Domain Decomposition Methods for Partial Differential Equations (Numerical Mathematics and Scientific Computation)

N U M E R I C A L MATHEMATICS A N D SCIENTIFIC COMPUTATION

*P. Dierckx: Curve and surface fittings with splines *H. Wilkinson: The algebraic eigenvalue problem *I. Duff, A. Erisman, and J. Reid: Direct methods for sparse matrices *M. J. Baines: Moving finite elements *J. D. Pryce: Numerical solution of Sturm-Liouvilleproblems K. Burrage: Parallel and sequential methods for ordinary differential equations Y. Censor and S. A. Zenios: Parallel optimization: theory, algorithms and applications M. Ainsworth, J. Levesley, M. Marietta, and W. Light: Wavelets, multilevel methods and elliptic PDEs W. Freeden, T. Gervens, and M. Schreiner: Constructive approximation on the sphere: theory and applications to geomathematics Ch. Schwab: p- and hp- finite element methods: theory and applications to solid andfluid mechanics J.W. Jerome: Modelling and computation for applications in mathematics, science, and engineering A. Quarteroni and A. Valli: Domain decomposition methods for partial differential equations Monographs marked with an asterix (*) appeared in the series 'Monographs in Numerical Analysis' which has been folded into, and is continued by, the current series.

Domain Decomposition Methods for

Partial Differential Equations ALFIO QUARTERONI Professor of Numerical Analysis, Politecnico di Milano, Italy, and Ecole Polytechnique Federate de Lausanne, Switzerland ALBERTO VALLI Professor of Functional Analysis, University ofTrento, Italy

C L A R E N D O N PRESS

1999

• OXFORD

CONTENTS

1 1.1 1.2 1.3 1.4

1.5

1.6 1.7 2 2.1

2-2

2-3

2-4

The Mathematical Foundation of Domain Decomposition M e t h o d s Multi-domain formulation and the Steklov-Poincare interface equation Variational formulation of the multi-domain problem Iterative substructuring methods based on transmission conditions at the interface Generalisations 1.4.1 The Steklov-Poincare equation for the Neumann boundary value problem 1.4.2 Iterations on many subdomains The Schwarz method for overlapping subdomains 1.5.1 The multiplicative and additive forms of the Schwarz method 1.5.2 Variational interpretation of the Schwarz method 1.5.3 The Schwarz method as a projection method 1.5.4 The Schwarz method as a Richardson method 1.5.5 A characterisation of the projection operators 1.5.6 The Schwarz method for many subdomains The fictitious domain method The three-field method

1 2 5 10 18 22 24 26 26 28 29 31 32 33 34 38

Discretised Equations and Domain Decomposition Methods Finite element approximation of elliptic equations 2.1.1 The multi-domain formulation for finite elements 2.1.2 Algebraic formulation of the discrete problem Finite element approximation of the Steklov-Poincare operator 2.2.1 Eigenvalue analysis for the finite element SteklovPoincare operator Algebraic formulation of the discrete Steklov-Poincare operator: the Schur complement matrix 2.3.1 Preconditioners of the stiffness matrix derived from preconditioners of the Schur complement matrix The case of many subdomains

41 41 43 45 46 48 49 51 55

xii 2.5

3 3.1 3.2 3.3

3.4 3.5 3.6

3.7

4

CONTENTS Non-conforming domain decomposition methods 2.5.1 The mortar method 2.5.2 The three-field method at the finite dimensional level Iterative Domain Decomposition Methods at the Discrete Level Iterative substructuring methods at the finite element level The link between the Schur complement system and iterative substructuring methods Schur complement preconditioners 3.3.1 Decomposition with two subdomains 3.3.2 Decomposition with many subdomains The Schwarz method for finite elements Acceleration of the Schwarz method 3.5.1 Inexact solvers Two-level methods 3.6.1 Abstract setting of two-level methods 3.6.2 Multiplicative and additive two-level preconditioners 3.6.3 The case of the Schwarz method 3.6.4 Convergence of two-level methods Direct Galerkin approximation of the Steklov-Poincare equation

4.4 4.5 4.6

Convergence Analysis for Iterative Domain Decomposition Algorithms Extension theorems and spectrally equivalent operators 4.1.1 Extension theorems in i ? 1 ^ ) 4.1.2 Extension theorems in iJ(div; Qj) 4.1.3 Extension theorems in .ff (rot; fi;) Splitting of operators and preconditioned iterative methods 4.2.1 The finite dimensional case 4.2.2 The case of symmetric matrices 4.2.3 The case of complex matrices Convergence of the Dirichlet-Neumann iterative method 4.3.1 An alternative way to prove convergence Convergence of the Neumann-Neumann iterative method Convergence of the Robin iterative method Convergence of the alternating Schwarz method

5 5.1

Other Boundary Value Problems Non-symmetric elliptic operators

4.1

4.2

4.3

59 60 66

71 71 73 77 77 79 86 91 96 96 97 98 99 99 100

103 104 104 111 114 117 122 125 128 133 133 135 135 137 141 141

5.1.1 5.1.2 5.1.3 5.2

5.3

5.4

5.5

5.6

5-7

6 6-1 6-2 6-3

tJUlNlijJlNTS

xiii

Weak multi-domain formulation and the SteklovPoincare interface equation Substructuring iterative methods The finite dimensional approximation

142 146 147

The problem of linear elasticity 5.2.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.2.2 Substructuring iterative methods 5.2.3 The finite dimensional approximation The Stokes problem 5.3.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.3.2 Substructuring iterative methods 5.3.3 Finite dimensional approximation: the case of discontinuous pressure 5.3.4 Finite dimensional approximation: the case of continuous pressure 5.3.5 Methods based on the Uzawa pressure operator The Stokes problem for compressible flows 5.4.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.4.2 Substructuring iterative methods 5.4.3 The finite dimensional approximation The Stokes problem for inviscid compressible flows 5.5.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.5.2 Substructuring iterative methods 5.5.3 The finite dimensional approximation First-order equations 5.6.1 Weak multi-domain formulation and the SteklovPoincare interface equation 5.6.2 Substructuring iterative methods The time-harmonic Maxwell equations 5.7.1 Weak multi-domain formulation 5.7.2 The finite dimensional Steklov-Poincare interface equation 5.7.3 Substructuring iterative methods Advection—Diffusion Equations The advection-diffusion problem and its multi-domain formulations Iterative substructuring methods for one-dimensional problems Adaptive iterative substructuring methods: ADN, A R N and AR^N

147 148 151 151 153 156 170 173 185 189 190 191 194 195 197 198 201 202 203 204 208 210 211 213 214 219 220 224 227

CONTENTS

xiv 6.3.1

6.4

6.5 7 7.1

7.2

7.3

8 8.1

8.2

8.3

8.4

The damped form of the iterative algorithms: dADN, d - A R N and d-AR^N Coercive iterative substructuring methods: 7 - D R and 7 RR 6.4.1 The 7 - D R iterative method 6.4.2 Convergence of the 7 - D R iterative method 6.4.3 The 7 - D R iterative method for systems of advection-diffusion equations 6.4.4 The 7 - R R iterative method 6.4.5 Convergence of the 7 - R R iterative method The finite element realisation of the iterative algorithms Time-Dependent Problems Parabolic problems 7.1.1 Multi-domain formulation and space discretisation 7.1.2 Implicit time discretisation and subdomain iterations Hyperbolic problems 7.2.1 Multi-domain formulation 7.2.2 Implicit time discretisation and subdomain iterations Non-linear time-dependent problems 7.3.1 Navier-Stokes equations for incompressible flows 7.3.2 Navier-Stokes equations for compressible flows 7.3.3 Euler equations for compressible flows Heterogeneous Domain Decomposition Methods Heterogeneous models for advection-diffusion equations 8.1.1 The Steklov-Poincare reformulation 8.1.2 The coupling for non-linear convection-diffusion equations Heterogeneous models for incompressible flows 8.2.1 The coupling for the linearised Stokes problem 8.2.2 The coupling for the Navier-Stokes equations in exterior domains Heterogeneous models for compressible flows 8.3.1 The coupling between the Navier-Stokes and Euler equations 8.3.2 The coupling between the Euler equations and the full potential equation The coupling for the compressible Stokes equations 8.4.1 Variational formulation and finite element approximation 8.4.2 An alternative formulation: variational setting and finite element approximation

232 234 234 237 239 240 242 244 251 252 253 256 261 263 266 272 273 275 277 285 287 290 295 296 297 300 305 306 309 317 320 322

CONTENTS 8.5 9 9.1 9.2

The coupling for the time-harmonic Maxwell equations Appendix Function spaces Some properties of the Sobolev spaces

xv

328 ^33 3^ 339

References Index

357

1

THE MATHEMATICAL FOUNDATION OF DOMAIN DECOMPOSITION METHODS In this chapter the reader is encouraged to discover the mathematical foundation of domain decomposition methods, which are based on partitions of the computational domain into subdomains with or without overlap. We introduce the concept of transmission conditions at subdomain interfaces and the Steklov-Poincare problem for the interface variables. Both differential and variational formulations are addressed. Then we present substructuring iterative methods for disjoint subdomains, and the Schwarz alternating method for overlapping subdomains. The convergence analysis of these methods will be carried out in Chapter 4. We also comment on two other approaches: the fictitious domain method and the so-called three-field method. We deal mainly with symmetric linear elliptic boundary value problems, and, in particular, with the Poisson problem: f-Au = /

in fi

Iu = 0

on dfl.

(11) Here, and in the rest of this book, f2 is a d-dimensional domain (d = 2,3), with a Lipschitz boundary dCl, whose outer unit normal direction is denoted by n*, / is a given function of L 2 ( Q ) , A : = J 2 j = i D j D i is the Laplace operator and D j denotes the partial derivative with respect to Xj, j = 1,... ,d. To start

FIG. 1.1. Non-overlapping partition of the domain fi into two subdomains.

2

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

with, we assume that 0 is partitioned into two non-overlapping subdomains fii and 0 2 ) and denote by T : = fii n Q 2 (see Fig. 1.1). We also assume that T is a Lipschitz (d — l)-dimensional manifold. The generalisation to other boundary value problems will be done in later chapters, particularly Chapters 5, 6 and 8. The finite dimensional approximation is addressed in the next two chapters.

1.1

Multi-domain formulation and the Steklov-Poincare interface equation

We indicate by Ui the restriction to fij, i = 1,2, of the solution u to (1.1), and by n ' the normal direction on dVLt fi T, oriented outward. For simplicity of notation we also set n = n 1 . It is easily seen that the Poisson problem (1.1) can be reformulated in the equivalent multi-domain form:

(1.1.1)

k

-Aui = /

in f2i

ui = 0

on a o i n an

Ui

on

r

on

r

=

1l2

du2

dui

dn

dn

u2=0

on dfi 2 n dn

- AU2 = f

in fi2-

Equations (1.1.1)3 and (1.1.1)4 are the transmission conditions for iti and on r. The physical meaning of this split formulation is clear as soon as the original solution of problem (1.1) is smooth enough (say, u G C 1 ( f i ) ) . In a more general framework the equivalence between (1.1) and (1.1.1) is shown in the next section by resorting to the weak formulation of both problems. We will see in Section 1.3 that domain decomposition methods are generally amenable to iterative procedures for an interface equation that is associated with the given differential problem. This interface problem can be defined in terms of the Steklov-Poincare operator that we are going to introduce. Let us refer to the model problem (1.1) and its multi-domain formulation (1.1.1), which corresponds to the domain partition of Fig. 1.1. The same arguments apply to the other boundary value problems that will be described in Section 1.4 (for further details see also Quarteroni and Valli 1991a, b). Let A denote the unknown value of u on T. We consider the two Dirichlet problems:

tji.l

lViU-Lil

r WJrtiViUijAlIUiN AINU S f liN 1 MI.F AUB ii^UAX lUlN -AWi = f

in fij

Wi = 0

on dQi D dtl

(

. Wi = A

3

on r ,

f o r i = 1,2. We can obviously state that Wi = u? + u.

(1.1.3) where we have defined problems:

and u* to be the solutions of the following Dirichlet

-Au? = 0 (1.1.4)

in Q.i on d£li fl dfl

u° = 0

on r,

U? = A and -Au* = /

on 8Q,i n 80,

«* = 0

(1.1.5)

u:

in fij

on r.

0

For each i — 1,2, it' is the harmonic extension of A into and will be denoted HiX. We will also write Q t f instead of u*. If we proceed formally, comparing (1.1.1) with (1.1.2), it follows that (1.1.6)

Wi = Ui for i = l , 2 if and only if

^ ^ = -tt— on T. on on

The latter condition amounts to the requirement that A satisfies the Poincare interface equation (1.1.7)

5A =

on T,

X

where 9

(1.1.8)

r 2

t

dn

Qif

d

i=l a nd

S is the Steklov-Poincare

operator, which is formally defined as

Steklov-

4

THE MATHEMATICAL FOUNDATION OK DL> MhiThLuus d

d_ dn

(1.1.9) =

£

i=i In particular, S can be split as S — (1.1.10)

SiT1 : =

+ S2, with

d

i = 1

>2-

This operator, which was introduced a century ago (1896-1900), has been more recently analysed by Agoshkov and Lebedev (1985) in the framework of iterative methods. It is, however, worthwhile to point out that Agoshkov and Lebedev in fact considered the inverse operators S^1 and S2 1 , and called them Poincare-Steklov operators. Remark 1.1.1 (Generalisation) The analysis that we are going to carry out on the Poisson equation will be applied to a far more general differential problem of the form (1.1.11)

Cu = f

in 0 ,

where £ is a partial differential operator, / is a given datum, and u is the unknown solution. A broad range of problems can be considered (see Chapter 5), including non-symmetric elliptic problems, the linear elasticity problem, the Stokes problem for incompressible flows, the viscous and inviscid Stokes problem for compressible flows, and the time-harmonic Maxwell system. Should fi be partitioned into two disjoint subdomains fii and CI2 as indicated in Fig. 1.1, we can go along the same lines presented above to generate a split version of problem (1.1.11). Denoting again for i = 1,2 by m the restriction of u to Oi, it follows from (1.1.11) that

(1.1.12)

= f •£^2 = /

in H], in fl2-

To guarantee the equivalence with (1.1.11) we need to enforce transmission conditions between u\ and u^ across T. In an abstract form, such conditions can be expressed by the two relationships (1.1.13)

$ ( u i ) = $(1x2) on T tf(ui)

= ${U2) on r ,

where the functions $ and ^ will depend upon the nature of the problem. Typically, for second-order elliptic operators, (1.1.13) expresses the continuity across r of u and of the normal 'flux' (namely, the normal stress) involving firstorder derivatives of iti and ?i2. More generally, these interface conditions are most often determined noting that:

§L2

VARIATIONAL FORMULATION OF THE MULTI-DOMAIN PROBLEM

5

• The solution u belongs to a space of functions defined over the whole Q,. This requires that in fix and in 0,2 enjoy a certain regularity therein, and in addition that they satisfy a suitable matching on F. • The restrictions UjQ1 andu^2 are distributional solutions to the given equation in fli and O2> respectively. Another interface condition between them comes from the fact that u in fact satisfies the equation in the sense of distributions in the whole Q; namely, through the interface T and not only separately in Qi and Q.2 • For the Poisson problem (1.1.1) the obvious identification $(v) = v,

=

^

holds true. Keeping in mind this correspondence, all iterative substructuring methods that we are going to introduce for the Poisson problem in Sections 1.3, 1.4 can actually be extended to the more general problems (1.1.12), (1.1.13) in a straightforward manner. This is the case, in particular, for the classical methods like Dirichlet-Neumann, Neumann-Neumann or Robin, originally introduced for the Laplace operator, and here applied to a very general family of boundary value problems (see Chapter 5). • Remark 1.1.2 In transforming the Dirichlet boundary value problem (1.1) into an equation on F, we have chosen as interface unknown the physical variable that has to match on T as described in the first condition in Remark 1.1.1, while the interface equation (1.1.7) for the Steklov-Poincare operator S is based on ensuring that the second condition is satisfied. The same procedure will be constantly followed in the remainder of this book, but clearly other choices of interface equation could be devised. These are amenable to different forms of the Steklov-Poincare operator, and, correspondingly, different iterative substructuring methods. •

1.2

Variational formulation of the multi-domain problem

In this section we formulate (1.1) in a variational way. This requires us to introduce Sobolev spaces and to take into account some of their properties. We will not dwell here on this argument, and refer the interested reader to the comprehensive presentation of this theory that can be found, for example, in J.-L. Lions and Magenes (1972) (see also Chapter 9). By integrating by parts in fi, it is easily seen that the weak formulation of (1.1) reads (1-2.1) where

find

u G V : a{u, v) = (/, v)

V v G V,

6

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

{w,v)

:= / w v Jn a(w, v) : = ( V W , Vv) := {u e L2{9,)\DjV

€ L2(n),

:={v£H1(n)\vldn=

Hl0({l)

l,...,d}

0}

fi)

V

j =

and v\qq denotes the trace of v (that is, its restriction) on dft. The norm of H1(Q) will be denoted by || • ||i,n, while || • ||o,fj will indicate the norm of L2(fl). We recall that IMIo,Q =

1

M

/

2

,

while

1/2

d Mli,n =

I IMlo,n +

Sll^i«llo,n

j=i for each v e H1(fi). The Poincare inequality states that there exists a constant Cq > 0 such that (1.2.2)

f v Jn

2

< c

u

[ V p ^ ) Jq -

2

v . e

= 1

Therefore, the norm ||u||I,n is equivalent to the norm ||VU||o,n FOR EACH v € //q (fi). It is worthwhile to note that the same result is true for functions that vanish only on an open and non-empty subset S of a\\r,\\2A

V 77 G A,

for a suitable constant a > 0, therefore S is a coercive operator. Proceeding in an analogous way, from (1.1.10) we have

i

^ 2

V A K I A T I U I N A L , f ' U l t M U L A T l U l N U F T-tiii iVIUJ^Ti-JUUMAilN F R U B L K M

(1.2.8)

(SiV, ^

=

fn

'

= ai(HiV, Hin)

9

V 77, fj, € A.

Clearly, each Si is symmetric and coercive; that is, it satisfies (Stv,v)

(1.2.9)

VryeA.

> "iIMIa

Another relevant property of the Steklov-Poincare operators Si is that they are continuous; that is, there exist constants (3i > 0 such that (1.2.10)

(Siv,v) < A I M U I H I A -

In fact {SiT],ij) < H-Hi^Hi,^ llffiMlli.Oi, and from well known estimates for the solution of elliptic boundary value problems (see, for example, J.-L. Lions and Magenes 1972) it follows that (1.2.11)

\\Hiv\\i^ < C M a

y r] G a,

hence (1.2.10) holds. These properties are of great interest, because they can be exploited to obtain a numerical solution to the Steklov-Poincare problem (1.1.7). Clearly, as soon as an approximation of A is available (1.1.2) can be reduced to the solution of two independent Dirichlet problems. Finally, we can also give a variational interpretation of the right-hand side % in (1.1.7). From (1.1.8) it can be expressed through the functions / and Qif as follows: 2 (1.2.12)

i = 1

2 X

i=1

'

2

Qi

= f^{(f,Tllfi)ni-ai(gif,1liii)} i=1

V > G A.

Therefore, the Steklov-Poincare equation (1.1.7) can be written in a variational form as (1-2.13)

findAeA

: (SX,n) = (x,m)

V p e A.

Let us also note that, from a variational point of view, the functions it? = HiX and u* = Q^ introduced in (1.1.4) and (1.1.5) are the solutions to the following Problems: HiXsVi (1-2.14)

{

ai(Hi\,Vi)

H, A = A

: = 0 on F

\/VieV?

L/tl. 1

i t l t IViAinaiVIAli^Alj r UUiNJJAliUiN Ut UU MllitlWJJa

IU and (1.2.15)

: ai(gif,vi)

Gif

V v, e V°.

= (f,vi)

Remark 1.2.2 The variational form (1.2.13) of the Steklov-Poincare equation could also be obtained directly from the interface relationship (1.2.6)4. Indeed, from this relation using the splitting m = + u* we obtain 2

2

ai{u°,TZifj,) = Z—1

2

Tli^Qi ~ 1

ai(ui'nilJ)

V /u e A.

2=1

Integrating by parts on each side, and using (1.1.4) and (1.1.5), we deduce that ai{ui,

f du° T^-iH) = J

a,i(u*,1liLx) = (f,niii)Qi

+ J

(the integrals over T are formal expressions: indeed, they should be replaced by the duality pairing between A' and A). We therefore conclude with the following integral equations at the interface P: f fdu°

du°2 \

f (8u\

du*2\

or, equivalently,

I ( I * * " which coincides with (1.2.13).

" = " /r ( l

S l f

- l ^ ' ) "

V

"

e K

•

In Table 1.2.1 we summarise the relation between the differential arid the variational forms of the single domain problem, the multi-domain problem and the Steklov-Poincare equation.

1.3

Iterative substructuring methods based on transmission conditions at the interface

We face now the task of solving the multi-domain problem (1.1.1) by iterative procedures that will then be replicated at the finite dimensional level. These methods are traditionally referred to as iterative substructuring methods. Typically, they introduce a sequence of subproblems in fix and $^2 for which conditions (1.1.1)3 and (1.1.1) 4 provide, respectively, Dirichlet or Neumann data at the internal boundary P. This can be accomplished in several ways, some of which are presented below.

ITERATIVE SUBSTRUCTURING METHODS

r §1-3

11

Table 1.2.1 The differential and the variational forms of the single domain, the multi-domain and the Steklov-Poincare problems Differential vs variational form

—Au = f in fi

Single domain

u

: Vd£

— 0 on dQ,

a(u,v)

in fii

Ui€VI

: V «1 e V°

ai(u!,vi)

= (f,v i)n x

-Aui = f

Multidomain

u e ^(fi)

on

72-jM)

- Gzf) ~Ei

ai(0if,

Hill)

In general, two sequences of functions { u j } , { u ^ } are generated starting from an initial guess uj, u2, and will converge to and u2, respectively. 1- The Dirichlet-Neumann

method

Given A 0 , solve for each k > 0: ' -

A fc+i

(1.3.1)

.u

=

= = 0 Afc

f

in f2i on dQi on r,

n dQ

THE MATHEMATICAL FOUNDATION OF DD METHODS

12

<Jh. 1

then ' -

A

=

< u\+1 = 0

(1.3.2)

/

in 0 2 on dfl2 n dO on r

with A ^ 1 : = ffu^t1 + (1 - 6)\ k

(1.3.3)

6 being a positive acceleration parameter. This method was considered, for example, by Bj0rstad and Widlund (1986); Bramble et al. (1986a); Funaro et al. (1988); and Marini and Quarteroni (1988, 1989). The same method without relaxation (that is, with 0 = 1) does not necessarily converge, unless special assumptions are made about f2j and This can be easily seen already for one-dimensional problems: if the length of Qi (the Dirichlet subdomain) is larger than that of Q 2 we have convergence, otherwise the unrelaxed method diverges. An example is furnished in Fig. 1.3.1, where the method is applied to approximate the (null) solution of the problem -u"(x)=0 in (0,1) u(0) = u ( l ) = 0. Note that for this particular problem, whatever partition is chosen, a suitable parameter 6 yields exact convergence in two iterations.

o

FIG. 1.3.1. Divergence (left) and convergence (right) for the unrelaxed Dirichlet-Neumann iterations. Let us point out that a similar iterative procedure can be obtained by applying a relaxation procedure on the Neumann boundary condition, defining v!{ + l

rrmATivt; «ubstku<jtukijng m e t h o d s

§1.3 and U2 +l

as

^

solutions

13

to

r -Auk+1 = f -.fc+1 =_ 0 du k 2 +1

on dVt2 D 5 0 k

= /

' -Auk+X < uk+1

in n 2

=0

in fii

n dQ,

on

^ uk+1 = uk+1

on T,

setting then

When we present the finite dimensional version of the Dirichlet-Neumann scheme and its algebraic formulation (see Sections 3.2 and 3.3), it will be useful to resort to the weak formulation of the iterative procedure (1.3.1), (1.3.2). It reads as follows: ' find u k + 1 e Vi : < a1(uk+1,v1) ^ uk+1

' find uk+1

=Xk

= (f,v1)Ql

VvieV?

on T

G V2 :

< Muk2+1,v2) . a2{uk2+\ll2iA

= (f,v2h2 = (/,n2i^h2

vv2ev2° + (f,niAi)0l

- ai{uk+1

V n e A,

w here

a ; (-,-) have been introduced in (1.2.4) and Hi denotes any possible extension operator from A to Vi, and the notations are the same as in Section 1.2. Let us show how this iterative scheme can be interpreted as a preconditioned Richardson procedure. For the sake of exposition, we refer to the differential formulation (1.3.1)—(1.3.3); however, the same result can be proved for the weak formulation introduced above. Since we have chosen n = n 1 = —n2, we have

14

T-tihJ M A T i l E M A T i f A L f UUiNJJ/Vl IUiN U f

L)U

' - A ( < 4 + 1 - g2f) = o

Ju+

k 1 -g2f

Noting that uk+1

oil. i

in ft2 on 1. a

18

THE MATHEMATICAL FOUNDATION OF DD METHODS

<Jh. 1

On the other hand, the issue of parallelism is relevant in the case of partitions of ft using many (more than two) subdomains, and will be addressed in a more general framework in Section 1.4.2. Q

1.4

Generalisations

The whole set of considerations that we have developed so far for the Poisson equation can be straightforwardly extended to the homogeneous Dirichlet boundary value problem

{

Lu = /

in ft

u = 0

on a0|f|2

L°°(Q).

V £ £ R d , for almost all x £ ft

i,j=i for a suitable constant «o > 0. The operator L is symmetric if its coefficients satisfy aij(x) = ajt(x) V l,j -l,...,d, for almost all x £ ft. The associated bilinear form is (1.4.3)

a*(w,v):=

f

( Y] \i,j=i

aijDjwDiv

do wv

We assume that ao(x) > 0 for almost all x £ ft, hence a*(-,-) turns out to be symmetric, continuous, and coercive in Hq(Q), because from the Poincare inequality (1-2.2) it follows that a*(v,v)

>ao|!V«||2!n > a 0 ( l + C y ^ l M l l a

V v £ if^ft).

In particular, the form a*(-, •) induces a scalar product in Hq (ft). The week formulation of (1.4.1) reads: (1.4.4)

find u £ i ^ f t )

: a*(u,v) = (f,v)

VuGJ^fi),

and, as a consequence of the Lax-Milgram lemma, there exists a unique solution'

19

ULILN EK.AL1S A'L'LUJN S

Denoting by Ui the restriction of u to fij, i = 1,2, u being the solution to (1.4.1), the interface conditions satisfied on T = dfii fl ° : O 5 K , T ; 2 ) = (f,v2) uk

- a j V , ^ )

29

ufc + l/2

wk

a2 - a*2{uk+1/2, v2)

V u2 G

^

where wk denotes the extension of wk by 0 in ft \ fij. Similarly, method (1.5.3), (1.5.4) is obtained by solving Wk (1.5.9)

G V°

Wk G

: a*1(Wk,v1)

= ( / , Vl)0l

:

= (f,v2)n2-a*2(Uk,v2)

j/fc+i = uk + Wf +

- oj(tf*, ^ )

V

Vl

V

G V^0 G

W£,

having set U° :—u°. The proof that these variational formulations are equivalent to the original ones (1.5.1), (1.5.2) and (1.5.3), (1.5.4) is obtained via the verification of the following relations: uk+1/2

=

in

f

(1.5.10) ,k+1

_ J «2+1

^

and

(1.5.11)

1.5.3

C / ^ 1 = I U k +\

+ U ^

2

- U

k

^

2

in fix,2

T h e Schwarz m e t h o d as a projection m e t h o d

Concerning the alternating Schwarz method (1.5.1), (1-5.2), using (1.5.8)i and (1-5.8)2 we see that for each v G V* a*(uk+1/2

_

— al(wk,v\ni)

_

= U,v\n1)n1 =

-

al(uk,v|nJ

(f,v)-a*(u",v)

— a*(u — uk, v). Similarly, from (1.5.8)3, (1.5.8)4 it follows that a*{uk+1

- uk+1/2,v)

= a*(u — uk+1/2,v)

Vv G

.

30

THE MATHEMATICAL FOUNDATION OF DD METHODS

Therefore, the sequences u f c + 1 / 2 and uk+l uk

(1 5 12^

satisfy

+ l/2_uk

uk+1

<Jh. 1

=V*iu_uk')

-uk+1'2

=rz{u-u>

where V*, i = 1, 2 is the orthogonal projection of V onto V* with respect to the scalar product induced by the bilinear form a*(-,-); that is, for any w 6 V it holds that V*w e V* : a*(V*w -w,v) = 0 V v £ V?. Let us denote by I the identity operator, by Ji, i = 1,2, the (non-dense) immersion of V* into V (that is, J^v — v for each v 0.

= {I-V2)(I-Vi)ek

These equations are the basis of the proof of the convergence of uk and uk+1/2 to u in i? 1 (f2) (see Section 4.6). Similarly, setting Ek := u — Uk and using (1.5.19), for the Schwarz method (1.5.3), (1.5.4) it holds that (1.5.23) 1.5.4

Ek+1

= (I - Vi - V2)Ek

V k > 0.

The Schwarz method as a Richardson method

On the basis of (1.5.17), the alternating Schwarz method (1.5.1), (1.5.2) can also be regarded as a Richardson iterative procedure for the solution of the new problem (1.5.24)

Qmu = g-.=

QmQf.

The multiplicative term V2V\ is responsible for Qm being a second-degree polynomial operator, and prevents the parallelisation of method (1.5.1), (1.5.2). As a matter of fact (1.5.1), (1.5.2) is a sequential algorithm. The presence of the term ^Ti justifies the adjective multiplicative, which is attributed to the alternating Schwarz method. On the other hand, the algorithm (1.5.3), (1.5.4) is called the additive Schwarz method since it can be regarded as a Richardson iterative procedure for solving a nother problem that reads

32

THE MATHEMATICAL FOUNDATION OF DD METHODS QaU = g* :=

(1.5.25)

It is easy to see that g* = g{ + a*{g%v,)

QaQf.

where g{ £ V]*,

= {f,Vi)

<Jh. 1

£ V2* and

VviGV*,

i = 1,2,

hence g* can be computed by two local solves. The projection operators T% = J{P* are symmetric with respect to the scalar product induced by the form a*{-, •), because a*(ViV,w)

=

a*(V*v,w)

= a*(V*v,T*w)

(1.5.26)

=a*(v,V*w)

= a*(v, Viw)

V

v,u> £ V.

Consequently, the operator Q a is symmetric and positive definite with respect to the bilinear form a*{-, •). This property suggests basing on Q~ 1 the construction, at the finite dimensional level, of a preconditioner of the original problem, see Section 3.5. 1.5.5

A characterisation of the projection operators

In view of the discretisation of the alternating Schwarz method that will be discussed in Section 3.4, it is useful to rewrite the projection operators and, consequently, Qm in a different form. This involves the original differential operator L, and its restrictions to the subspaces V L e t us introduce the operators Li : —» (V®)' associated with the bilinear forms a*(-, •): (1-5.27)

{ L i w l , v i ) : = a*(Wi,Vi)

V

wuVi £

as well as the extension operators pj :

—>• V*

(1.5.28)

V F ^ V ,

PJVI:=VI

r0 V°, i'

0

and the transpose restriction operators pi : (V*)' —» (Vf)1 (1.5.29)

(PiG,vt)

:= (G,pjvi)

V

G £ (V*y,Vl

£

It is easily seen that (1.5.30)

U =

piJ^LJipf.

We claim that the following identities hold: (1.5.31)

V*=pjL~lPlJ?L,

i =

l,2.

In fact, define and j*, the identification operators between V° and its dual (Vf)' and V* and its dual {V*)', respectively. In particular, we have that

§1.5

THE SCHWARZ METHOD FOR OVERLAPPING SUBDOMAINS

(1.5.32) and

=

pij*p[,

u*)-1

=

33

pJ(j°r1pt,

that, from (1.5.30),

(1.5.33)

J? LJi =

jtplirfr'L.iXj^po;.

Then, using (1.5.32) and (1.5.33) for each v £ V*,w G V it holds that a'ipjL^piJTLw^)

=

a'iJipjL^piJ^Lw^iv)

= (LJipfL-'piJ^Liu, =

Jtv)

{J^LJipfL-'p^Lw^)

= (JTLw, v) = (.Lw, Jiv) = a*(w, Jiv) — a*(w, v), and the proof of (1.5.31) is complete. As a consequence, we also obtain (1.5.34) 1.5.6

Vi = JipTLT1

PiJ?L,

V%g = ;7:p! L,

.

The Schwarz method for many subdomains

The generalisation of the Schwarz method to the case of when Q is partitioned into M > 2 subdomains (see Fig. 1.5.3 for an example) is straightforward.

FIG. 1.5.3. Overlapping decomposition into 16 subdomains. Precisely, the multiplicative Schwarz method is given by (1.5.35)

uk+^ = ( I - V i ) u k + ^ + J i p T L - l P i J ? f ,

i = l,...,M

(here we have used (1.5.34)), and the additive Schwarz method reads

THE MATHEMATICAL FOUNDATION OF DD METHODS

34

<Jh. 1

FIG. 1.6.1. Domain embedding. (

M

1=1

\

M

/

4=1

The corresponding error equations read (1.5.37)

u - uk+1

= {I — Pm) •••(/ — Vi){u -

uk)

for the multiplicative case, and

(1.5.38)

u-Uk+1

=

{u-Uk)

for the additive case.

1.6

The fictitious domain method

This is one of the earliest ideas closely related to domain decomposition. The main motivation is that whenever a problem needs to be solved on a domain 0 having a complex boundary, it may be useful to embed it into a larger domain f2 of a simpler shape; say, for instance, a rectangle as in Fig. 1.6.1, and then solve a problem of similar type in 0 (see, for example, Buzbee et al. 1971; Matsokin 1972; Proskurowski and Widlund 1976; Kuznetsov 1989; and Glowinski et al 1994). Suppose, for instance, that the given problem in Cl is the following elliptic problem with a non-homogeneous Dirichlet boundary condition { L i t = /

in 0

u = (pr> on dCt. Here, we assume that / G L2(Cl), (pD G H1^2(dCl). The operator L is given by

THE FICTITIOUS DOMAIN METHOD

§1.6

35

d Lw := -

(1.6-2)

^

Di(aijDjw)

+

a0ii,

1,3 = 1 where the coefficients a^, a 0 belong to L°°(ft) and satisfy the ellipticity condition d ^

a i j ( x >

a0|£|2

V £ G R d , for almost all x £ A

1,3 = 1 for a suitable constant ao > 0. Moreover, we assume that aij(x)

= aji(x)

V l,j =

l,...,d

for almost all x £ ft, and that ao(x) > 0 for almost all x £ ft. As a consequence, the associated bilinear form

i * ( w , v ) •= JCl

\ \i,j=i

ai

i D •wD[V + a0 w v

turns out to be symmetric, continuous, and coercive in iJg(ft). Therefore, there exists a unique solution u of the corresponding weak formulation find u £ i f 1 (ft) : (1.6.3)

Vi) Gilo 1 (ft)

a* (u, v) = f fv Jn MiSq = ipjy

on u\dn =

= [ fv Jn

WvG on yUo > 0 for almost all x 6 H, to guarantee the existence and uniqueness of the solution. Furthermore, we suppose that tp^ G L2(dQ). It is convenient to formulate problem (1.6.7) in a different form. Setting Pi := £ . aijDjU, we have

THE FICTITIOUS DOMAIN METHOD

§1-6

divp + aou = f

37

in fi

and aljpj - D[U = 0

in fi, V 1 = 1,...,

d,

j=l where are the entries of the inverse matrix of { thus the function u\ in Oi w := > . u2 m U2 belongs to Hq(Q) and satisfies = A. Taking p, = A in ( 1 . 7 . 1 ) 3 and summing up (1.7.1)i (with vi = ui) and (1.7.1)5 (with v2 = u2) we obtain that ai(ui,ui)

+ a2(u2,u2)

= a*(w,w)

= 0,

whence w — 0 (by the coerciveness of the bilinear form a*(•,•)), and therefore Ui = 0 and A = 0. From (1.7.l)i we have now (o"i,Wi|r)r = 0 for each £ V±, which implies 01 = 0 on F. The last equality, a 2 = 0 on F, follows similarly.

• There is a clear equivalence between the three-field method and the classical way of relaxing Dirichlet boundary conditions through Lagrange multipliers. Indeed, let us consider, for instance, equations (1.7.1)i, (1.7.1)2 for known values °f / and A. The problem reads: find ui £ Vi, <j\ £ A' such that al(u\,vi) (1.7.4)

- (

1,

where P R ( K ) denotes the set of polynomials defined in K and of degree less than or equal to r globally with respect to all space coordinates. Then we set (2.1.3)

VH : = {vHE

XRH | vhldn

= 0}

§2.1

FINITE ELEMENT APPROXIMATION OF ELLIPTIC EQUATIONS

43

When the reference element K is the unit d-cube, the space Vh is defined in the same way, but in this case the space XJ^ is given by (2.1.4)

XRH : = {vh G C ° ( f t ) | vh]Ko

TK G Q R(K)

V i f G TH},

where Qr(K) denotes the set of polynomials defined in K and of degree less than or equal to r with respect to each variable Xi,... , Xd. The family of triangulations Th is said to be regular if there exists a constant a > 1 such that

— 0,

where hpc denotes the diameter of K and pk the maximum diameter of a ball contained in K. Under this assumption, denoting by 7Th,v £ Vh the interpolant of a continuous function v at the nodes of Th, the following interpolation error estimate holds: (2.1.5)

||u - 7Thu||o,n + h\\u -

n
respectively. B y the interpolation error estimate (2.1.5), we finally find that (2.1.6)

||u - uh||i,fi
h \ vh]r = 0}.

(2-1.8)

By repeating the lines of the proof of Lemma 1.2.1, we see that the finite element problem (2.1.1), after identifying u i ^ with and u2,h with uh|n2, is equivalent to the multi-domain problem ' ai(ui,h,vi,h) Ui,h = u2,h (2"L9)

= {f,v\,h)Q1 on r

a2(u2,h, v2lh) = (f,v2,h)n2 2

V v2,h G V2°h

2 ai(ui,h,

i=1

V v1:h G

) = £ ( / , Ki,hHh)ni i=1

V flh G Ah,

where 7Zith, i = 1,2, is any extension operator from A^ into VU}L. In practical implementation, these extension operators will be taken equal to the finite element interpolant ni^fth, which belongs to Vi.h, equals /J,h at the nodes on the interface P, and vanishes at the internal nodes in fi;. The formulation (2.1.9) may be generalised to many subdomains, possibly including cross-points. Note, in particular, that if IZi^^h is the restriction to fii of the finite element shape function associated with a cross-point P, then (2.1.9)4 enforces the continuity of the 'normal' derivative in P in a natural form.

§2.1

tiiNiTUttLEMHilN1 ArrxvUAlMArlUlN Uf KLbiFTIO ECj U ATKJJNS

45

R e m a r k 2 . 1 . 1 A multi-domain approach based on a dual variational formulation and the use of Lagrange multipliers, called FETI (Finite Element Tearing and Interconnecting), has been proposed by Farhat and Roux (1991, 1994), and is extensively used nowadays. •

2.1.2

Algebraic formulation of the discrete problem

The unknowns of the finite dimensional problem (2.1.1) are given by the pointvalues of Uh at the finite element nodes a r In fact, denoting by TV/1 the total number of the nodes and by ipj the basis functions of V/,; that is, the unique functions in Vh satisfying hr)h' i=l

2 Hi

>hlJLh}

=

i=1

Y^i^h^h),

and 2 t=l where the discrete extension operators 7?.^ were introduced in (2.1.9). Therefore, the operators Sh and S^h, i = 1,2, are symmetric. Moreover, due to the Poincare inequality (1.2.2) and the trace inequality (1.2.5) it follows that (Si,hr]h,Vh) = \\VHithr]h\\ln. > ^

+

^

II^IIA.

namely, the Steklov-Poincare operators are coercive, uniformly with respect to h. In Theorem 4.1.3 it will be shown that there exist two positive constants C2, independent of h, such that (2-2.5)

C i l M U < H f f i . / ^ l k f i , 0, independent of h, such that, for all rjh £ Ah (2.2.6) (Si,hVh,Vh) < (Shrjh,r)h) < C{SithVh,Vh), i = 1,2. In other words, the operators and Sh are spectrally equivalent, and therefore either Si^h ° r S2th can serve a§. an optimal preconditioner of Sh (see also the related result (2.3.14) below, which deals with the corresponding matrix Problem).

48 2.2.1

DISCRETISED EQUATIONS AND DOMAIN DECOMPOSITION

<Jh.

2

Eigenvalue analysis for the finite element Steklov-Poincare op. erator

Let us introduce the eigenvalue problem for the discrete Steklov-Poincare operator Sh• find j ( , £ R and wj, 6 A),, w j / 0, such that (2.2.7)

{ShVh,t*h) =

J uhjj,h

Vfj.fi £

Let us indicate by Nr the dimension of the interface space Ah- For example, in the case of piecewise-linear finite elements (r = 1), Nr is given by the number of finite element nodes lying on P, excluding the nodes on the boundary dfl (see Section 2.3). We have the following result: Proposition 2.2.1 Since the bilinear form a(-, •) is symmetric, continuous, and coercive in V, the associated discrete Steklov-Poincare operator Sh has real and positive eigenvalues Sj^, j — 1, • • -, Nr, which obey the following bounds

Kx < 5j,h


where notation is as in Sections 1.2 and 2.2, and 7Z±^ is any extension operator from A/i into V±^ (as previously noted, in practice 1Zi,h is an interpolant operator); then update (3.1.3)

\ k h + 1 =9u k 2 + h ] r + ( l ~ e ) \ k h

onT.

72

Oh. 3

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Note that (3.1.1) is the finite element approximation of the Dirichlet problem (1.3.1), whereas (3.1.2) is the finite element approximation of the mixed problem (1.3.2). As such, (3.1.2) can be equivalently rewritten as

«2(W2X1

(3.1.4)

' V2'h)

U2,/i)fi2

=

V V2,h

£

V£h

-aiiu^ilh^fih)

V ^

£ Ah,

where 1l 2 ,h is any extension operator from A^ into V2thIf the iterative method converges, the limit solutions uith •= lim u{h, '

k—too

u2:h :=

lim u2

k—too

h

satisfy (2.1.9).

The other iterative methods introduced in Section 1.3 can be similarly reformulated at the finite element level. Let us focus, for instance, on the N e u m a n n Neumann method. Again, let A° G Ah be an initial guess and for each k > 0, define and u ^ 1 to be the solution of the following problems: ,k+1 find u^ G Vi,h (3.1.5)

a

i(

U

,k+1 iji

>Vi,h)

1; see Brenner 19986). In three dimensions, the condition number grows faster than H/h. The constant C, in addition to being independent of h and H, is also independent of the coefficients of the differential operator L if they are constant in each subdomain flj. From (3.3.10) we deduce that the number of iterations of a Krylov subspace method to achieve a given tolerance grows like log With the aim of removing this residual (albeit mild) dependence on ^ , additional coupling between the edges and vertex points is necessary. The Schur complement £/,, introduced in (3.3.4), is not block diagonal in the permutation { e i , e2, • • •, e m , V } , since, in general, T, ekej 0 whenever et and ej are edges of the same subdomain. Having ignored this coupling in both preconditioners Ph and P ® p s the result is the logarithmic growth factor in the condition number, see (3.3.7) and (3.3.10). Some overlap in the decomposition of interface T = (li™ = 1 {ek}) U (U" = 1 {wj}) is obtained by introducing vertex regions {ri,r2, • • • ,rn}. The j t h vertex region

§3-3

SCHUR COMPLEMENT PRECONDITIONERS

41

-

it

i

Qi+

l>j>i

••• + ( - 1

QiAQjAQi

)M'1QMA---Q2AQ1

Since this is no longer symmetric, rather than using CG iterations we should use GMRES or other Krylov procedures that are suitable for non-symmetric matrices (see, for example, Saad 1996). R e m a r k 3.5.2 The restriction operator Ri can be regarded as being the sum of two terms Ri = R^ + R^, where accounts only for the nodes inside the domain excluding the overlapping region, while RI for those in the overlapping region.

96

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Oh.3

A new preconditioner has been introduced recently by Cai and Sarkis (1997), which is defined as follows: ( M PL

=

\i=l

and is called the restricted additive Schwarz preconditioner. The main motivation for designing PI s is that, in a parallel implementation, substantial communication cost can be saved because computing R(-v does not involve any data exchange with the neighbouring processors. • 3.5.1

Inexact solvers

If the Schwarz method is used as a preconditioner of an acceleration scheme, it is not necessary for the local subproblems to be solved exactly. In other words, in the preconditioning step of the GG iterations (3.5.13), P , ' 1 could be replaced by the following approximation: M P^1

—

^RjA^Ri, i=1

where Ai is a convenient approximation of A.t on the domain ft,;. A practical implementation of this idea consists of replacing the bilinear form a*(-, •) in f2j by a simpler one a*(-, •) (for example, freezing to a constant the coefficients in flj if the original elliptic problem has variable coefficients, or else by approximating A{ by an inexact factorisation). In any case, if a*(-, •) and •) are spectrally equivalent-, that is, there exist two positive constants K\ and K2 such that K{a*(Vi,Vi)

< a* {vi,

< K2a* (vi,Vi)

V vt G V*,

then the corresponding preconditioned system - P ^ A retains the same spectral properties of P ^ A , and the 'inexact' Schwarz algorithm converges with the same rate of the exact one.

3.6

Two-level methods

The convergence rate of the preconditioned iterative domain decomposition methods deteriorates when the number of subdomains becomes large. As has already been pointed out, this is due to the fact that in Schwarz algorithms, as well as in substructuring iterations, information is exchanged only between neighbouring subdomains. This weakness of the method has been overcome in previous sections; for instance, by introducing a coarse global problem set over the whole domain in order to guarantee a mechanism of global communication among all subdomains.

1

TWO-LEVEL METHODS

§3.6

97

This is not the only way, though; indeed, the coarse grid solver can be regarded as a special case of a two-level method, which is based on the simultaneous use of a 'fine' problem (the original one) and an auxiliary 'coarse' problem. 3.6.1

Abstract setting of two-level methods

Two-level methods involve the smoothing of the original problem and the solution (or preconditioning) of an auxiliary problem on a related mesh that is coarser than the original one. They can be thought of as an additive version of the general two-level multigrid algorithm. On a given grid, the preconditioner for the original problem is obtained as a superposition of the solution of an auxiliary problem on a related grid and a smoother on the original grid. The auxiliary problem should be simpler to solve, which is the case if it originates from a coarser grid or a numerical method simpler than the one on the original grid (for example, it uses piecewise polynomials of lower degree, or involves averaging the coefficients of the original operator). To formulate our problem in an abstract setting, we consider two finite dimensional problems, the original one (3.6.1)

find

uh G Vh

: ah(uh,vh)

V vh G Vh,

= Th{vh)

and the auxiliary one (3.6.2)

find

uH G VH

• CIH{UH,VH) = FH(Vh)

V vH G VH,

where A/,(•, •) and O,H(•, •) denote suitable bilinear forms on VH and VH, respectively. Given a basis m)-

The goal is t o find a preconditioner for Ah using either AH or a g o o d preconditioner of AH • The two spaces Vh and VH are related by an operator // t : VH —> Vh. Typically, Ih is the linear interpolation from the coarse grid to the fine grid and its matrix representation is RT (see Section 3.4). In these cases, h and H are the maximum diameter of the elements of the triangulations %, and TH , respectively. T w o different adjoint operators can be associated with Ih-

(3.6.4)

and

f Ih - - V h - ^ V H : 1 I (llwh,vH)vH = (Wh,IhVH)v h

v wh& Vh,vH

€ VH

98

(3.6.5)

Oh.3

ITERATIVE DD METHODS AT THE DISCRETE LEVEL f Jl --V^VH I [ aH(Jlwh,vH)

: = ah(wh,

V wh G Vh,vH

JhvH)

G VH-

In matrix form, keeping the same notation, this last definition reads AHJ% f £ A h , yielding = A'^ljAhA preconditioner Ph for Af, can be constructed in the following way: (3.6.6)

PH1

••= Q~H + QI1,

=

Q~H-=HA-£LZ,

where Qh is any convenient symmetric positive definite matrix, much simpler than Ah itself. The preconditioned matrix becomes p-'Ah^hJl

+

Qh'Ah.

If Qh is a g o o d preconditioner for Ah, Ph remains a good preconditioner for Ah even replacing in (3.6.6) A^1 with a good preconditioner on the coarse grid. In a two-level multi-grid context, Qh is the smoothing operator on the fine grid (in general, a few Jacobi iterations to damp the high frequencies of the error), while A ^ 1 is a coarse grid solution operator. In general, it is not enough to use Qjj 1 alone as the preconditioner, because it has a large null space. As a matter of fact, the rank of Q^1 is equal to the dimension of AH, which is lower than the dimension of AH- Therefore, any component of the residual lying in the null space of Qj/ would never be corrected. For this reason, Q ^ 1 must have full rank. For an elliptic partial differential equation, QH is designed to account for the long-range effects, Qh for the local ones. At the lowest level, Qh coincides with the diagonal part of the stiffness matrix Ah- In the domain decomposition context, the application of Qf1 will involve subdomain solves. 3.6.2

Multiplicative and additive two-level preconditioners

Define the vector f as f j : = a two-step preconditioner containing both the long-range and local-range components, QH and QH, respectively, is QH(U"

Qh(un+1

+ 1

/2_U")

= f - A f t U "

- uN+1/2) = f _

Ahun+1/2.

It can be written as a one-step method: (3.6.8)

u n + 1 = u " + {QH 1 + Q ^ 1 -

AhQ-£)(f

- Au").

This is the multiplicative two-level method. Its corresponding additive form, the additive two-level method, replaces (3.6.8) with the following equation: (3.6.9)

un+1 = u " + ( g ^ 1 + Q f t 1 ) ( f - A u n ) .

1

TWO-LEVEL METHODS

§3.6

99

This is one step of the Richardson method for the fine grid problem AHu = f with preconditioner PH (see (3.6.6)). It can be made more efficient when it is used as part of a Krylov subspace accelerator; namely, if the same preconditioner PH is used within a G M R E S or a conjugate gradient iterative procedure. Parallel multi-level preconditioners have been developed by Bramble et al. (1990); see also Zhang (1992) and Griebel (1994). 3.6.3

T h e case o f the Schwarz m e t h o d

When Qh is either a multiplicative or an additive overlapping Schwarz preconditioner (QH = P m s or Qh — Pas, see Section 3.5), we say that (3.6.8) or (3.6.9) are a two-level Schwarz methods. The generation of a coarse grid for the Schwarz method requires extra care with respect to the case of non-overlapping partitions. A commonly used technique for constructing an overlapping decomposition of fi into M subdomains Oi, • • •, fiju assumes that a non-overlapping partition • • •,LOM of CI is available. One possibility is to choose each subregion Wi as an element from a coarse finite element triangulation TH of fi of size H. Next, each UJ1 is extended to a larger domain fij, consisting of all points in fi at a distance not larger than SH from Wi, with 0 < 5 < 1. The restriction and extension maps, Ri and RF, as well as the local matrices Ai, are defined accordingly. Assume now that the finite element triangulation Th is a refinement of the coarse grid partition TH- Accordingly, let us denote by RJQ the interpolation map of coarse grid functions to fine grid functions. When using piecewise-linear elements, R j j interpolates the nodal values from the coarse grid vertices (of co,J to all the vertices of the fine grid. Its transpose RH is a weighted restriction map. Correspondingly, let AH denote the stiffness matrix of our elliptic problem on t h e c o a r s e m e s h TH, i.e. AH

=

RHAR

In the additive case, we have M (3.6.10)

Q^1

= Y,RJAJ1RI,

Q-J

=

RthA„1RH,

where the index i refers to the zth subdomain, i — 1 ,...,M, and the index H refers to the coarse grid (where subdomains play the role of elements). The resulting preconditioner is therefore

(3.6.11) where we have set, for notational convenience, RQ := RH and AO := AH- The multiplicative preconditioner is obtained similarly. 3.6.4

Convergence of two-level m e t h o d s

Let us assume that the two grids underlying problems (3.6.1) and (3.6.2), say TH and TH, are comparable, so that there exist two positive constants CO and C\

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

100

such that for any KH £ TH and KH £ TH with KH H KH (3.6.12)

Oh. 3

0 it holds that

C0 diam KH < diam Kh < G\ diam KH-

If the two grids are quasi-uniform and of comparable size (i.e. Co, Ci — 1), using either a block-Jacobi or a symmetric block-Gauss-Seidel iteration as a smoother, the preconditioner Ph is uniformly optimal for the operator Ah (the spectrum of P^xAh is uniformly bounded with respect to h). If, instead, TH is genuinely coarser than Th, which means that there are triangles KH and KH, KH n KH 0, for which (3.6.12) holds only for extremely small Co, then the above smoothers are no longer sufficient to guarantee that Ph is uniformly optimal for Ah- However, if the smoothing operator is based on the overlapping Schwarz methods (Qh given by the symmetric multiplicative Schwarz preconditioner, see (3.5.7), or Qh = Pas, see (3.5.8)), then P^1 is uniformly optimal, provided the coarse grid TH is 'comparable' with the subdomain partition. This means that there should exist positive constants Co and C\ such that for each KH £ TH such that KH D fij ^ 0 it holds that Co diam KH < diam 0 , < C\ diam KHFor the proofs of these results, see Bramble et al. (1996), and the references therein. In particular, we point out that the spectrum is bounded independently of h, H and the linear measure of the overlapping region, provided the latter is kept proportional to H, say, given by 5H. The convergence of the iterative procedures is poor for very small values of but improves rapidly as the overlap increases. Finally, it is worthwhile mentioning that the number of iterations for the symmetric multiplicative Schwarz method is roughly half of that needed for the additive Schwarz method. For numerical evidence see Smith et al. (1996); we refer to the same monograph for the analysis of multi-level methods with more than two levels. Remark 3.6.1 The generation of coarse grid preconditioners on unstructured grids is a difficult task, especially for three-dimensional problems. An algebraic way, inspired by agglomeration multi-grid techniques, is often adopted. A discussion can be found in Chan and Mathew (1994a); Bank and J. Xu (1995); and Chan and Smith (1995). See also Smith et al. (1996), Section 2.6 and the s references therein. •

3.7

Direct Galerkin approximation of the Steklov-Poincare equation

All the methods described in Section 1.3 share the property of being derived by a suitable splitting of the given boundary value problem over the subdomains. Thus, they are amenable to iterative procedures that, at each step, require the solution of at least as many independent boundary value problems as the number of subdomains. On the other hand, from a merely speculative point of view,

1

§3.7

DIRECT GALERKIN APPROXIMATION OF THE SP EQUATION

101

these iterative processes on the primitive variables Ui = (u being the solution of the given boundary value problem) can be interpreted as suitable iterative schemes for the dual variable X = u\r, which is the solution of the interface equation (1.1.7). More precisely, the iterative procedure on the primitive variables U{ induces a preconditioned iterative procedure on A. The same interpretation clearly holds at the finite dimensional level, which has been dealt with in Section 3.1. A suitable approximation is introduced first for the given boundary value problem (for example, a Galerkin finite element approximation), then an appropriate domain decomposition iterative procedure is chosen for the discrete primitive variables u^/j = U^Q., and the latter can be regarded as an iterative scheme for the dual variable Ah = Uh\T, which in turn is the solution of the discrete interface equation (2.2.3). It has to be pointed out, however, that the driving mechanism of the domain decomposition procedure is the subdomain iterative method acting on the discrete primitive variables. A different approach consists of attacking directly the interface equation (1.1.7) by a Galerkin method on a suitable subspace Ah of the space of traces A. More precisely, we start by reformulating (1.1.7) in a variational way as follows: (3.7.1)

find

A € A : S(X,p)

= (x,fA

V / i £ A,

where S(r],fj,) := (Sr],p) is the bilinear form associated with S, which is symmetric and coercive in A. Then we associate with (3.7.1) the following internal Galerkin approximation (3.7.2)

find

Aft £ A h : S{Xh,ph)

= <Xh,A»*>

V ph 6 Ah,

where \h is a convenient approximation to the right-hand side x of (1.1.7). The latter problem is completely defined after having chosen the finite dimensional subspace Ah of A. Obviously, (3.7.2) yields an algebraic problem that is symmetric and positive definite. This approach has been introduced by Agoshkov and Ovtchinnikov (1994), and has been given the name of Projection Decomposition method (PDM). Its interest relies mainly on the possibility of constructing well-conditioned, piecewisepolynomial bases for the space A^, an option that has been successively pursued by Ovtchinnikov. In a series of papers (Ovtchinnikov 1993, 1995; Gervasio et al. 1997; and Xantis and Ovtchinnikov 1994) it has been proved how to obtain these bases in a fast and accurate way for both Laplace and Stokes operators (through a Gram-Schmidt orthogonalisation procedure). The good conditioning of these bases makes it possible to solve the algebraic problem by a non-preconditioned conjugate gradient method. At each step, independent boundary value problems have to be solved in each subdomain. The accuracy with which these piecewise-polynomial functions are obtained allows us to maintain the order of accuracy of the solution of local problems in the subdomains, even when high-order methods are adopted therein. In several

102

ITERATIVE DD METHODS AT THE DISCRETE LEVEL

Ch. 3

instances, the P D M has been seen to be very effective compared with other domain decomposition methods.

4 CONVERGENCE ANALYSIS FOR ITERATIVE DOMAIN DECOMPOSITION ALGORITHMS In this chapter we present some abstract theorems that are useful for proving the convergence of iteration-by-subdomain methods. The main part of our analysis will cover the case of substructuring procedures (for disjoint subdomains). The convergence analysis of the Schwarz method for overlapping partitions will be addressed in the last section of this chapter. For a complete analysis of Schwarz methods we refer to Dryja and Widlund (1990, 1995), J. Xu (1992) and Smith et al. (1996). The chapter is organised as follows. We begin in Section 4.1 by providing some extension theorems in different function spaces H(div; fi,), i?(rot; fij)), as well as in the corresponding finite element subspaces. As outlined in the previous chapters, the local extension operators represent the basic mathematical ingredients of the Steklov-Poincare operator. On the other hand, iterative substructuring methods are amenable to preconditioned iterative algorithms on the Steklov-Poincare equation. For this reason, in Section 4.2 we focus on operators in Hilbert spaces, given in split form, and consider preconditioners whose inverses are made up of suitable combinations of inverses of suboperators. Then we provide some abstract convergence theorems for iterative methods of a preconditioned Krylov type in this framework. The analysis is also particularised to the algebraic finite dimensional counterpart of our equations. As of our paradigmatic elliptic boundary value problem, we address its convergence analysis in this chapter, immediately after having carried out the abstract analysis of Section 4.2. Since we also cover the non-symmetric and complex cases, our abstract results will be applied (through the following chapters) to a wide class of boundary value problems. In particular, Theorems 4.2.2 and 4.2.5 can be applied to Dirichlet-Neumann and Neumann-Neumann iterations, respectively, for a wide variety of situations, including general elliptic problems, the elasticity operator, the Stokes problem, and its generalisations, as well as for advection-diffusion equations. The first theorem will also be useful for some of the heterogeneous domain decomposition procedures that we will describe in Chapter 8. Theorems 4.2.10 and 4.2.13 (and Corollaries 4.2.11 and 4.2.14) are concerned with iterations for complex matrices, with a non-symmetric preconditioner, and can be applied to the Dirichlet-Neumann and Neumann-Neumann iterative scheme, respectively, for situations like those arising with the time-harmonic Maxwell equations.

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CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

The proof of the convergence of the Robin iterative substructuring method is based on an ad hoc argument and is presented in Section 4.5.

4.1

Extension theorems and spectrally equivalent operators

As anticipated in the previous chapters, typical domain decomposition preconditioners are expressed in terms of the sum of operators, each one related to a certain subdomain ft; of the (bounded) computational domain ft C R d , d = 2,3. This is the case for the Steklov-Poincare operator S for the Laplace operator (or, similarly, for the symmetric elliptic operator L introduced in (1.4.2)), which is defined as M {.Sr),n) =

Y^iSil'V)

(4.1.1) {Si-q, y) = a,i (Hn/, Hifj.) = / VHiri • VHi/j,, JQi where rj and fj. are trace functions on the interface F, and Hi denotes the harmonic extension operator from T into ft, (see Section 1.2). The preconditioners are constructed by assembling the local operators Si. To analyse the spectral properties of a preconditioner based on substructuring; namely, based on the local operators Si, it is necessary to show that the extension operators Hi induce an equivalent norm on A, the space of traces on

r.

In the next section we will focus on the equivalence of these extension operators. Then in Sections 4.1.2 and 4.1.3 we will consider extension operators in different function spaces, that will then be used in Chapter 5 for the analysis of many other boundary value problems. As we have already mentioned in the Preface, we restrict our attention to the case of a two-domain decomposition of the domain ft. 4.1.1

Extension theorems in i f 1 (ft,)

The first result we are interested in concerns the harmonic extension. Proposition 4.1.1 Let the space A be defined in (1.2.4) and the extension operators Hi in (1.2.14). Then there exist two positive constants C\ and C2 such that (4.1.2)

(AIMIa < H ^ U i , n i R | ip = (v • n)|r, v G # 0 ( d i v ; ft)}-

The trace space ^ coincides with the dual space of Hl/2(T) (see Girault and Raviart 1986, p. 27-9), and therefore is the dual of A if V n 3ft = 0, as in Fig. 1-1 (right), whereas \I> is strictly included in the dual space of A if T fl 0ft ^ 0, as in Fig. 1.1 (left). The norm in 'J' will be denoted by || • For each ip G the vector function Qi%p e Wi, i = 1,2, is the solution to (Qi^.vOn, + (div(Qi^),divvi)ni = 0 (4.1.21) (Qi4> • n)jr = ijj

on r ,

V v< G W °

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CONVERGENCE ANALYSIS FOR ITERATIVE DD ALGORITHMS

Ch. 4

where (•, -)q ; denotes the L 2 (fi;)-scalar product. In differential form, Qiift satisfies the following problem in the sense of distributions ' Qi^

—

V div Qiip = 0

(Qiip •

= 0

in Orrii on 1 / 2 , and that \\Miiph\\i+s*