The Finite Element Method for Elliptic Problems
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The Finite Element Method for Elliptic Problems
SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem *First time in print.
Classics in Applied Mathematics (continued) Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerised Tomography Avinash C. Kak and Malcolm Slaney, Principles of Computerised Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimisation and Economic Theory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lancsos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the NavierStokes Equations
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The Finite Element Method for Elliptic Problems Philippe G. Ciarlet Universite Pierre et Marie Curie Paris, France
Siam Society for Industrial and Applied Mathematics Philadelphia
Copyright © 2002 by the Society for Industrial and Applied Mathematics This SIAM edition is an unabridged republication of the work first published by North-Holland, Amsterdam, New York, Oxford, 1978.
1098765432 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Ciarlet, Philippe G. The finite element method for elliptic problems / Philippe G. Ciarlet. p. cm. — (Classics in applied mathematics ; 40) Includes bibliographical references and index. ISBN 0-89871-514-8 (pbk.) 1. Differential equations, Elliptic—Numerical solutions. 2. Boundary value problems—Numerical solutions. 3. Finite element method. I. Title. II. Series. QA377 .C53 2002 515'.353--dc21
2002019515
Siam
is a registered trademark.
To Monique
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TABLE OF CONTENTS PREFACE TO THE CLASSICS EDITION
xv
PREFACE
xix
GENERAL PLAN AND INTERDEPENDENCE TABLE 1. ELLIPTIC BOUNDARY VALUE PROBLEMS Introduction
xxvi
1 1
1.1. Abstract problems The symmetric case. Variational inequalities The nonsymmetric case. The Lax-Milgram lemma Exercises 1.2. Examples of elliptic boundary value problems The Sobolev spaces Hm(l3). Green's formulas First examples of second-order boundary value problems . . . . The elasticity problem Examples of fourth-order problems: The biharmonic problem, the plate problem Exercises Bibliography and Comments 2. INTRODUCTION TO THE FINITE ELEMENT METHOD Introduction
2 2 7 9 10 10 15 23 28 32 35 36 36
2.1. Basic aspects of the finite element method The Galerkin and Ritz methods The three basic aspects of the finite element method. Conforming finite element methods Exercises 2.2. Examples of finite elements and finite element spaces Requirements for finite element spaces First examples of finite elements for second order problems: nSimplices of type (k), (3') Assembly in triangulations. The associated finite element spaces n-Rectangles of type (k). Rectangles of type (2'), (3')- Assembly in triangulations First examples of finite elements with derivatives as degrees of freedom: Hermite n-simplices of type (3), (3'). Assembly in triangulations First examples of finite elements for fourth-order problems: the ix
37 37 38 43 43 43 44 51 55 64
X
CONTENTS Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations Exercises 2.3. General properties of finite elements and finite element spaces ... Finite elements as triples (K, P, £). Basic definitions. The P-interpolation operator Affine families of finite elements Construction of finite element spaces Xh. Basic definitions. The Xhinterpolation operator Finite elements of class u Uniform boundedness of the mapping u -» uk with respect to appropriate weighted norms Estimates of the errors \u - Mjo.»,n a°d |« — "hli,ocjj- Nitsche's method of weighted norms Exercises Bibliography and comments
110 112 112 116
4. OTHER FINITE ELEMENT METHODS FOR SECOND-ORDER PROBLEMS s Introduction 4.1. The effect of numerical integration Taking into account numerical integration. Description of the resulting discrete problem Abstract error estimate: The first Strang lemma
69 77 78 78 82
122 126 131 131 134 136 139 143 147 147 155 163 167 168
174 174 178 178 185
CONTENTS Sufficient conditions for uniform V h -ellipticity Consistency error estimates. The Bramble-Hilbert lemma Estimate of the error ||u - unlin Exercises 4.2. A nonconforming method Nonconforming methods for second-order problems. Description of the resulting discrete problem Abstract error estimate: The second Strang lemma An example of a nonconforming finite element: Wilson's brick Consistency error estimate. The bilinear lemma Estimate of the error (2 K 6 T jH-«,,H. K ) I / 2 Exercises 4.3. Isoparametric finite elements Isoparametric families of finite elements Examples of isoparametric finite elements Estimates of the interpolation errors \v - f J K v \ m q K Exercises 4.4. Application to second order problems over curved domains Approximation of a curved boundary with isoparametric finite elements Taking into account isoparametric numerical integration. Description of the resulting discrete problem Abstract error estimate Sufficient conditions for uniform V h -ellipticity Interpolation error and consistency error estimates Estimate of the error jju - «Ji./D, Exercises Bibliography and comments Additional bibliography and comments Problems on unbounded domains The Stokes problem Eigenvalue problems 5. APPLICATION OF THE FINITE ELEMENT METHOD TO SOME NONLINEAR PROBLEMS Introduction 5.1. The obstacle problem Variational formulation of the obstacle problem An abstract error estimate for variational inequalities Finite element approximation with triangles of type (1). Estimate of the error \\u - wj, „ Exercises 5.2. The minimal surface problem A formulation of the minimal surface problem Finite element approximation with triangles of type (1). Estimate of the error ||u - MA||,A Exercises 5.3. Nonlinear problems of monotone type
xi 187 190 99 201 207 207 209 211 217 220 223 224 224 227 230 243 248 248 252 255 257 260 266 270 272 276 276 280 283
287 287 289 289 291 294 297 301 301 302 310 312
xii
CONTENTS A minimization problem over the space Wo"((l), 2 0, the affine mapping is a contraction. To see this, we observe that
since, using inequalities (1.1.3) and (1.1.21),
Therefore the mapping defined in (1.1.23) is a contraction whenever the number p belongs to the interval ]0,2a/M 2 [ and the proof is complete. D Remark 1.1.3. It follows from the previous proof that the mapping A: V-> V is onto. Since the mapping A has a continuous inverse A"', with
Therefore the variational problem (1.1.18) is well-posed in the sense that its solution exists, is unique, and depends continuously on the data f (all other data being fixed). Exercises 1.1.1. Show that if M,, / = 1,2, are the solutions of minimization problems (1.1.1) corresponding to linear form /, G V, i = 1, 2, then
10
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1, § 1.2.
(i) Give a proof which uses the norm reducing property of the projection operator. (ii) Give another proof which also applies to the variational problem (1.1.15). 1.1.2. The purpose of this exercise is to give an alternate proof of the Lax-Milgram lemma (Theorem 1.1.3). As in the proof given in the text, one first establishes that the mapping stf = T • A: V-+V is continuous with \\d\\ *s M, and that a|H| «= \\stv\\ for all v G V. It remains to show that d(V)= V. (i) Show that s&(V) is a closed subspace of V. (ii) Show that the orthogonal complement of s#(V) in the space V is reduced to {0}. 1.2. Examples of elliptic boundary value problems The Sobolev spaces Hm(fl). Green's formulas Let us first briefly recall some results from Differential Calculus. Let there be given two normed vector spaces X and Y and a function v: A-* Y, where A is a subset of X. If the function is k times differentiate at a point a G A, we shall denote Dkv(a), or simply Dv(a) if k = 1, its fc-th (Frechet) derivative. It is a symmetric element of the space J£fc(X; Y), whose norm is given by
We shall also use the alternate notations Dv(a) = v'(a) and D2v(a) v"(a). In the special case where X — R" and Y = R, let eh l^i^n, denote the canonical basis vectors of R". Then the usual partial derivatives will be denoted by, and are given by, the following:
Occasionally, we shall use the notation Vt>(a), or grad v ( a ) , to denote the gradient of the function v at the point a, i.e., the vector in R" whose components are the partial derivatives diV(a), l^i^n.
Ch. 1, § 1.2.]
EXAMPLES
11
We shall also use the multi-index notation: Given a multi-index a = (a,, a2, • • • , ««) £ N", we let |a| = 2?=i a,. Then the partial derivative d a v ( a ) is the result of the application of the |aj-th derivative DMv(a) to any |a[-vector of (R")1"1 where each vector e± occurs a, times, 1 «s / ^ n. For instance, if n = 3, we have div(a) = d"M}v(a), dmv(a)= d < 1 < U ) t>(a), dmv(a) = d(™Mv(a), etc... There exist constants C(m, n) such that for any partial derivative d a v ( a ) with |a| = m and any function v,
where it is understood that the space R" is equipped with the Euclidean norm. As a rule, we shall represent by symbols such as Dkv, v", dtv, dav, etc. . . , the functions associated with any derivative or partial derivative. When h\ ~ h2 = • • • - hk - h, we shall simply write Thus, given a real-valued function u, Taylor's formula of order k is written as
for some 6 £ ]0,1[ (whenever such a formula applies). Given a bounded open subset fl in R", the space 3)(fi) consists of all indefinitely differentiate functions v: /2-»R with compact support. For each integer m 5*0, the Sobolev space Hm(fi) consists of those functions v G L2(H) for which all partial derivatives dav (in the distribution sense), with |cr|*sra, belong to the space L 2 (/2), i.e., for each multi-index a with |a|*£w, there exists a function d"v G L2(fl) which satisfies
Equipped with the norm
the space Hm(fl) is a Hilbert space. We shall also make frequent use of the semi-norm
12
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1, § 1.2.
We define the Sobolev space the closure being understood in the sense of the norm ||-||m>/j. When the set ft is bounded, there exists a constant C(fl) such that
this inequality being known as the Poincare-Friedrichs inequality. Therefore, when the set fl is bounded, the semi-norm \-\m,n is a norm over the space H0m(/2), equivalent to the norm \\~\\m,n (another way of reaching the same conclusion is indicated in the proof of Theorem 1.2.1 below). The next definition will be sufficient for most subsequent purposes whenever some smoothness of the boundary is needed. It allows the consideration of all commonly encountered shapes without cusps. Following NECAS (1967), we say that an open set (I has a Lipschitzcontinuous boundary F if the following conditions are fulfilled: There exist constants a > 0 and (3 > 0, and a finite number of local coordinate systems and local maps ar, l^r^R, which are Lipschitz-continuous on their respective domains of definitions {x r GR"~'; |jcr|s£a}, such that (Fig. 1.2.1):
where xr = (jc 2 r ,..., x£), and \xr\ < a stands for |x/j < a, 2 «£ i «s n. Notice in passing that an open sef w/f/i a Lipschitz-continuous boundary is bounded. Occasionally, we shall also need the following definitions: A boundary is of class tH? if the functions ar: \xr\ ^ a -*R are of class % (such as (€m or ,trd, for functions in the space H2(fi), the following characterization holds: Given two functions u, v E H\fl), the following fundamental Green's formula
holds for any i E [1, n]. From this formula, other Green's formulas may be easily deduced. For example, replacing u by d,u and taking the sum from 1 to n, we get
for all u G. H2(O), v E H'(/2). As a consequence, we obtain by subtraction:
for all u, v E H2(fl). Replacing u by Au in formula (1.2.6), we obtain
for all u E H\fl), v E H2(O). As another application of formula (1.2.4), let us prove the relation which implies that, over the space H02(/2), the semi-norm v-^\Av\9 /j = 0, we deduce that v = 0, which is in contradiction with the equalities ||uji,/j= 1 for all k. From this theorem, we infer that the bilinear form of (1.2.23) is V-elliptic since we have a(v, v)z*p\v\],a for all vEHl(O), as an application of the inequalities of (1.2.24) and (1.2.25) shows. By the Lax-Milgram lemma (Theorem 1.1.3), there exists a unique function u E V which satisfies the variational equations
Referring once again to formula (1.2.4), we obtain another Green's formula:
valid for all functions u E H\fl), v E H\fl), provided the functions ajy are smooth enough so that the functions a^diu belong to the space Hl(Cl} (for example, a^ £ 0, /u, > 0. We define the bilinear form
24
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1, § 1.2
and the linear form
where / = (/,,/2,/3)e(L2(/2))3 and g = (g^^EO^r,))3, with F, = F - FO are given functions. It is clear that these bilinear and linear forms are continuous over the space V. To prove the V-ellipticity of the bilinear form (see Exercise 1.2.4), one needs Korn's inequality: There exists a constant C(fl) such that, for all v = (t>i, t>2, t>3) e (H\f)rf,
This is a nontrivial inequality, whose proof may be found in DUVAUT & LIONS (1972, Chapter 3, $3.3), or in FICHERA (1972, Section 12). From it, one deduces that over the space V defined in (1.2.30) the mapping
is a norm, equivalent to the product norm, as long as the dy-measure of F0 is strictly positive, which is the case here (again the reader is referred to Exercise 1.2.4). The V-ellipticity is therefore a consequence of the inequalities A > 0, M > 0 , since by (1.2.33) We conclude that there exists a unique function « G V which minimizes the functional
over the space V, or equivalently, which is such that
Ch. 1,§1.2.]
EXAMPLES
25
Since relations (1.2.37) are satisfied by all functions v G ((/}))3, they could yield the associated partial differential equation. However, as was pointed out in Remark 1.2.1, it is equivalent to proceed through Green's formulas, which in addition have the advantage of yielding boundary conditions too. Using Green's formula (1.2.4), we obtain, for all u E(H 2 (/2)) 3 and all t>E(H'(/2)) 3 :
so that, using definitions (1.2.31) and (1.2.32), we have proved that the following Green's formula holds:
for all functions Arguing as in the previous examples, we find that we are formally solving the equations
It is customary to write these equations in vector form: which is derived from (1.2.39) simply by using relations (1.2.32). Taking equations (1.2.39) into account, the variational equations (1.2.37) reduce to
since To sum up, we have formally solved the following associated boundary value problem:
26
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1,§ 1.2.
which is known as the system of equations of linear elasticity. Let us mention that a completely analogous analysis holds in two dimensions, in which case the resulting problem is called the system of equations of two-dimensional, or plane, elasticity, the above one being also called by contrast the system of three-dimensional elasticity. Accordingly, the variational problem associated with the data (1.2.30), (1.2.33) and (1.2.34) is called the (three- or two-dimensional) elasticity problem.
Fig. 1.2.3
Assuming "small" displacements and "small" strains, this system describes the state of a body (Fig. 1.2.3) which occupies the set fi in the absence of forces, u denoting the displacement of the points of ll under the influence of given forces (as usual, the scale for the displacements is distorted in the figure). The body 17 cannot move along jT0, and along F\, surface forces of density g are given. In addition, a volumic force, of density /, is prescribed inside the body /2. Then we recognize in (e/,(ii)) the strain tensor while (0^(11)) is the stress tensor, the relationship between the two being given by the linear equations (1.2.32) known in Elasticity as Hooke's law for isotropic bodies. The constants A and ft are the Lame coefficients ts of the material of which the body is composed.
Ch. 1,§1.2.]
EXAMPLES
27
The variational equations (1.2.37) represent the principle of virtual work, valid for all kinematically admissible displacements v, i.e., which satisfy the boundary condition v = 0 on F0. The functional / of (1.2.36) is the total potential energy of the body. It is the sum of the strain energy:
and of the potential energy of the exterior forces: This example is probably the most crucial one, not only because it has obviously many applications, but essentially because its variational formulation, described here, is basically responsible for the invention of the finite element method by engineers. Remark 1.2.2. It is interesting to notice that the strict positiveness of the dy-measure of F0 has a physical interpretation: It is intuitively clear that in case the dy-measure T0 would vanish, the body would be free and therefore there could not exist an equilibrium position in general. Remark 1.2.3. The membrane problem, which we have already described, the plate problem, which we shall soon describe in this section, and the shell problem (Section 8.1), are derived from the elasticity problem, through a process which can be briefly described as follows: Because such bodies have a "small" thickness, simplifying a priori assumptions can be made (such as linear variations of the stresses over the thickness) which, together with other assumptions (on the constitutive material in the case of membranes, or on the orthogonality of the exterior forces in the case of membranes and plates), allow one to integrate the energy (1.2.36) over the thickness. In this fashion, the problem is reduced to a problem in two variables, and only one function (the "vertical" displacement) in case of membranes and plates. All this is at the expense of a greater mathematical complexity in case of plates and shells however, as we shall see. Remark 1.2.4. Since problem (1.2.40) is called system of linear elasticity, the linearity being of course that of the mapping (/, g)-»u, it is worth saying how this problem might become nonlinear. This may happen in three nonexclusive ways:
28
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1, § 1.2.
(i) Instead of minimizing the energy over the space V, we minimize it over a subset U which is not a subspace. This circumstance, which we already commented upon (Remark 1.1.1) is examined in Exercise 1.2.5 for a simpler model. Another example is treated in Section 5.1. (ii) Instead of considering the "linearized" strain tensor (1.2.31), the "full" tensor is considered, i.e., we let
Actually, it suffices that for at least one pair (i, /), the above expression be considered. This is the case for instance of the von Karmann's model of a clamped plate. (iii) The linear relation (1.2.32) between the strain tensor and the stress tensor is replaced by a nonlinear relation. D Examples of fourth-order problems: The biharmonic problem, the plate problem Whereas in the preceding examples the spaces V were contained in the space H'(/2), we consider in the last examples Sobolev spaces which involve second-order derivatives. We begin with the following data:
Since the mapping v-*\Av\w is a norm over the space H02(/2), as we showed in (1.2.8), the bilinear form is H02(/2)-elliptic. Thus there exists a unique function u E H$(fl) which minimizes the functional
over the space H02(/2) or, equivalently, which satisfies the variational equations
Ch. 1 , § 1 . 2 . ]
EXAMPLES
29
Using Green's formula (1.2.7):
we find that we have formally solved the following homogeneous Dirichlet problem for the biharmonic operator A1:
We shall indicate a physical origin of this problem in the section "Additional Bibliography and Comments" of Chapter 4. As our last example, we let, for n = 2,
These data correspond to the variational formulation of the (clamped) plate problem, which concerns the equilibrium position of a plate of constant thickness e under the action of a transverse force, of density F = (Ee3l\2(\ - o-2))/ per unit area. The constants E = /A(3A + 2^)/(A + jx.) and a - A/2(A +/u,) are respectively the Young's modulus and the Poisson's coefficient of the plate, A and /u, being the Lame's coefficients of the plate material. When / = 0, the plate is in the plane of coordinates (x,,jc 2 ) (Fig. 1.2.4). The condition u G Hl(fl) takes into account the fact that the plate is clamped (see the boundary conditions in (1.2.48) below). As we pointed out in Remark 1.2.3, the expressions given in (1.2.45) for the bilinear form and the linear form are obtained upon integration over the thickness of the plate of the analogous quantities for the elasticity problem. This integration results in a simpler problem, in that there are now only two independent variables. However, this advantage is compensated by the fact that second partial derivatives are now present
30
ELLIPTIC BOUNDARY VALUE PROBLEMS
[Ch. 1, § 1.2.
Fig. 1.2.4
in the bilinear form. This will result in a fourth-order partial differential equation. See (1.2.48). The Poisson's coefficient a satisfying the inequalities 0,,K is the /-th component of the unit outer normal vector along dK. By summing over all finite elements, we obtain
and the proof follows if we notice that the sum 2jce^,/a/cuU^»'i,K dy vanishes: Either a portion of dK is a portion of the boundary F of fl in which case $ = 0 along this portion, or the contribution of adjacent elements is zero. The boundary F being Lipschitz-continuous by assumption, the second inclusion follows from the characterization which was mentioned in Section 1.2. Assuming Theorem 2.1.1 applies, we shall therefore use the finite element space Vh = Xoh if we are solving a second-order homogeneous Dirichlet problem, or Vh = Xh if we are solving a second-order homogeneous or nonhomogeneous Neumann problem. The proof of the next theorem is similar to that of Theorem 2.1.1 and, for this reason, is left to the reader as an exercise (Exercise 2.1.1). Theorem 2.1.2. Assume that the inclusions PK C H\K) for all K G 3~h and Xh C/)) is symmetric and positive definite, which is an advantage for the numerical solution of the linear system (2.1.4). By contrast, this is not generally the case for standard finite-difference methods, except for rectangular domains. Assuming again the symmetry of the bilinear form, one could conceivably start out with any given basis, and, using the Gram-Schmidt orthonormalization procedure, construct a new basis (w^)^=i which is orthonormal with respect to the inner product «(•,-)• This is indeed an efficient way of getting a sparse matrix since the corresponding matrix (a(w*, w>,*)) is the identity matrix! However, this process is not recommended from a practical standpoint: For comparable computing times, it yields worse results than the solution by standard methods of the linear system corresponding to the "canonical" basis. It was mentioned earlier that the three basic aspects were characteristic of the finite element method in its simplest form. Indeed, there are more general finite element methods: (i) One may start out with more general variational problems, such as variational inequalities (see Section 5.1) or various nonlinear problems (see Sections 5.2 and 5.3), or different variational formulations (see Chapter 7). (ii) The space Vh, in which one looks for the discrete solution, may no longer be a subspace of the space V. This may happen when the boundary of the set fl is curved, for instance. Then it cannot be exactly triangulated in general by standard finite elements and thus it is replaced by an approximate set fih (see Section 4.4). This also happens when the functions in the space Vh lack the proper continuity across adjacent finite elements (see the "nonconforming" methods described in Section 4.2 and Section 6.2). (iii) Finally, the bilinear form and the linear form may be approximated. This is the case for instance when numerical integration is used for computing the coefficients of the linear system (2.1.4) (see Section 4.1), or for the shell problem (see Section 8.2). Nevertheless, it is characteristic of all these more general finite element methods that the three basic aspects are again present. To conclude these general considerations, we shall reserve the terminology conforming finite element methods for the finite element
Ch. 2, § 2.2.]
FINITE ELEMENTS AND FINITE ELEMENT SPACES
43
methods described at the beginning of this section, i.e., for which Vh is a subspace of the space V, and the bilinear form and the linear form of the discrete problem are identical to the original ones. Exercises 2.1.1. Prove Theorem 2.1.2. 2.1.2. The purpose of this problem is to give another proof of the Lax-Milgram lemma (Theorem 1.1.3; see also Exercise 1.1.2) in case the Hilbert space V is separable. Otherwise the bilinear form and the linear form satisfy the same assumptions as in Theorem 1.1.3. (i) Let Vh be any finite-dimensional subspace of the space V, and let uh be the discrete solution of the associated discrete problem (2.1.2). Show that there exists a constant C independent of the subspace Vh such that HwfcU^sC (as usual, there is a simpler proof when the bilinear form is symmetric). (ii) The space V being separable, there exists a nested sequence (Vv)veN of finite-dimensional subspaces such that (U V(=N Vv)~ = V. Let (*/„)„
with
From these two theorems, we derive the definition of the rectangle of type (2;) (Fig. 2.2.12) and of the rectangle of type (3') (Fig. 2.2.13). If it happens that the set Cl C R" is rectangular, i.e., it is either an n-rectangle or a finite union of n-rectangles, it can be conveniently "triangulated" by finite elements which are themselves /i-rectangles: The fifth condition (5*5) on a triangulation now reads: (5*5) Any face of any n-rectangle KI in the triangulation is either a subset of the boundary f, or a face of another n-rectangle /C2 in the triangulation.
Ch. 2, i 2.2.]
FINITE ELEMENTS AND FINITE ELEMENT SPACES
Fig. 2.2.12
Fig. 2.2.13
63
64
INTRODUCTION TO THE FINITE ELEMENT METHOD
[Ch. 2, § 2.2.
In the second case, the n-rectangles X, and K2 are said to be adjacent. An example of a triangulation made up of rectangles is given in Fig. 2.2.14. With such a triangulation, we may associate in a natural way a finite element space Xh with each type of the rectangular finite elements which we just described. Since the discussion is almost identical to the one concerning n-simplices, we shall be very brief. In particular, one can prove the following analog of Theorem 2.2.3. Theorem 2.2.7. Let Xh be the finite element space associated with n-rectangles of type (k) for any integer k > 1 or with rectangles of type (2') or (3'). Then the inclusion holds. Finally, arguing as before, it is easily seen that such finite element spaces possess a basis whose functions have "small" support (FEM 3). First examples of finite elements with derivatives as degrees of freedom: Hermite n-simplices of type (3), (3'). Assembly in triangulations So far, the degrees of freedom of each finite element K have been "point values", i.e., of the form p ( a ) , for some points a G K. We shall next introduce finite elements in which some degrees of freedom are partial derivatives, or, more generally, directional derivatives, i.e.,
Fig. 2.2.14
Ch. 2, § 2.2.]
FINITE ELEMENTS AND FINITE ELEMENT SPACES
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expressions such as Dp(a)b, D2p(a)(b, c), etc..., where b, c are vectors inR". The first example of this type of finite element is based on the following theorem. Theorem 2.2.8. Let K be an n-simplex with vertices a,, 1 *£ / *£ n + 1, and let aiik = 3(0, + Oj + a^, 1 ^ / < / < fc«£ n + 1. TTiew any polynomial in the space P3 is uniquely determined by its values and the values of its n first partial derivatives at the vertices a,, 1 «£ i *s /i + 1, and its values at the points ank, l^i<j