrilateral mesh and four different quadrature formulas. All of them guarantee that quadratic functions on T are integrated exactly, but only the first three yield stable numerical results. The quadrature nodes are given as shown in Figure 7.6. Case I follows the specified construction principle. The weights for this special macro-element are given by w1 = w3 = 5/9|tT |, w2 = 20/9|tT |, w4 = w6 = −4/9|tT |, w5 = −16/9|tT |, w7 = w9 = 2/9|tT | and w8 = 8/9|tT | in Case I. Cases II and III are based on Gauss nodes in the tangential direction, whereas in the normal direction we use equilibrated spaced nodes. In Case II, the weights are w1 = w2 = 2/3|tT |, w3 = w4 = −4/3|tT |, and w5 = w6 = 5/3|tT |. The weights for Case III are given by w1 = w2 = 4|tT |, w3 = w4 = −15/2|tT |, w5 = w6 = 6|tT | and w7 = w8 = −3/2|tT |. Case II can only be used if the element mappings are affine. For the more general case of det Ft−1 ∈ Q1 (tT ), T Case I or Case III should be used. Both cases can also be easily applied in the

686

B. Wohlmuth

Table 7.1. Energy and displacement results for Case IV. Time

Kinetic energy

Elastic energy

Total energy

x2 -displacement

4.0 · 10−6 5.0 · 10−6 6.0 · 10−6 6.5 · 10−6 7.0 · 10−6

9.3 · 10−2 −3.1 −1.4 · 105 −2.9 · 107 −6.1 · 109

7.3 · 10−3 3.2 1.4 · 105 2.9 · 107 6.1 · 109

1.0 · 10−1 1.0 · 10−1 1.0 · 10−1 1.0 · 10−1 1.0 · 10−1

−3.4 · 10−2 −4.1 · 10−2 1.8 · 10−1 −3.4 4.9 · 101

3D setting. The negative weights do not disturb the computation as long as all local mass matrices MT for T ∈ TH are positive semi-definite. Case IV is based on triangular-distributed second-order Lagrange interpolation nodes. The weights are defined by w1 = w3 = 1/3, w2 = 1, w4 = w5 − 2/3 and w6 = 2/3, and give rise to the following local mass matrix:   23 9 −5 −3 0 0  9 23 −3 −5 0 0   |tT |  −5 −3 3 5 0 0  . MT = 3 0 0 24  −3 −5 5  0 0 0 0 0 0 0 0 0 0 0 0 A straightforward computation shows that MT has rank four but has one negative eigenvalue. Moreover, a closer look reveals that the global mass matrix can also have a negative eigenvalue. Thus, even for simple contact problems a non-physical negative kinetic energy can occur: see Table 7.1. Although the total energy is preserved, the numerical results are of no use. From the very beginning the kinetic energy is negative and exponentially increasing. The vertical displacement at a selected node is highly oscillating and far too big. This effect is a result of the negative eigenvalue of the local mass matrix MT . Thus Case IV cannot be used for numerical computations. 7.2. Analysis in terms of an interpolation operator We do not provide a full analysis for the mass modification. In Hager and Wohlmuth (2009a) it has been shown that for a linear elasticity problem, one can show O(hl + ∆t2 ) a priori estimates for the fully discretized problem in the H 1 (Ω)-norm in space and the discrete L∞ -norm in time for the displacement and in the L2 (ΩSl )-norm in space and the discrete L∞ -norm in time for the velocity. Moreover, under some additional regularity, an order (h2l + ∆t2 ) can be obtained in the L2 (Ω)-norm in space and the discrete L∞ -norm in time for the displacement. We restrict ourselves to families of quasi-uniform shape-regular triangulations. Figure 7.7 shows a qualitative

687


(b) Energy 0.01

0.025

0

0.02

−0.01

0.01 0.005 0 0

Displacement in x −direction

1

0.03

0.015

(c) Displacement in x −direction

2

1

standard mass matrix modified mass matrix

0.5 −0.02

total elastic kinetic total (mod) elastic (mod) kinetic (mod) 5

0

−0.03

−0.5

−0.04

10

15

20

25

−0.05 0

5

10

15

20

25

Time

Time

standard mass matrix modified mass matrix

−1 0

5

10

15

20

25

Time

Figure 7.7. Energy (a), horizontal (b) and vertical displacement at a selected node (c). (a)

(b)

Error at time t = 3e−4

Error difference at time t = 3e−4

−1

10

−6

10

−2

10

L2 error H1 error c*h 2 c*h

−3

10

4

diff L2 diff H1 c*h4

−8

10

5

c*h 8

16

32

1/h

4

8

16

32

1/h

Figure 7.8. Error decay in the L2 - and H 1 -norm in space (a) and decay of the difference of the two formulations (b).

comparison between the modified mass approach and the standard one for a geometrically non-linear elasticity problem without contact: see Hager et al. (2008) for details. As can be observed, there is no significant difference in the displacement and in both settings the energy is preserved. The same parameter and geometry setting but for the linearized strain formulation is considered for a quantitative comparison in Figure 7.8. In Figure 7.8(b) we show that the difference between the two approaches can be asymptotically neglected and is of higher order than the discretization error. For the discretization error in space an order h2l and an order hl in the L2 - and H 1 -norm can be observed, respectively, whereas the difference decreases with order h5l and order h4l , respectively. The analysis of the modified formulation applies ideas from the analysis of the influence of quadrature errors as well as of the influence of a standard mass lumping. Here we only provide two results that are essential to obtaining a priori estimates. In the previous subsection, the modified bilinear form mH (·, ·) was introduced in terms of a quadrature formula based on the macro-triangulation. Now, we define an interpolation operator IH such that mH (vl , wl ) = m(IH vl , IH wl ),

wl , vl ∈ Vl .

(7.2)

688

B. Wohlmuth s 1 (b) (c) 0

J

J

J

J

J J

J

J

Jr

J

tT J

tT J J

J J J

s

J

0 0 0 1 J J

J

J

J

J

J

J

J

J

J

J

J −1 J 1 Figure 7.9. (a,b) Nodal values of two modified basis functions on the macro-element T ; (c) support of (a).

(a)

To do so, we introduce a set of modified basis functions φmod which are p possibly discontinuous. The new basis functions are defined for each macros . If T ∈ T ∩T , then element and are associated with the vertices p ∈ Pl \PC;l H l mod | := E(φ | ) φmod | := φ | for all vertices p of T . If T ∈ T \T , then φ p T p tT T H T l p p for all vertices p of tT , where E stands for the polynomial extension of φp |tT onto T . Figures 7.9 and 7.10 illustrate the two-dimensional case for a simplicial mesh. In Figure 7.9 the macro-element T is the union of two elements in Tl , whereas in Figure 7.10 the macro-element T is the union of four elements in Tl . In both cases, only three basis functions are locally associated with the macro-element. The support of the modified basis function is still local but can be enlarged: see Figures 7.9(c) and 7.10(c). In terms of these modified basis functions, we define our interpolation s } by operator IH : Vl → span {φmod ei ; 1 ≤ i ≤ d, p ∈ Pl \ PC;l p vl (p)φmod . IH vl := p s p∈Pl \PC;l

The construction of the quadrature formula and the operator IH are both based on the macro-elements such that it is easy to see that (7.2) holds. In terms of the properties of the operator IH , the semi-discrete system (7.1) can be analysed. We refer to the recent contributions of Doyen and Ern (2009) and to Hager and Wohlmuth (2009a) in the case of a linear problem with given surface traction on ΓC . We do not provide any details but remark that the analysis follows the lines of mass lumping techniques. We refer to Thomée (1997) for the parabolic case and to Baker and Dougalis (1976) for the second-order hyperbolic case. The main difference is that in our situation Mlmod is singular and does not define a matrix that is spectrally equivalent to Ml . Thus the analysis is more technical and relies on some additional arguments. Firstly, the semi-discrete system has to be considered and Gronwall’s lemma plays an important role. Secondly the fully discrete system has to be analysed and Taylor expansion with respect to time enters into the proof. Although these two steps are quite technical they are well

689

Numerical algorithms for variational inequalities s 1 (b) (c) 0

J

J

J

J

J J

J

J

J

J r

tT J

tT J J

J

J

J s

J

0 0 0 1 J

J

J

J

J J

J

J

J

J

J

J

J

J J

J

J J −1 −1 J 1 −1 0 2 J

J

J

J

J

J Figure 7.10. (a,b) Nodal values of two modified basis functions on the macro-element T ; (c) support of (a).

(a)

established; see also Baker and Dougalis (1976), Evans (1998), Dautray and Lions (1992), Raviart and Thomas (1983) and Thomée (1997). One crucial ingredient is the following lemma, which bounds the quadrature error introduced by the bilinear form mH (·, ·): ∆m(vl , wl ) := mH (vl , wl ) − m(vl , wl ). If vl , wl ∈ Vl then

|∆m(vl , wl )| ≤ Chl vl 0;Dl wl 1;Dl + wl 0;Dl vl 1;Dl ,

Lemma 7.3.

(7.3a)

and if v, w ∈ V then

|∆m(Zl v, Zl w)| ≤ Ch2l v1;Ω w2;Ω + w1;Ω vl 2;Ω ,

(7.3b)

where Zl is a locally defined Scott–Zhang-type operator (Scott and Zhang 1990). Proof. The proof is based on the properties of the operator IH . Using (7.2) and noting that (IH vl )|ΩSl = vl |ΩSl , we find ∆m(vl , wl ) = m(IH vl − vl , IH wl ) + m(vl , IH wl − wl ) ≤ C(IH vl − vl 0;Sl IH wl 0;Sl + vl 0;Sl IH wl − wl 0;Sl ) ≤ Chl (vl 1;Dl wl 0;Dl + vl 0;Sl wl 1;Dl ). To show (7.3b), we apply (7.3a), the local L2 - and H 1 -stability of Zl , and a 1D Sobolev embedding ∆m(Zl v, Zl w) ≤ Chl (Zl v1;Dl Zl w0;Dl + Zl v0;Sl Zl w1;Dl ) ≤ Chl (v1;D˜ l w0;D˜ l + v0;D˜ l w1;D˜ l ) ≤ Chl ( hl v2;Ω hl w1;Ω + hl v1;Ω hl w2;Ω ), ˜ l , and the diameter of the strip D ˜ l perpendicular to ΓC is where Dl ⊂ D bounded by Chl .

690

B. Wohlmuth

Remark 7.4. The bound (7.3b) can be weakened by using Besov space norms with index 1/2 and 3/2; see, e.g., Li et al. (2010). Remark 7.5. In the proof of Lemma 7.3, we do not use that IH reproduces macro-element-wise affine functions. The same arguments hold true if IH is replaced by a locally defined operator which reproduces vl ∈ Vl on ΩSl , and on t ⊂ Sl it reproduces vl if it is constant. This observation motivates the use of a more simple quadrature formula based on the triangulation Tl to define the bilinear form mH (·, ·); see also Section 7.4. As can be easily seen, the modified bilinear form mH (·, ·) is continuous and coercive with respect to the L2 (ΩSl )-norm but not coercive with respect to the L2 (Ω)-norm. Thus Aubin–Nitsche-type arguments provide only a priori estimates in the L2 (ΩSl )-norm, which is a semi-norm on L2 (Ω). The following lemma shows that a priori estimates in the L2 (Ω)-norm can also be obtained and have the same order. Lemma 7.6. For v ∈ V, we have v0;Ω ≤ C(v0;ΩSl + hl v1;Ω ). Proof. We start with the non-overlapping decomposition of Ω into Sl and ΩSl . To bound v0;Sl , we apply element-wise a Poincaré–Friedrichs-type inequality and a scaling argument. In terms of

v20;t ≤ C h2l v21;t + hl v20;f , 1 2 2 v0;t + |v|1;t · v0;t , v0;f ≤ C hl where f ⊂ ∂t is a face of the element t ∈ Tl , we find

v20;Sl ≤ C h2l v21;Sl + hl v20;∂Sl ∩∂ΩS ≤ C h2l v21;Ω + v20;ΩS . l

l

7.3. Energy-preserving time integration For many applications energy is one of the quantities of interest to preserve. Here, we present an energy-conserving time-integration scheme based on the standard Newmark method (Hughes 1987, Kane, Marsden, Ortiz and West 2000) in combination with a persistency condition introduced in Laursen and Chawla (1997); see also Bajer and Demkowicz (2002), Chawla and Laursen (1998), Demkowicz and Bajer (2001), Laursen and Meng (2001) and Laursen and Simo (1993b). The discrete displacement at time tk := t0 + k∆t is given by ukl and the velocity by vlk . The Newmark scheme with γ := 12 and β := 14 applied to the first line of


691

(7.1), where we replace Ml by Mlmod , then yields 2 1 2 k+ 12 k+ 21 k+1 mod M + + B λ = f + ∆u A M mod vlk − Al ukl , l l l l l l (∆t)2 2 ∆t l (7.4a) 2 − vlk , (7.4b) vlk+1 = ∆uk+1 l ∆t where the time increment ∆uk+1 of the displacement is defined by ∆uk+1 := l l k+1/2 k+1 1 k+1 k k ul − ul , and we set fl := 2 (fl + fl ). To obtain an energy-conserving scheme for frictionless contact problems, we have to discretize the non-penetration condition in a suitable way: see the second line in (7.1). As is well known, the complementarity condition λn ([un ] − g) = 0 is not suitable, but has to be replaced by the persistency condition λn ([u˙ n ] − g) ˙ = 0: see Laursen and Chawla (1997). Letting gpk be the space- and time-discretized gap function, we replace the non-penetration condition by gpk > 0 ⇒ (γpn )k+1/2 = 0,  n k+1/2 (γ ) ≥ 0,    p gpk ≤ 0 ⇒ ∆(unp )k+1 ≤ gpk ,    ∆(unp )k+1 (γpn )k+1/2 = 0. This discrete version of the persistency condition can then be rewritten in the NCP function framework, and reads as ! Cpn ((γpn )k+1/2 , ∆(unp )k+1 ) := (γpn )k+1/2 − max 0, (γpn )k+1/2 + cn g˜pk = 0, (7.4c) s , where for all p ∈ PC;l

  − (γpn )k+1/2 − g k p k cn g˜p :=  ∆(un )k+1 p

if gpk > 0, if gpk ≤ 0.

The tangential part of the NCP function (see the third line of (7.1)) is discretized in time by

k+ 1 s (7.4d) Cpt γ p 2 , ∆ukp = 0, p ∈ PC;l (see also Chawla and Laursen (1998)). Now the space- and time-discretized system of a two-body contact problem with Coulomb friction is given in each time step by the non-linear system of equations (7.4a)–(7.4d). Introducing the discrete energy Elk = (Elkin )k + (Elpot )k at time tk as the sum of the kinematic (Elkin )k := 12 vlk Mlmod vlk and the potential energy

692

B. Wohlmuth

(Elpot )k := 12 ukl Al ukl , we can show that the time-integration scheme preserves energy. Lemma 7.7. The contact algorithm defined by (7.4a)–(7.4d) guarantees energy preservation in the sense that k+ 21

Elk+1 − Elk = ∆tvl k+ 21

where vl k+ 21

vl

:=

k+ 12

Bl λ l

k+1 1 2 (vl

k+ 12

(fl

1

− Bl (λtl )k+ 2 ),

+ vlk ). Moreover, in the case of ν = 0, we have

= 0.

Proof. We start with the observation that (7.4b) yields for the mass contribution k+1 1 k+ 21 mod ∆ul k − vl = (vlk+1 + vlk )Mlmod (vlk+1 − vlk ) v l Ml ∆t 4 1 = (vlk+1 Mlmod vlk+1 − vlk Mlmod vlk ) 4 1 kin k+1 1 kin k − (El ) , = (El ) 2 2 and for the stiffness term uk+1 − ukl 1 k+ 21 k+1 k ∆ul + ul = l Al (uk+1 vl Al + ukl ) l 2 2∆t 1 − ukl Al ukl ) = (uk+1 Al uk+1 l 2∆t l 1 pot k+1 (El ) − (Elpot )k . = ∆t k+ 21

Using vl

as the test function in (7.4a), we then obtain k+ 21

Elk+1 − Elk = ∆tvl k+ 1

k+ 12

(fl

k+ 12

− Bl λ l

).

k+ 1

In the last step, we consider vl 2 Bl λl 2 in more detail. It can be decomposed into its normal and tangential contribution, i.e., k+ 21

vl

k+ 21

Bl λ l

k+ 21

= vl

1 1 Bl (λtl )k+ 2 + (λnl )k+ 2 .

For the normal part, we find, in terms of the discrete persistency condition, which is realized by (7.4c), that it vanishes. In the case ν = 0, (5.4) k+ 1

1

yields that Cpt (γp 2 , ∆ukp ) = (γ tp )k+ 2 and thus (7.4d) guarantees that the tangential part is equal to zero for ν = 0.


693

7.4. Numerical results In this subsection, we provide some numerical results which illustrate the performance of the applied mass modification. We refer to Hager et al. (2008), H¨ ueber (2008) and Hager and Wohlmuth (2009a) for the problem setting and parameter choices. These techniques can also be generalized to an overlapping two-scale domain decomposition approach: see Hager (2010) and Hager, Hauret, Le Tallec and Wohlmuth (2010a). The introduction of the macro-element triangulation is motivated by theoretical and computational aspects. It allows a local assembling process while at the same time the properties (M0)–(M2) can easily be satisfied. Our theoretical considerations show however that the same order of convergence in the a priori estimates can be obtained with less restrictive assumptions: see Remark 7.5. Therefore, we use a second type of quadrature formula associated with the elements of the original mesh Tl . If t ∈ Tl is in ΩSl , we use a standard quadrature rule such that t ρφp φq dx is exactly evaluated by it. If t ∈ Tl is in Sl , we use a quadrature formula such that all nodes are placed on ∂t ∩ (∂Sl ∩ ΩSl ). Moreover, we require that on each element constants are integrated exactly and that the resulting element mass matrices are positive semi-definite. Figure 7.11 illustrates the situation for simplicial elements in 2D. (a)

(b) Case I q2 q 1 q1 v v v @ @ t@ t @ @ @ @ @

q3 v

(c) Case II q2 q1 v v @ t @ @ @

(d) Case III q1 v @ t @ @ @

Figure 7.11. Different positions of t with respect to ∂Sl ∩ ΩSl .

In the situation in Figure 7.11(a), i.e., the element has one face on the contact boundary, there is no other option than placing the quadrature node on the opposite vertex and setting the weight w1 to |t|. If the element t shares only one vertex with ΓsC , we have several options: see Cases I–III. For Case I, we define w1 := w3 := 1/12|t| and w2 := 5/6|t|. The weights in Case II are set equal and thus are 1/2|t|. And in Case III, we have w1 := |t|. Then in all cases constant functions are integrated exactly and the local mass matrices are given by       7 5 0 5 4 0 1 1 0 |t|  |t|  |t|  5 7 0, MT = 4 5 0, MT = 1 1 0 MT = 24 0 0 0 18 0 0 0 18 0 0 0 for Case I to Case III. It is obvious that only the first two matrices have rank two and are positive semi-definite. Thus Case III is not recommended.

694

B. Wohlmuth

Associated with this type of quadrature rule, we can define an interpolation operator such that (7.2) and Lemma 7.3 hold true. As we will see in the following, this type of quadrature formula also gives rise to good numerical results and can be applied as well to obtain a stable space–time integration scheme. Influence of the choice of the quadrature formula In this subsection, we compare the influence of the choice of the quadrature formula on our numerical results. Figure 7.12 shows the problem setting and two different meshes. One is based on simplicial elements and the other one on quadrilaterals, which are not necessarily affine equivalent to the reference square. Figure 7.13 shows the discrete kinetic, the potential and the total energy at time tk . As seen in the previous subsection, the total energy is preserved for all time steps. v0 = 800 E = 100 ν = 0.3 ρ = 2e−08

v0 = 800

Figure 7.12. Initial grids and effective stress for a contact problem with ν = 0.

Energy for triangular grid

Energy for quadrilateral grid

3

3

2.5

2.5

2

2

1.5 1 0.5 0 0

1.5

total const elastic const kinetic const total mod elastic mod kinetic mod 0.2

0.4

1 0.5 0.6 Time

0.8

1 −3

x 10

0 0

total const elastic const kinetic const total mod elastic mod kinetic mod 0.2

0.4

0.6 Time

Figure 7.13. Energy results for the two-circle contact problem without friction.

0.8

1 −3

x 10

695


As the initial conditions are given by a constant velocity and zero displacement, the total energy can be captured exactly by all our discussed quadrature rules on simplicial meshes. The situation is different for the presented quadrilateral mesh. Here, we have a non-constant Jacobian for the element mappings and thus a small difference is obtained if the simplified quadrature rule based on the original mesh is applied. However, this difference is not significant, in particular on fine meshes. Normal LM for triangular grid

Normal LM for quadrilateral grid

10

10

8

8

6

6

4

4

2

2 standard constant

0 −2

2

4

6 Time

standard constant modified

0 8

10 −4

x 10

−2

2

4

6 Time

8

10 −4

x 10

Figure 7.14. LM for simplicial and quadrilateral grid at the bottom slave node.

In Figure 7.14 we compare the results in the Lagrange multiplier. For the standard discretization with no mass modification, a highly oscillating Lagrange multiplier in the normal direction is obtained. The amplitude and frequency is rather independent of the applied mesh and is not reduced for smaller time steps. The numerical results are drastically improved if the modified mass matrix approach is applied. The numerical results do not show a significant difference between the different proposed quadrature rules. Index reduction In the original mass modification approach, the mass modification was only carried out with respect to the normal components. From the theoretical point of view this is sufficient to reduce the index. We recall that the algebraic constraints are given in the displacement for the non-penetration which involves the normal components. The friction law works on the tangential velocity, and these constraints result in an index-two system, which, compared to the original index-three system, has better stability properties. However, as our numerical results show, the index-two system still shows oscillations in the Lagrange multiplier. As can be seen from Figure 7.15, only the mass modification in both directions is able to remove the oscillations from the Lagrange multiplier. However, the mass modification in the normal direction not only removes

696

B. Wohlmuth

(a)

(b)

(c)

Multiplier at point (0.47, 0.14) 5 4

Multiplier at point (0.47, 0.14) 5

F*λn λt

4


F*λn λt

3

3

2

2

2

1

1

1

0

0

0

−1

−1

−1

4

6 8 10 Time steps

12

14 −4 x 10

(d)

2

4

6 8 10 Time steps

12

14 −4 x 10

(e) Multiplier at point (0.00, 0.14) 15

F*λn λt

5

5

5

6 8 10 Time steps

12

14 −4 x 10

0

12

14 −4 x 10

λt 10

4

6 8 10 Time steps

F*λn

λt 10

2

4


F*λn

10

0

2

(f)


λt

4

3

2

F*λn

2

4

6 8 10 Time steps

12

14 −4 x 10

0

2

4

6 8 10 Time steps

12

14 −4 x 10

Figure 7.15. Normal and tangential Lagrange multiplier with respect to time at two different selected nodes: mass modification in both directions (a,d), in the normal direction (b,e) and no mass modification (c,f).

the oscillations in normal directions but also reduces the oscillations in the tangential direction compared to the unmodified approach. Finite deformations As we have seen in Section 6.8 for the adaptive refinement process, the proposed algorithms naturally generalize to finite deformations. The same holds true for the time-integration scheme. The simple Newmark method has to be replaced by a generalized scheme: see, e.g., Chung and Hulbert (1993), Gonzalez (2000) and Hulbert (1992). We refer to Hesch and Betsch (2006) for a comparison between a simple node-to-node and a Lagrange multiplier-based simulation of dynamic large-deformation contact problems. Figure 7.16 shows the influence of the friction on the numerical results. We consider the two cases ν = 0 and ν = 0.3. In the long range the results are quite different, whereas in the short range almost no difference can be observed. The total contact work up to time tk is set to be equal to (Wlcon )k :=

k−1 j=0

j+ 21

∆tvl

1

Bl (λtl )j+ 2 .

697


Figure 7.16. Van Mises stress at four different time steps for ν = 0 and ν = 0.3. (a)

(b) Without friction

5

2.5

x 10

total energy elastic energy kinetic energy

2

(c) 5

1.5

2.5

x 10

Coefficient of friction F=0.3

5

x 10

total energy elastic energy kinetic energy

2

1.55

Coefficient of friction F=0.3 total energy total energy + contact work

1.5

1.5 1.45

1

1

0.5

0.5

0 0

10

20 Time

30

40

0 0

1.4

10

20 Time

30

40

1.35 0

10

20 Time

30

40

Figure 7.17. Energy ν = 0 (a) and ν = 0.3 (b) and contact work (c).

Then Lemma 7.7 guarantees that Elk + (Wlcon )k is constant provided that there is no source term. Figure 7.17 shows the energy for the two different situations. In Figure 7.17(a), it can be seen that the total energy is constant over time. For ν = 0.3, we observe that the energy is decreasing due to the loss in the friction: see Figure 7.17(b). In Figure 7.17(c), we observe that this loss is in balance with the total contact work. Coulomb friction in the three-dimensional setting As a final test, we consider two different three-dimensional settings and include Coulomb friction with ν = 0.5. In both cases, we apply a simple quadrature formula based on the elements of the mesh Tl , which is exact for element-wise constants and does not have nodes on Sl \ ∂ΩSl . In the first setting, a ball comes in contact with a hexahedron. The evolution of the energy and the contact work is presented in Figure 7.18(a,b). Figure 7.18(c) shows the value of the Lagrange multipliers in normal direction at the lowest point of the ball over time.

698

B. Wohlmuth Energy

15

Total energy and contact work total energy elastic energy kinetic energy

10

9 8.995

Normal Lagrange multiplier

total energy total energy + contact work

4000

8.99 5

2000

8.985 0 0

0.02

0.04 0.06 Time

0.08

0.1

8.98 0

standard mass modified mass

6000

0.02

0.04 0.06 Time

0.08

0.1

0 0

0.02

0.04 0.06 Time

0.08

0.1

Figure 7.18. Energy (a), contact work (b) and normal Lagrange multiplier in the normal direction (c).

The last example illustrates that also for sliding geometries with many nodes in contact and a contact set which varies widely, the algorithm is numerically stable and no spurious oscillations occur. A two-dimensional cross-section is depicted in Figure 7.19(a), and the initial condition is illustrated in Figure 7.19(b,c). Here, the outer tube is assumed to be the slave side. In Figure 7.20 the two-dimensional cross-sections of the situations with the effective stress σeff at four different time steps are shown. (a)

(b)

u=0

(c)

Ωs Ωm uD Ωm Ωs u=0 Figure 7.19. Problem definition and initial configuration u0 at t0 .

Figure 7.20. Situation at t15 , t30 , t45 and t60 .


699

7.5. Conclusion Although the Lagrange multiplier-based formulation has many attractive features, it requires a careful handling of time-integration schemes. A naive application may result in high oscillation in the contact stresses and thus the non-linear solver possibly breaks down. As shown, a local modification of the mass matrix at the contact nodes in both normal and tangential directions reduces the oscillations significantly. In many applications, the quadrature rule based on the original mesh gives satisfying results and is easier to handle. On the other hand, the first- and second-order moments can only be preserved if a quadrature rule associated with a macro-triangulation is applied. We note that neither proposed mass matrix modification requires any global operator, and both fit into standard assembling procedures for finite elements. Moreover, in the case of linear elasticity problems an a priori analysis shows that optimal convergence can be obtained under suitable regularity, and numerical results indicate that the difference from a standard finite element scheme in space is negligible and asymptotically of higher order. The analysis follows the lines of variational crimes and standard mass lumping techniques. However, we point out that in contrast to lumping techniques, the resulting modified mass matrix is singular and thus the proof of the theoretical results is more involved and technical. From the differential-algebraic point of view the reduction of the DAE system from index three to one results in a stable algorithm.

8. Further applications from different fields In this section, we provide some more complex applications with inequality constraints from different application areas. For each problem a brief introduction into its physical or financial interpretation is given; the details about the physical and mathematical models are, however, omitted. For each selected example, it is characteristic that the solution of a partial differential equation system has to satisfy additionally an inequality constraint. For the discretization in space we use volume- and/or surface-based Lagrange multipliers such that in space we have a variationally consistent discretization. In time, we apply a suitable finite difference scheme, possibly modified according to Section 7. As in Section 5, the fully discretized variational inequality system in terms of a pair of variables can be rewritten as a non-linear equality system. The constraints are taken nodally into account by suitable problem-dependent non-linear complementarity functions. 8.1. Mathematical finance: American options Our first application stems from the field of financial economics. The application of a semi-smooth Newton method as a numerical solver for obstacle-

700

B. Wohlmuth

type variational inequalities obtained by the mathematical model of an American option can be found in Hager, H¨ ueber and Wohlmuth (2010b), where more numerical results are also presented, including sparse grid techniques. We refer to the monograph by Achdou and Pironneau (2005) and to Pironneau and Achdou (2009) for an introduction to mathematical models for option pricing and for a discussion of numerical and implementational issues. An American option is a contract which permits its owner to receive a certain pay-off ψ = ψ(x) ≥ 0 at any time τ between 0 and the expiry date T , depending on the value of the underlying assets x at time τ . In this subsection, we consider options on a set of two assets x = (x1 , x2 ) and ask for its fair price. A simple mathematical model is based on the Black– Scholes equation (Black and Scholes 1973) and the no-arbitrage principle (Hull 2006, Wilmott, Dewynne and Howison 1997). The symmetric and positive definite volatility matrix " # 2 σ12 σ σ 1+2 1 2 Ξ= , 2 σ σ σ22 1+2 1 2 the volatilities σk , the correlation rate ∈ (−1, 1), the interest rate r and the dividend rates qk on the asset xk , k ∈ {1, 2}, enter as parameters into the model. In the case of an American option, the no-arbitrage principle implies that its fair value can never be below its pay-off as the option can always be exercised. Further, a hedging argument yields that the Black– Scholes equation becomes an inequality (Hull 2006, Wilmott et al. 1997). Thus, the price P of an American put with pay-off function ψ satisfies the following set of conditions for x ∈ R2+ , t ∈ (0, T ] with t := T − τ : P˙ − LP ≥ 0,

P − ψ ≥ 0,

(P˙ − LP )(P − ψ) = 0,

(8.1)

with the initial conditions P |t=0 = ψ. Here the partial differential operator L is given by 2 2 ∂2 ∂ 1 Ξk,l xk xl + (r − qk )xk − r. (8.2) L := 2 ∂xk ∂xl ∂xk k,l=1

k=1

To solve this problem numerically, we truncate the semi-infinite domain R2+ to a bounded one Ω := (0, X1 ) × (0, X2 ) and impose artificial boundary conditions on it. We refer to the monograph by Achdou and Pironneau (2005) for a discussion of possible choices for these boundary conditions depending on the pay-off function ψ. Here we apply a strategy where on the boundary a 1D variational inequality has to be solved, and the solution of it imposes appropriate Dirichlet boundary conditions at time ti : see Hager et al. (2010b) for details. In contrast to our contact formulations, the inequality constraints are not imposed on part of the boundary but in the domain itself. As a con-


701

sequence, we have to replace the surface-based Lagrange multiplier by a volume-based one, and the H 1/2 -duality by the H 1 -duality. However, for the numerical solution strategy, this difference does not matter. As before, we use a biorthogonal Lagrange multiplier and transform (8.1) into a non-linear equality system based on a weak variational formulation. Using low-order conforming finite elements on a family of simplicial meshes, we arrive at the discrete system in saddle-point form. Find (Pil , λil ), 1 1 i ) + Al (Pil + Pi−1 (8.3) Ml (Pil − Pi−1 l l ) − Dl λl = 0, ∆t 2 with Ml , Al and Dl being the lumped mass, the stiffness and the diagonal duality matrix associated with the mesh on level l, respectively. Given Pi−1 l , i i (8.3) has to be solved for (Pl , λl ) together with the node-wise complementarity condition λil − max(0, λil − c(Pil − ψ l )) = 0,

(8.4)

where ψ l is a finite element representation of the pay-off function on level l and c a fixed positive constant. From the algebraic point of view, there is no structural difference to a contact formulation without friction, and thus a semi-smooth Newton method can be easily applied as solver. We note that the situation here is simple. Firstly the inequality is the only source of a non-linearity, secondly the NCP function given by (8.4) is piecewise affine, and thirdly we can use the solution from the previous time step as initial guess. Thus the implementation of the solver is directly based on the equivalent primal–dual active set strategy. The adaptive refinement strategy follows the same lines as discussed in Section 6. Here two essential differences have to be taken into account. In the case of an obstacle problem, the value of the Lagrange multiplier is a priori known if the actual zone of contact between solution and obstacle is known. Thus this extra information can be used to redistribute element-wise the computed discrete Lagrange multiplier. Using such a post-processed Lagrange multiplier on the right side of the vertex-based equation system for the flux moments gives much better results. Details can be found in Weiss and Wohlmuth (2010). Following the construction principle of Section 6 and applying mixed RT0 , RT1 or BDM1 elements will result in robust and reliable adaptive mesh refinement in the case of smooth obstacles. From the theoretical point of view, the use of RT0 elements is sufficient. Then the divergence and the face fluxes are obtained from the right side of the PDE and the moments by the element-wise and face-wise L2 -projection onto constants, respectively. For RT1 or BDM1 elements, the face-wise linear moments will be exactly reproduced by the face fluxes of the mixed element. Moreover, for RT1 elements, we obtain that the divergence is given by the element-wise L2 projection onto polynomials of degree at most one.

702

B. Wohlmuth

However, a naive application of the proposed construction principle fails in the case of a non-smooth obstacle. To get a better understanding, we consider firstly a simple obstacle problem on the unit square where the obstacle has the form of a pyramid. Obstacle-type problem The obstacle is non-differentiable at the two axes. Figure 8.1(a) shows the solution and the obstacle. Ignoring the kinks in the obstacle, the definition of ηL results in a non-optimal estimated error decay: see Figure 8.1(b). (a)

(b) 0

Error

10

ηL O(h) ηmod

−1

10

−2

10

2

4

10

10 No. of elements

Figure 8.1. Non-smooth obstacle: obstacle and solution (a), and comparison of the unmodified and the modified estimated error (b).

To see what goes wrong, we consider additionally the adaptively refined meshes on Level 3 and Level 6. In Figure 8.2(a,b), we observe a strong over-refinement at the kinks of the obstacle, where the solution is actually in contact with the obstacle. This highly over-refined zone results from the fact that the local contribution ηL;T measures the distance between the finite element solution and a globally H(div)-conforming mixed finite element, although the solution is not in H(div; Ω). Thus one cannot expect ηL to be efficient. A possible remedy can be quite easily constructed. The Lagrange multiplier is additively decomposed into a volume part in H −1 (Ω) and an interface part in H −1/2 (γ), where the obstacle ψ has kinks on γ. The interface part depends only on the obstacle and is given by the jump of its normal fluxes. Then the lifting of the fluxes is not globally H(div; Ω)-conforming but does correctly reflect the jump. In Figure 8.2(c,d), we illustrate the positive effect of the decomposition of the Lagrange multiplier on the adaptive refinement process. In terms of the proposed modification, the estimated error in the interior of the contact zone is zero, and therefore no overestimation of the error occurs. Moreover, as can be seen in Figure 8.1(b), the obtained error decay has the correct slope.


(b)

(c)

703

(d)

Figure 8.2. Non-smooth obstacle: refined mesh on Level 3 and Level 6, naive application of ηL (a,b) and with suitable modification (c,d).

American basket option We are now in the setting to apply an adaptive algorithm for the numerical solution of pricing American basket options. An error estimator in space and time designed for parabolic variational inequalities with special focus on American options is introduced in Moon et al. (2007); see also the more recent contribution of Nochetto et al. (2010). As is standard for time-dependent systems, we include a coarsening strategy in the adaptive refinement process. In addition, we apply the previously discussed modification in the error indicator, because of the kinks in the pay-off function, and take note of the different structure of the PDE (8.2) compared to the Laplace operator. Two different pay-off functions are tested, ψmax := max(0, K − max(x1 , x2 )) and ψmin := max(0, K − min(x1 , x2 )), and we refer to Weiss and Wohlmuth (2010) for the problem specification.

(a)

(b)

(c)

(d)

Figure 8.3. American put option: solution (a,b) and adaptive mesh (c,d) at times t = 0.5 and t = 0.9; (a,c) ψmax , (b,d) ψmin .

The adaptively refined meshes in Figure 8.3 show that the error estimator does not over-refine at the kinks of the pay-off functions and that the proposed modification also works well for much more complex situations.

704

B. Wohlmuth

8.2. Porous media: multi-phase flow problems As second example, we consider an incompressible multi-phase flow process in porous media. Here we can identify two different sources for inequalities in the mathematical model. To simplify the notation, both of them will be described separately. The first one results from heterogeneous media and is associated with interior interfaces: see Helmig, Weiss and Wohlmuth (2009). The second one is related to a phase transition process and yields inequality constraints on the simulation domain: see Lauser, Hager, Helmig and Wohlmuth (2010). Interface inequalities: heterogeneous media with entry pressure The mathematical model for a two-phase one-component system we are using here lives on the macro-scale and is based on mass conservation, momentum balance and Darcy’s law for each phase: see, e.g., Helmig (1997). Here we consider two phases in isothermal equilibrium, the wetting phase (α = w) and the non-wetting phase (α = n). Originally Darcy’s law was obtained for slow laminar flow of a single phase but can be easily extended to the two-phase setting by using the relative permeability (Scheidegger 1960). Then the phase velocity vα , α ∈ {w, n}, is given by vα = −ξα (Sα )K(∇pα − ρα g),

ξα (Sα ) :=

krα (Sα ) , µα

where K, g, krα , µα and ρα stand for the intrinsic permeability, the gravity, the relative permeability, the dynamic viscosity and the density of phase α, respectively. Moreover, pα denotes the unknown phase pressure. Then the mass balance yields ∂(Φρα Sα ) + div(ρα vα ) = ρα qα , α ∈ {w, n}, (8.5) ∂t where Sα is the unknown saturation of the phase α, Φ is the porosity, and qα denotes the source/sink term. To close the system, we have to add two additional relations: a capillary pressure-saturation relation, i.e., pn −pw = pc (Sn ), and a saturation balance, i.e., Sn + Sw = 1. Here we use a non-standard dynamic capillary pressure relation including a retardation term (Hassanizadeh and Gray 1993, Hassanizadeh, Celia and Dahle 2002): pn − pw = pc (Sn ) = pstat c (Sn ) + τ

∂Sn , ∂t

τ ≥ 0.

(8.6)

is assumed to be continuously The static capillary pressure function pstat c entry differentiable, non-negative, strictly increasing and pstat c (Sn ) tends to pc for Sn → 0. Typical choices for pstat are the Brooks–Corey (Brooks and c Corey 1964) or the Van Genuchten model (Van Genuchten 1980). We note

705


(b)

1

0.6

krn

krw

0.4 0.2

0

0.5

Brooks−Corey Van Genuchten

8 Capillary pressure

Relative permeability

0.8

0

10

Brooks−Corey Van Genuchten

1

6 4 2 0

0

Saturation: Sw

0.5

1

Saturation: Sw

Figure 8.4. Comparison of the Van Genuchten and Brooks–Corey model: relative permeability (a) and capillary pressure (b).

that the Van Genuchten model with zero entry pressure can be regarded as a regularization of the Brooks–Corey approach (see Figure 8.4), and the parameters in both models are related (Lenhard, Parker and Mishra 1989). Equation (8.5) for α = n and α = w yields a strongly coupled highly non-linear system. Next, we describe how the heterogeneity of the material is accounted for. For simplicity, we assume that the domain Ω is split into two subdomains Ωm , Ωs with the interface Γ := ∂Ωm ∩ ∂Ωs , such that the parameters Φ and K are constant on each subdomain. Further, the domains are chosen such that the master subdomain has a lower entry pressure, i.e., a higher relative permeability, than the slave domain. The flow at the interface Γ has to be modelled correctly. Here, we describe only the more interesting case when the non-wetting phase penetrates into the subdomain with the higher entry pressure. The mathematical model introduced in de Neef (2000) gives rise to the following transmission conditions at the material interface: [pc ] ≥ 0,

Sns ≥ 0,

[pc ]Sns = 0,

(8.7)

where Sns = 1 − Sws stands for the saturation of the non-wetting phase on the slave side, and [pc ] denotes the jump of the capillary pressure. Then, (8.7) states that the capillary pressure at the interface is continuous if the non-wetting phase is present on the side with the higher entry pressure. In order to solve the above problem numerically, we apply a node-centred conservative finite volume scheme in space in combination with upwind techniques: see Huber and Helmig (2000). We remark that the meshes used do not need to be matching at the interface Γ. Then, in terms of the mortar projection, we can define node-wise inequality constraints for the saturation Sn and the capillary pressure pc on the slave side. In contrast to the previous application, the non-linearity of the system is not restricted to the inequality constraints (8.7). A popular approach to reduce the complexity is based on a fractional flow formulation. It is equivalent to the original system but can be more efficiently solved by block decoupling strategies. In the case of the

706 (a)

B. Wohlmuth (b)

(c)

Figure 8.5. Comparison of different time-integration and decoupling schemes.

classical fractional flow formulation, the total velocity vt is introduced by vt := vw + vn . Then a coupled but considerably simplified system with a much more moderate non-linearity is obtained for the so-called global pressure and the saturation: see, e.g., Binning and Celia (1999), Chavent and Jaffré (1986), Rivière (2008) and Wooding and Morel-Seytoux (1976). Here, we cannot directly apply this approach since the interface model has no equivalence in terms of the non-physical variable of the global pressure. We work with an alternative fractional flow formulation which is based on a pressure equation for pw and a saturation equation for Sw . The interface condition is then directly formulated in these primary variables, and we obtain, by replacing (8.7) by the equality to zero of an NCP function, a fully coupled system for (pw , Sw ) having possibly two different pressure values on the interface. For the discretization in time, different strategies can be applied; see Figure 8.5 for a comparison of the numerical results. Here, we illustrate the algorithm for a matrix with three inclusions of lower relative permeability. In Figure 8.5(a), the wetting velocity vw is not at all updated in time, resulting in a significant different solution compared to the two alternative approaches shown in Figure 8.5(b,c). This strategy is the most simple one, and we have to solve in a pre-process a linear elliptic pressure equation and then in each time step a non-linear equation for the saturation. In Figure 8.5(b), the fully non-linear and coupled system for the pressure and saturation is solved by an implicit Euler scheme. This approach is the most expensive one since in each time step a fully coupled system has to be solved, where the non-linearities result from the PDE and the inequality constraints at the interfaces. In Figure 8.5(c) a suitable decoupling strategy is applied. An explicit time integration is used for the pressure equation, whereas an im-

707


No. of active nodes

60

τ = 60 τ=0

40

20

0 0

2

4 Time

6

8

Figure 8.6. Pair of solutions for τ = 0 and τ = 60 at three different time steps and number of active nodes with respect to time.

plicit scheme is selected for the saturation equation. This explicit–implicit method can also be regarded as one step of a non-linear block Gauss–Seidel solver applied to the fully coupled implicit time-integration system. As can be seen, this inexact approach is quite attractive. It gives highly accurate results and is considerably less expensive than the solution of the fully coupled system. The sequential solution of a linear pressure equation and a non-linear saturation equation is required. Figure 8.6 shows the influence of the dynamic parameter τ on the solution and on the active set. Here we denote the faces on which we have continuity of the pressure as active and mark these faces by white squares. In the short and middle time range there is a significant difference in the results. First of all, a non-zero τ has a retardation effect on the wave front. Thus the penetration of the non-wetting phase into the subdomains with lower regularity starts later, and in the short range we observe a smaller number of active faces. Secondly, due to the dynamic capillary pressure, a nonmonotonous wave profile is created with a sharper wave front, resulting in a larger active set in the middle time range. In the long range, we will reach a stationary equilibrium, and thus there is no difference between τ = 0 and τ = 60. This is reflected by the fact that, for sufficiently large times, the number of active faces is equal. So far we have described the mathematical model of a heterogeneous material interface resulting in a surface-based inequality. In the next step, we describe how a volume-based inequality enters into the model.

708

B. Wohlmuth

Volume inequalities: phase transition processes We now extend the model from the simple two-phase one-component situation to an M p–N c system with N different components and with M different phases. In the following, we use the lower index α = 1, . . . , M for the phase, ordered by their wettability, i.e., α = 1 denotes the gas phase, and the upper index j = 1, . . . , N stands for the component. Assuming that the fugacity of any component is the same in all phases, we have a priori M N + 2M + N + 1 unknowns. As before, pα and Sα stand for the phase pressure and for the saturation of phase α, α = 1, . . . , M . In addition, we have for a non-isothermal system the temperature T . Related to the different components is the fugacity f j and the mole fraction xjα of component j in phase α, j ∈ {1, . . . , N }, α ∈ {1, . . . , M }. We refer to Acosta, Merten, Eigenberger, Class, Helmig, Thoben and M¨ uller-Steinhagen (2006), Class and Helmig (2002), Class, Helmig and Bastian (2002) and Niessner and Helmig (2007) for the description of general non-isothermal multi-phase systems. For each component one mass balance equation has to hold, and for the temperature the energy balance equation has to be satisfied, resulting in a coupled highly non-linear system of (N + 1) partial differential equations. In addition to the coupled PDE system, we have to observe suitable constitutive relations, such as, for the saturations, M

Sα = 1,

(8.8)

α=1

and for the phase pressures pα , pα−1 − pα = pc,(α−1)α ,

2 ≤ α ≤ M,

with the capillary pressure pc,(α−1)α = pc,(α−1)α (Sα ) depending on the saturation Sα of the phase with higher wettability (Niessner and Helmig 2007). As in the first example (see (8.6)), different models can be used to define pc,(α−1)α (·). In addition M N constitutive relations between fugacities and mole fractions have to be provided. These relations are in general quite complex and rely on additional assumptions on the nature of the system. In many applications from these relations, we can completely eliminate f j and obtain the mole fractions xjα , 1 ≤ j ≤ N and 2 ≤ α ≤ M explicitly in terms of p1 , xj1 , 1 ≤ j ≤ N , T , i.e., xjα = gαj (p1 , x11 , . . . , xN 1 , T ),

1 ≤ j ≤ N, 2 ≤ α ≤ M

(8.9)

with some given functions g(·) depending on the law of Henry and Raoult (Class 2001). In terms of the constitutive equations, the number of unknowns can then be reduced from M N + 2M + N + 1 to M + N + 1. One possibility is to set the pressure of the gas phase, its mole fractions with respect to the N


709

components, M −1 saturations and the temperature as primary variables X:

(8.10) X := p1 , x11 , . . . , xN 1 , S2 , . . . , SM , T . To close the (N + 1)-dimensional PDE system, we have to include compatibility conditions for the different phases. The component sum of the mole fractions xjα is equal to one if the phase α is actually present, i.e., Sα > 0. This observation yields the following complementarity conditions: " # N N xjα ≥ 0, Sα ≥ 0, Sα 1 − xjα = 0, 1 ≤ α ≤ M, (8.11) 1− j=1

j=1

where we have included the physical condition of a non-negative saturation. Replacing the inequality constraints (8.11) by the equivalent form " ## " N 1 N j = 0 (8.12) Cˆα (Sα , x , . . . , x ) := Sα − max 0, Sα − cα 1 − x α

α

α

j=1

with a fixed positive constant cα > 0, we obtain a highly non-linear system. Although at first glance the derivatives of the non-complementarity function seem to be as easy to calculate as those in the case of the normal contact conditions of Section 5, there is an essential difference. We note that in (8.12), the NCP functions Cˆα , 1 ≤ α ≤ M , depend on all variables and not only on the primary variable X. All unknowns that are not a primary variable (see (8.10)) have to be replaced by (8.8) and (8.9) before the Newton scheme is applied, and thus the partial derivatives of gαj (·) appear. In the primary variable X, we thus define the NCP function (1 − N xj1 ) j=1 C(X) := (Sα )M α=2 N j for X such that Sα − cα (1 − j=1 gα (X)) ≤ 0 for 2 ≤ α ≤ M , and otherwise we set # " M N j S − max(0, 1 − S − c (1 − x )) 1− M α α 1 α=2 α=2 j=1 1 . C(X) := j M (Sα − max(0, Sα − cα (1 − N j=1 gα (X))))α=2 Let us consider now the more simple case of a two-phase two-component system, where the phase index α = 1 stands for the non-wetting phase and α = 2 denotes the wetting phase. Moreover, we assume that component j = 1 is air and j = 2 is water. In this simplified setting, we have three PDEs to satisfy, two NCP functions have to be zero, and the primary variables are X = (p1 , x11 , x21 , S2 , T ). Assuming the gas phase behaves as an ideal gas, the fugacities are given by f 1 = x11 p1 ,

f 2 = x21 p1 .

710

B. Wohlmuth

(a)

(b)

(c)

20

300

18

250 Relative error

Newton steps

16 14 12 10

200 150 100

8 50

6 4

0

50

100

150 Time

200

250

0

300

1

2

3

4 5 Newton step

6

7

8

Figure 8.7. (a) Problem setting, (b) Newton iterations with respect to time and (c) error decay within one time step.

In terms of Raoult’s law we get, by assuming that x22 is close to one, f 2 = p2vap x22

x22 =

and thus

f2 p1 = 2 x21 . p2vap pvap

(8.13)

Here p2vap = p2vap (T ) stands for the vapour pressure of water. To obtain x12 , we use Henry’s law with the Henry coefficient H21 = H21 (T ): f 1 = H21 x12

x12 =

and thus

f1 p1 = 1 x11 . 1 H2 H2

(8.14)

Using (8.13) and (8.14), we obtain the explicit form for g21 (·) and g22 (·): g21 (p1 , x11 , x21 , S2 , T ) =

p1 x11 , H21 (T )

g22 (p1 , x11 , x21 , S2 , T ) =

p1 x21 . p2vap (T )

Furthermore, we consider the simplified model of a constant temperature, i.e., the Henry coefficient and the vapour pressure are constant. In the kth Newton step, we then have to consider the following three cases. • If Xk−1 such that I2k−1

:=

S2k−1

then

− c2

2 k−1 pk−1 (x11 )k−1 pk−1 1 (x1 ) − 1− 1 p2vap H21

≤ 0,

(x11 )k + (x21 )k = 1 . = 0 S2k

• If Xk−1 such that I2k−1 > 0 and I1k−1 := 1 − S2k−1 − c1 (1 − (x11 )k−1 − (x21 )k−1 ) ≤ 0, then " pk−1 (x11 )k 1 H21

S2k +

= pk−1 (x21 )k 1 p2vap

= 1−

1 (x11 )k−1 H21

+

#

k−1

(x21 ) p2vap

(pk1 − pk−1 1 )

.


711

Figure 8.8. Evolution of the different ‘active’ zones.

• If Xk−1 such that I2k−1 > 0 and I1k−1 > 0, then # " (x11 )k + (x21 )k = 1 . pk−1 (x11 )k pk−1 (x2 )k (x11 )k−1 (x2 )k−1 1 + 1 p2 1 = 1− + p12 (pk1 − pk−1 1 ) H1 H1 2

vap

2

vap

Figure 8.7 shows the geometry of the problem considered and the performance of the semi-smooth Newton applied to the fully coupled non-linear PDE system enriched by the algebraic NCP functions. Here, a polynomial capillary pressure function has been used: see Leverett (1941). In Figure 8.8, we plot the three possible cases for different time steps. The light grey circles mark the region where both phases are present. The dark grey ones show the region where only the water phase is present, and the grey ones mark the region where only the gas phase is present. During the simulation the gas phase is more and more displaced by the water phase. For a similar example in 2D and a more realistic three-phase seven-component example in 3D simulating the injection of CO2 into the soil and the subsequent extraction of methane, we refer to Lauser et al. (2010). 8.3. Structural mechanics: frictional contact of elasto-plastic bodies Our final example is the modelling of frictional contact between several elasto-plastic bodies (Hager and Wohlmuth 2009b). This application includes several pairs of complementarity conditions, a volume-based one describing the plastification process, and a surface-based one for the contact. We restrict ourselves to infinitesimal associative plasticity and linear hardening and point out that the framework is much more general and can be extended to non-linear material or hardening laws (Han and Reddy 1995, Han and Reddy 1999, Simo and Hughes 1998). We refer to Wieners and Wohlmuth (2011) for an application of a semi-smooth Newton

712

B. Wohlmuth

solver to non-local gradient plasticity. In contrast to linear elasticity, the stress is now decomposed additively into an elastic and plastic part σ := Cεel := C(ε(u) − εpl ), where εpl is assumed to be symmetric and trace-free. Now both contact and plasticity can be formulated within the same abstract framework. Here we use a combination of Tresca and Coulomb law with the friction bound given by F + νλn and apply the rules for linear isotropic or kinematic hardening, respectively. To see the structure, we recall on the left the contact and on the right we introduce the plasticity setting: λ := −σn, Y co (λn ) := F + νλn , f co (λn , λt ) := λt − Y co (λn ), u˙ t λt = γ co λt , γ co ≥ 0, −f co (λn , λt ) ≥ 0, co γ f co (λn , λt ) = 0, λn ≥ 0, g(u) := gn − un ≥ 0, λn g(u) = 0,

pl η := dev σ − a−2 0 Kε ,

(8.15a)

Y pl (α) := a−1 0 (σ0 + Hα), pl f (α, η) := η − Y pl (α),

(8.15b)

ε˙ pl η = γ pl η,

(8.15c)

γ pl ≥ 0, −f pl (α, η) ≥ 0, γ pl f pl (α, η) = 0,

(8.15d)

pl α˙ = a−1 0 γ .

(8.15e)

Comparing the contact relations with the rules of plasticity, many parallels can be seen. In (8.15a), the dual variable λ for the contact and the inner variable η for the plasticity is given. The yield functions defined in (8.15b) have the same structure, and depend on the friction parameters F and ν and the hardening parameter H and the yield stress σ0 , respectively. The flow rule specified in (8.15c) imposes in each case a condition on the direction. Furthermore, the yield function f pl and the consistency parameter γ pl satisfy the same complementarity conditions (8.15d) as f co and γ co . d The constant scaling factor a20 := d−1 is used in order to have a consistent notation for both the two- and three-dimensional case. One of the main differences between the conditions for contact and plasticity is the evolution law (8.15e), which causes the plasticity law to be associative, in contrast to the complementarity conditions for the normal contact. The discrete version of the system is derived similarly to the previous examples. The plastic inner variables (α, εpl ) are approximated by the discrete space Qpl h , spanned by the piecewise constant indicator functions χT , T ∈ Tl . Hence we have one degree of freedom per element, which is a special case of the widely used approach associating the plastic variables with

713


(b)

(c)

(d)

Figure 8.9. (a) Geometry, (b) inner variable, (c,d) active sets. (a)

(b)

(c) Time 5

2pi

pi

0 −2

(e)

−1

0

1

2

0 −2

pi

−1

0

1

2

(g) Time 5

2pi

Time 15

2pi

pi

(f) Convergence at times 5, 10 and 15

0

10

(d) Time 10

2pi

0 −2

−1

0

1

2

1

2

(h) Time 10

2pi

Time 15

2pi

−5

10

pi −10

10

−15

10

0

pi

pi

rel error time 5 rel error time 10 rel error time 15 plast AS found cont AS found 5

10

No. of iterations

15

0 −2

−1

0

1

2

0 −2

−1

0

1

2

0 −2

−1

0

Figure 8.10. (a) Section view of the geometry; (b–d) active sets for plasticity; (e) convergence history; (f–h) active sets for contact.

Gauss integration points (Simo and Hughes 1998, Wieners 2007). We refer to Alberty, Carstensen and Zarrabi (1999) for the convergence analysis of a similar discretization. This leads to the discrete inner variables αl = χT αT , εpl χT εpl dev σ l = 2µ χT dev (Π0 ε(ul )T ), T, l = T ∈Tl

T ∈Tl

T ∈Tl

on which the definition of the NCP function is based. Because of the similar structure of contact and plasticity, all results of Section 5 can be applied. We apply these discretization techniques to two examples in the threedimensional setting. In Figure 8.9, the stress and the active sets are illustrated. Here we have the plastification as well as the contact zone. The geometry of the setting is shown in Figure 8.9(a). Figure 8.9(d) illustrates the contact zone whereas in Figure 8.9(c) the region with plastification is depicted.

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B. Wohlmuth

In Figure 8.10, we apply an exponential hardening law, and thus an extra source of non-linearity appears. Figure 8.10(b–d) shows the nodes where plastification occurs for three different time steps. The volume nodes are projected onto the surface. In Figure 8.10(f–h), we show the actual contact nodes. Here we provide the results for t5 , t10 and t15 ; the results for t3 , t6 and t9 are given in Hager and Wohlmuth (2010), where details on the problem specification can also be found. In Figure 8.10(e), we show the convergence history of the semi-smooth Newton method. The iteration in which the correct active sets are detected for the first time are marked by a circle for plasticity and by a diamond for contact. We point out that it depends on the time step which set is found first. For all time steps a super-linear convergence rate can be observed. 8.4. Conclusion In this section, we have illustrated that variationally consistent Lagrange multiplier formulations for PDE systems with algebraic constraints provide a flexible and powerful discretization technique. Of special interest are applications where both types of constraint, surface- and volume-based, enter into the setting. Both types can be handled within the same abstract framework of generalized saddle-point-type problems. The use of NCP functions allows a consistent linearization of the inequality constraints and is thus of special interest in combination with Newton-type solvers. Global convergence can only be guaranteed in special situations; however, for most problems local super-linear convergence is obtained. Rescaling of the NCP function and regularization of the Jacobian might significantly improve the robustness of non-linear solvers in the pre-asymptotic range.

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