The Finite Element Method for Elliptic Problems

THE FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS

STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 4

Editors : J.L. LIONS, Paris G . PAPANICOLAOU, New York R.T. ROCKAFELLAR, Seattle

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW YORK * OXFORD

THE FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS

PHILIPPE G . CIARLET Universitk Pierre et Marie Curie, Paris

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM * NEW YORK * OXFORD

@ NORTH-HOLLAND PUBLISHING COMPANY

- 1978

All rights reserved. N o pat? of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Library of Congress Catalog Card Number 77-24471 North-Holland ISBN 0 444 85028 7

Published by: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK . OXFORD Sole distributors for the U.S.A. and Canada:

Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York, NY 10017

1s t edition 19 7 8 2nd printing 19 79

Library of Congress Cataloging in Publication Data

Ciarlet, Philippe G. The finite element method for elliptic problems. (Studies in mathematics and its applications; v. 4) Bibliography: p. 481 Includes index. 1. Partial differential equations, Elliptic-Numerical solutions. 2. Boundary value problems - Numerical solutions. 3. Finite element method. I. Title. 11. Series. QA377.CS3 515l.353 77-24477 ISBN 0-444-85028-7

PRINTED IN THE NETHERLANDS

TO Monique

This Page Intentionally Left Blank

PREFACE

The objective of this book is to analyze within reasonable limits (it is not a treatise) the basic mathematical aspects of the finite element method. The book should also serve as an introduction to current research on this subject. On the one hand, it is also intended to be a working textbook for advanced courses in Numerical Analysis, as typically taught in graduate courses in American and French universities. For example, it is the author’s experience that a one-semester course (on a three-hour per week basis) can be taught from Chapters 1, 2 and 3 (with the exception of Section 3.3), while another one-semester course can be taught from Chapters 4 and 6. On the other hand, it is hoped that this book will prove to be useful for researchers interested in advanced aspects of the numerical analysis of the finite element method. In this respect, Section 3.3, Chapters 5 , 7 and 8, and the sections on “Additional Bibliography and Comments” should provide many suggestions for conducting seminars. Although the emphasis is mathematical, it is one of the author’s wishes that some parts of the book will be of some value to engineers, whose familiar objects are perhaps seen from a different viewpoint. Indeed, in the selection of topics, we have been careful in considering only actual problems and we have likewise restricted ourselves to finite element methods which are actually used in contemporary engineering applications. The prerequisites consist essentially in a good knowledge of Analysis and Functional Analysis, notably: Hilbert spaces, Sobolev spaces, and Differential Calculus in normed vector spaces. Apart from these preliminaries and some results on elliptic boundary value problems (regularity properties of the solutions, for example), the book is mathematically self-contained. The main topics covered are the following: Description and mathematical analysis of linear second- and fourthvii

...

VLLL

PREFACE

order boundary value problems which are typically found in elasticity theory: System of equations of two-dimensional and three-dimensional elasticity, problems in the theory of membranes, thin plates, arches, thin shells (Chapters 1 and 8). Description and mathematical analysis of some nonlinear secondorder boundary value problems, such as the obstacle problem (and more generally problems modeled by variational inequalities), the minimal surface problem, problems of monotone type (Chapter 5). Description of conforming finite element methods for solving secondorder or fourth-order problems (Chapter 2). Analysis of the convergence properties of such methods for secondorder problems, including the uniform convergence (Chapter 3), and fourth-order problems (Section 6.1). Description and convergence analysis of finite element methods with numerical integration (Section 4.1). Description and convergence analysis of nonconforming finite element methods for second-order problems (Section 4.2) and fourth-order problems (Section 6.2). Description and interpolation theory for isoparametric finite elements (Section 4.3). Description and convergence analysis of the combined use of isoparametric finite elements and numerical integration for solving secondorder problems over domains with curved boundaries (Section 4.4). Convergence analysis of finite element approximations of some nonlinear problems (Chapter 5 ) . Description and convergence analysis of a mixed finite element method for solving the biharmonic problem, with an emphasis on duality theory, especially as regards the solution of the associated discrete problem (Chapter 7). Description and convergence analysis of finite element methods for arches and shells, including an analysis of the approximation of the geometry by curved and flat elements (Chapter 8). For more detailed information, the reader should consult the Introductions of the Chapters. It is also appropriate to comment on some of the omitted topics. As suggested by the title, we have restricted ourselves to elliptic problems, and this restriction is obviously responsible for the omission of finite element methods for time-dependent problems, a subject which would require another volume. In fact, for such problems, the content of this

PREFACE

ix

book should amply suffice for those aspects of the theory which are directly related to the finite element method. The additional analysis, due to the change in the nature of the partial differential equation. requires functional analytic tools of a different nature. The main omissions within the realm of elliptic boundary value problems concern the so-called hybrid and equilibrium finite element methods, and also mixed methods other than that described in Chapter 7. There are basically two reasons behind these omissions: First. the basic theory for such methods was not yet in a final form by the time this book was completed. Secondly, these methods form such wide and expanding a topic that their inclusion would have required several additional chapters. Other notable omissions are finite element methods for approximating the solution of particular problems. such as problems on unbounded domains. Stokes and Navier-Stokes problems and eigenvalue problems. Nevertheless, introductions to, and references for, the topics mentioned in the above paragraph are given in the sections titled “Additional Bibliography and Comments”. As a rule, all topics which would have required further analytic tools (such as nonintegral Sobolev spaces for instance) have been deliberately omitted. Many results are left as exercises, which is not to say that they should be systematically considered less important than those proved in the text (their inclusion in the text would have meant a much longer book). The book comprises eight chapters. Chapter n , 1 =sn 8. contains an introduction, several sections numbered Section n. I . Section n.2. etc. . . , and a section “Bibliography and Comments”, sometimes followed by a section “Additional Bibliography and Comments“. Theorems, remarks, formulas, figures, and exercises, found in each section are numbered with a three-number system. Thus the second theorem of Section 3.2 is “Theorem 3.3.3”, the fourth remark in Section 4.4 is “Remark 4.4.4”, the twelfth formula of Section 8.3 is numbered (8.3.12) etc. . . . The end of a theorem or of a remark is indicated by the symbol 0. Since the sections (which correspond to a logical subdivision of the text) may vary considerably in length, unnumbered subtitles have been added in each section to help the reader (they appear in the table of contents). The theorems are intended to represent important results. Their number have been kept to a minimum, and there are no lemmas, propositions, or corollaries. This is why the proofs of the theorems are

X

PREFACE

sometimes fairly long. In principle, one can skip the remarks during a first reading. When a term is defined, it is set in italics. Terms which are only given a loose or intuitive meaning are put between quotation marks. There are very few references in the body of the text. All relevant bibliographical material is instead indicated in the sections “Bibliography and Comments” and “Additional Bibliography and Comments”. Underlying the writing of this book, there has been a deliberate attempt to put an emphasis on pedagogy. In particular: All pertinent prerequisite material is clearly delineated and kept to a minimum. It is introduced only when needed. Compfeteproofs are generally given. However, some technical results or proofs which resemble previous proofs are occasionally left to the reader. The chapters are written in such a way that it should not prove too hard for a reader already reasonably familiar with the finite element method to read a given chapter almost independently of the previous chapters. Of course, this is at the expense of some redundancies, which are purposefully included. For the same reason, the index, the glossary of symbols and the interdependence table should be useful. It is in particular with an eye towards classroom use and self-study that exercises of varying difficulty are included at the end of the sections. Some exercises are easy and are simply intended to help the reader in getting a better understanding of the text. More challenging problems (which are generally provided with hints and/or references) often concern significant extensions of the material of the text (they generally comprise several questions, numbered (i), (ii), . . .). In most sections, a significant amount of material (generally at the beginning) is devoted to the introductive and descriptive aspects of the topic under consideration. Many figures are included, which hopefully will help the reader. Indeed, it is the author’s opinion that one of the most fascinating aspects of the finite element method is that it entails a rehabilitation of old-fashioned “classical” geometry (considered as completely obsolete, it has almost disappeared in the curriculae of French secondary schools). There was no systematic attempt to compile an exhaustive bibliography. In particular, most references before 1970 and/or from the engineering literature and/or from Eastern Europe are not quoted. The interested reader is referred to the bibliography of Whiteman (1975). An

PREFACE

XI

effort was made, however, to include the most recent references (published or unpublished) of which the author was aware. as of October, 1976. In attributing proper names to some finite elements and theorems, we have generally simply followed the common usages in French universities, and we hope that these choices will not stir up controversies. Our purpose was not to take issues but rather to give due credit to some of those who are clearly responsible for the invention, or the mathematical justification of, some aspects of the finite element method. For providing a very stimulating and challenging scientific atmosphere, I wish to thank all m y colleagues of the Laboratoire d’Analyse NumCrique at the UniversitC Pierre et Marie Curie, particularly Pierre-Arnaud Raviart and Roland Glowinski. Above all, it is my pleasure to express my very deep gratitude to Jacques-Louis Lions, who is responsible for the creation of this atmosphere. and to whom I personally owe so much. For their respective invitations to Bangalore and Montrtal, I express my sincere gratitude to Professor K.G. Ramanathan and to Professor A. Daigneault. Indeed, this book is an outgrowth of Lectures which I was privileged to give in Bangalore as part of t h e “Applied Mathematics Programme” of the Tata Institute of Fundamental Research, Bombay, and at the University of Montreal, as part of the “SCminaire de MathCmatiques Superieures”. For various improvements, such as shorter proofs and better exposition at various places, I am especially indebted to J. Tinsley Oden, Vidar ThomCe, Annie Puech-Raoult and Michel Bernadou, who have been kind enough to entirely read the manuscript. For kindly providing me with computer graphics and drawings of actual triangulations, I am indebted to Professors J. H. Argyris, C. Felippa, R. Glowinski and 0. C. Zienkiewicz, and to the Publishers who authorized the reprinting of these figures. For their understanding and kind assistance as regards the material realization of this book, sincere thanks are due to Mrs. Damperat, Mrs. Theis and Mr. Riguet. For their expert, diligent, and especially fast, typing of the entire manuscript, I very sincerely thank Mrs. Bugler and Mrs. Guille. For a considerable help in proofreading and in the general elaboration of the manuscript, and for a permanent comprehension in spite of a finite, but

xii

PREFACE

large, number of lost week-ends and holidays, I deeply thank the one to whom this book is dedicated. The author welcomes in advance all comments, suggestions, criticisms, etc. December 1976

Philippe G . Ciarlet

TABLE OF CONTENTS

PREFACE

. . . . . . . . .

GENERALPLAN

.

.

.

.

. . . . . . .

. . . .

vii

. .

.

. . . . .

.

xviii

. . . .

.

. . . . . . .

1

A N D INTERDEPENDENCE TABLE

1. ELLIPTICBOUNDARY V A L U E 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 H"(R). Green's formulas . . . .

.

. . . . . . . . . . . . . . . . . . . . . First examples of second-order boundary value problems . . . The elasticity problem . . . . . . . . . . . . . . . .

Introduction

1.1. Abstract problems

Examples of problem . Exercises Bibliography

. . . . . .

.

. .

i

2 2 7 9 10 10 15

23

fourth-order problems: The biharmonic problem, the plate

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Comments . . . . . . . . . . . . . .

28 32 35

. . . . . . . . .

36

Introduction . . . . . . . . . . . . . . . . . . . . . 2.1. Basic aspects of the finite element method . . . . . . . . . . The Galerkirl 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 a s degrees of freedom: Hermite n-simplices of type (3). (3'). Assembly in triangulations . . . . . . . . . . . . . . . . . . . . First examples of finite elements for fourth-order problems: the

36 31 31

2. INTRODUCTION

TO T H E F I N I T E ELEMENT METHOD

xiii

38 43 43 43

44 51 55

64

xiv

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. 2). Basic definitions . The P-inter. . . . . . . . . . . . . . . . . . polation operator AfFine families of finite elements . . . . . . . . . . . . . Construction of finite element spaces xh . Basic definitions . The Xhinterpolation operator . . . . . . . . . . . . . . . . . Finite elements of class Vo and V' . . . . . . . . . . . . Taking into account boundary conditions. The spaces xoh and x m h . Final comments . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . 2.4. General considerations on convergence . . . . . . . . . . . Convergent family of discrete problems . . . . . . . . . . Cia's lemma . First consequences . Orders of convergence . . . . Bibliography and comments . . . . . . . . . . . . . . .

3 . CONFORMINGFINITEELEMENT METHODS

FOR SECOND ORDER PROBLEMS

Introduction . . . . . . . . . . . . . . . . . . . . . 3.1. Interpolation theory in Sobolev spaces . . . . . . . . . . . . The Sobolev spaces Wmn.p(fl). The quotient space Wk".P(R)/Pk(fl) . Error estimates for polynomial preserving operators . . . . . . Estimates of the interpolation errors Iu - n K u ) m . q . K for affine families of finite elements . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . 3.2. Application to second-order problems over polygonal domains . . . Estimate of the error IIu - uhlll.0 . . . . . . . . . . . . . Sufficient conditions for limh+& - uhlll. = 0 . . . . . . . . . Estimate of the error Iu - u ~ (The ~ Aubin-Nitsche . ~ lemma . . . . Concluding remarks . Inverse inequalities . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . 3.3. Uniform convergence . . . . . . . . . . . . . . . . . A model problem . Weighted semi-norms I.l+:m.R . . . . . . . Uniform boundedness of the mapping u + uh with respect to appropriate weighted norms . . . . . . . . . . . . . . . Estimates of the errors Iu - uhlOpo.and Iu - uhl,... Nitsche's method of weighted norms . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . . . . 4 . OTHERFINITE ELEMENT

. . . . . . . . . . . . .

METHODS FOR SECOND-ORDER PROBLEMS

Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.1. The effect of numerical integration Taking into account numerical integration . Description discrete problem . . . . . . . . . . . . Abstract error estimate: The first Strang !emma . .

69 77 78 78 82 88 95

96 99 101

103 103 104 106 110

110

112 I12 116 122 126 131 131 134 136 139 143 147 147 155

163 167 168 I74 174 178

of the resulting

. . . . . . . . . . . .

178 185

xv

CONTENTS

. . . . . . . . Sufficient conditions for uniform V,. ellipticity Consistency error estimates . The Bramble-Hilbert lemma . . . . Estimate of the error J1u- uhll,. . . . . . . . . . . . . . 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 (PKESAu- u ~ I : . ~ ) ’ ’ * . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Isoparametric finite elements lsoparametric families of finite elements . . . . . . . . . . Examples of isoparametric finite elements . . . . . . . . . . Estimates of the interpolation errors Iu - 17Kulm.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,. ellipticity . . . . . . . . Interpolation error and consistency error estimates . . . . . . Estimate of the error Ili - &,l(l.nh . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . . . . . Additional bibliography and comments . . . . . . . . . . . . Problems on unbounded domains . . . . . . . . . . . . . The Stokes problem . . . . . . . . . . . . . . . . . Eigenvalue problems . . . . . . . . . . . . . . . . .

187 190

199 201 207 207 209 21 I 217 220 223 224 224 227 230 243 248 248 252 255 257 260 266 270 272 276 276 280 283

5 . APPLICATIONOF THE FINITE E L E M E N T METHOD TO SOME NONLINEAR

. . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . The obstacle problem . . . . . . . . . . . . Variational formulation of the obstacle problem . .

PROBLEMS

5.1.

. . . . . . . . . . . . . . .

An abstract error estimate for variational inequalities . . . . . . Finite element approximation with triangles of type (1). Estimate of . . the error (Iu - u,,ll,. . . . . . . . . . . . . . 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 11u - uhlll.nr . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . 5.3. Nonlinear problems of monotone type . . . . . . . . . . . .

287 287 289 289 291 294 297 301 301 302 310 312

xvi

CONTENTS

. .

A minimization problem over the space Wlj.(f2) 2 r p and its finite element approximation with n-simplices of type ( I ) . . . . . . Sufficient condition for lim, $lu - ~ ~ l = l~ 0 . ~ .. . . . . . . . The equivalent problem Au = f. Two properties of the operator A . Strongly monotone operators . Abstract error estimate . . . . . Estimate of the error I(u - Uhl1l.p.n . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . . . . Additional bibliography and comments . . . . . . . . . . . . Other nonlinear problems . . . . . . . . . . . . . . . The Navier-Stokes problem . . . . . . . . . . . . . . 6 . FINITEELEMENTMETHODSFORTHEPLATEPROBLEM . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . 6.1. Conforming methods . . . . . . . . . . . . . . . . . . Conforming methods for fourth-order problems . . . . . . . . Almost-affine families of finite elements . . . . . . . . . . A “polynomial” finite element of class V’:The Argyris triangle . . A composite finite element of class V1:The Hsieh-Clough-Tocher triangle . . . . . . . . . . . . . . . . . . . . . . A singular finite element of class V’:The singular Zienkiewicz triangle Estimate of the error IIu - ~ ~ l .l ~ . . ~ . . . . . . . . . . Sufficient conditions for limhJlu - uh1I2. = 0 . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . 6.2. Nonconforming methods . . . . . . . . . . . . . . . . Nonconforming methods for the plate problem . . . . . . . . An example of a nonconforming finite element: Adini’s rectangle . Consistency error estimate . Estimate of the error GKe .rJ~ ~ ~ 1 . K. ) 1 ‘ 2 Further results . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . . . . 7. A

MIXED FINITE ELEMENT METHOD

. . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . 7.1. A mixed finite element method for the biharmonic problem . . . . Another variational formulation of the biharmonic problem . . . . The corresponding discrete problem . Abstract error estimate . . . Estimate of the error (lu - uhll. R + IAu + &,lo .n) . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . 7.2. Solution of the discrete problem by duality techniques . . . . . . Replacement of the constrained minimization problem by a saddle. . . . . . . . . . . . . . . . . . . point problem Use of Uzawa’s method . Reduction to a seauence of discrete Dirichlet . . . . . . . . . . . . . . problems for the operator - A

312 317 318 321 324 324 325 330 330 331 333 333 334 334 335 336 340 347 352 354 354 356 362 362 364 367 373 374 376 381 381 383 383 386 390 391 392 395 395 399

xvii

CONTENTS

Convergence of Uzawa’s method . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . Additional bibliography and comments . . . . . . . . . Primal dual and primal-dual formulations . . . . . . . Displacement and equilibrium methods . . . . . . . . Mixed methods . . . . . . . . . . . . . . . . Hybrid methods . . . . . . . . . . . . . . . . An attempt of general classification of finite element methods E L E M E N T METHODS FOR S H E L L S

. . . . . . . . . .

. . . . . . . . .

402 403 404 406 407 407 412 414 417 42 1

. . . . . . . . . . . .

425

.

8. FINITE

. . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . 8.1. The shell problem . . . . . . . . . . . . . . . . . . . Geometrical preliminaries . Koiter’s model . . . . . . . . . . . . . . . . Existence of a solution . Proof for the arch problem Exercises . . . . . . . . . . . . . . . . . . . . . 8.2. Conforming methods . . . . . . . . . . . . . . . . . . The discrete problem . Approximation of the geometry . Approximation of the displacement . . . . . . . . . . . . . . . . . . . . . Finite element methods conforming for the displacements Consistency error estimates . . . . . . . . . . . . . . . Abstract error estimate . . . . . . . . . . . . . . . . . . . . . Estimate of the error (Xi=, IJu, - U.hII?.fj+II& - U3hlli.fl)”’ Finite element methods conforming for the geometry . . . . . . . . . . . . . . Conforming finite element methods for shells 8.3. A nonconforming method for the arch problem . . . . . . . . . The circular arch problem . . . . . . . . . . . . . . . A natural finite element approximation . . . . . . . . . . . Finite element methods conforming for the geometry . . . . . . A finite element method which is not conforming for the geometry . Definition of the discrete problem . . . . . . . . . . . . Consistency error estimates . . . . . . . . . . . . . . . Estimate of the error (lu, - U l h l : . j + Iu2- ~~~l:.,)”’. . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . Bibliography and comments . . . . . . . . . . . . . . .

425 426 426 431 437 439

. . . . . . .

469

EPILOGUE: Some “real-life” finite element model examples BIBLIOGRAPHY

. . . . . . . . . . . . . .

GLOSSARYOF SYMBOLS

INDEX

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

439 440

443 447 448 450 450 451 451 452 453

453 461 465 466 466

481

512 52 I

GENERALPLAN ANDINTERDEPENDENCETABLE

r--

FUNDAMENTALS

t o the f.e.m.

Confonmng

1

II

sect. 4 . 1 Numrical lntriration

II

Sect. 4 . 3 . Isoparametric

-\7f.e.

'\

/'

Problems 4th-order p r .

BASIC THEORY

I I

MORL SPECIALIZED TOPICS

“A mathematician’s nightmare is a sequence n, that tends to 0 as becomes infinite.”

E

Paul R. HALMOS: How to Write Mathematics, A.M.S., 1973.

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CHAPTER

1

ELLIPTIC BOUNDARY VALUE PROBLEMS

Introduction Many problems in elasticity are mathematically represented b y the following minimization problem: T h e unknown u, which is the displacement of a mechanical system, satisfies

uE U

and

J ( u )= hf,J(v),

where the set U of admissible displacements is a closed convex subset of a Hilbert space V, and the energy J of the system takes the form J ( u )=ia(u, u>-f(u),

where a ( . . is a symmetric bilinear form and f is a linear form, both defined and continuous over the space V. In Section 1.1, we first prove a general existence result (Theorem 1.1.1), the main assumptions being the completeness of the space V and the V-ellipticity of the bilinear form. We also describe other formulations of the same problem (Theorem l.l.2), known as its uariational formulations, which, in the absence of the assumption of symmetry for the bilinear form, make up uariational problems on their own. For such problems, we give an existence theorem when U = V (Theorem l.l.3), which is the well-known L a x 0

)

Milgram lemma.

All these problems are called abstract problems inasmuch as they represent a n “abstract” formulation which is common to many examples, such as those which are examined in Section 1.2. From the analysis made in Section 1.1, a candidate for the space V must have the following properties: It must be complete on the one hand, and it must be such that the expression J ( u ) is well-defined for all functions u E V on the other hand ( V is a “space of finite energy”). The Soboleu spaces fulfill these requirements. After briefly mentioning some of their properties (other properties will be introduced in later sections, I

2

ELLIPTIC BOUNDARY VALUE PROBLEMS

[Ch. 1, 0 1.1.

as needed), we examine in Section 1.2 specific examples of the abstract problems of Section 1.1, such as the membrane problem, the clamped plate problem, and the system of equations of linear elasticity, which is by far the most significant example. Indeed, even though throughout this book we will often find it convenient to work with the simpler looking problems described at the beginning of Section 1.2, it must not be forgotten that these are essentially convenient model problems for the system of linear elasticity. Using various Green’s formulas in Sobolev spaces, we show that when solving these problems, one solves, at least formally, elliptic boundary value problems of the second and fourth order posed in the classical way.

1.1. Abstract problems

The symmetric case. Variational inequalities All functions and vector spaces considered in this book are real. Let there be given a normed vector space V with norm 11.11, a continuous bilinear form a ( - , : V x V + R , a continuous linear form f: V + R and a non empty subset U of the space V. With these data we associate an abstract minimization problem: Find an element u such that a)

uEU

and J(u) = hiJ ( u ) ,

(1.1.1)

where the functional J : V + R is defined by J : u E V + J ( u ) = :a(u, u ) - f ( u ) .

( 1.1.2)

As regards existence and uniqueness properties of the solution of this problem, the following result is essential. Theorem 1.1.1. Assume in addition that

(i) the space V is complete, (ii) U is a closed convex subset of V , (iii) the bilinear form a(., .) is symmetric and V-elliptic, in the sense that Sa > o , t l u E V , allul12sa(u, u ) .

(1.1.3)

Ch. 1. 0 1.1.1

3

ABSTRACT PROBLEMS

Then the abstract minimization problem (1.1.1) has one and only one solution. Proof. The bilinear form a(., .) is an inner product over the space V, and the associated norm is equivalent to the given norm 11-11. Thus the space V is a Hilbert space when it is equipped with this inner product. By the Riesz representation theorem, there exists an element af E V such that Vv E

v,

f ( v )= a ( d , 01,

so that, taking into account the symmetry of the bilinear form, we may rewrite the functional as

J ( u ) = ta(u, u) - a(af,u) = ta(u - af,u - a f )- f a ( a f ,af).

Hence solving the abstract minimization problem amounts to minimizing the distance between the element uf and the set U,with respect to the norm Consequently, the solution is simply the projection of the element uf onto the set U,with respect to the inner product a(., By the projection theorem, such a projection exists and is unique, since U is a closed convex subset of the space V. 0

w.

a).

Next, we give equivalent formulations of this problem.

Theorem 1.1.2. A n element u is the solution of the abstract minimization problem (1.1.1) if and only if it satisfies the relations uEU

and V u E U , a ( u , u - u ) a f ( u - u ) ,

(1.1.4)

in the general case, or (1.1.5)

if U is a closed conuex cone with vertex 0, or uE

U and Vu E U, a(u, u ) = f ( u ) ,

(1.1.6)

if U is a closed subspace. Proof. The projection u is completely characterized by the relations

uEU

and VuEU, a ( u f - u , u - u ) S O ,

(1.1.7)

ELLIPTIC BOUNDARY VALUE PROBLEMS

4

[Ch. 1, 0 1.1.

Fig. 1.1.1

the geometrical interpretation of the last inequalities bemg that the angle between the vectors (uf-u)and ( u - u ) is obtuse (Fig. 1.1.1) for all u E U.These inequalities may be written as vuE

u,

a(u, u - u ) 3 a(uf,u - u ) = f ( u - u ) ,

which proves relations (1.1.4). Assume next U is a closed convex cone with vertex'0. Then the point ( u + u ) belongs to the set U whenever the point u belongs to the set U (Fig. 1.1.2).

Fig. 1.1.2

Ch. 1. 0 1.1.1

ABSTRACT PROBLEMS

5

Therefore, upon replacing u by ( u + u ) in inequalities (1.1.4), we obtain the inequalities v u E U, a(u, u ) a f ( u ) ,

so that, in particular, a(u, u ) a f ( u ) . Letting u = 0 in (1.1.4), we obtain a(u, u ) Q f(u), and thus relations (1.1.5) are proved. The converse is

clear. If U is a subspace (Fig. 1.1.3), then inequalities (1.1.5) written with u and - u yield a(u, u ) B f ( u ) and a(u, u ) C f ( u ) for all u E U,from which relations (1.1.6) follow. Again the converse is clear.

-V

Fig. 1.1.3

The characterizations (1.1.4), (1.1.5) and (1.1.6) are called variational formulations of the original minimization problem, the equations (1.1.6) are called variational equations, and the inequalities of (1.1.4) and (1.1.5) are referred to as uariational inequalities. The terminology “variational” will be justified in Remark 1.1.2.

Remark 1.1.1. Since the projection mapping is linear if and only if the subset U is a subspace, it follows that problems associated with uariational inequalities are generally non linear, the linearity or non linearity being that of the mapping f E V’+ u E V,where V’is the dual space of V,all other data being fixed. One should not forget, however, that if the resulting problem is linear when one minimizes over a subspace this is also because the functional is quadratic i.e., it is of the form (1.1.2). The

6

ELLIPTIC BOUNDARY VALUE PROBLEMS

[Ch. 1, 5 1.1.

minimization of more general functionals over a subspace would correspond to nonlinear problems (cf. Section 5.3). 0 Remark 1.1.2. The variational formulations o f Theorem 1.1.2 may be also interpreted from the point of view o f Differential Calculus, as follows. We first observe that the functional J is differentiable at every point u E V, its (Frkchet) derivative J ’ ( u ) E V‘ being such that V u E V , J’(u)u = a(u, v ) - f ( u ) .

(1.1.8)

Let then u be the solution of the minimization problem ( 1 . 1 . l), and let v = u + w be any point of the convex set U. Since the points ( u Ow)

+

belong to the set U for all 8 E [0, 11 (Fig. 1.1.4), we have, by definition of

Fig. 1.1.4

the derivative J ’ ( u ) ,

o s J ( U + e w ) - J ( U ) = eJ’(u)w + ellwlle(e) for all 8 E [0,1], with limo-.0r(O)= 0. As a consequence, we necessarily have J’(u)w 3 0 ,

( 1.1.9)

since otherwise the difference J(u + Ow) - J(u) would be strictly negative for 8 small enough. Using (1.1.8), inequality (1.1.9) may be rewritten as

J’(u)w = J‘(u)(v - u ) = a ( u , u - u ) - f ( u - u ) 3 0, which is precisely (1.1.4). Conversely, assume we have found an element u E U such that vuE

u,

J’(u)(u- u ) 3 0.

( 1.1.10)

Ch. 1, 8 1.1.1

7

ABSTRACT PROBLEMS

The second derivative J”(u)€L&(V;R) of the functional J is independent of u € V and it is given by tlu,w €

v,

J”(u)(v,w ) = a(v, w).

(1.1.11)

Therefore, an application of Taylor’s formula for any point u = u + w belonging to the set U yields 1 2

J(u + w )- J ( u ) = J‘(u)(w )+ - a ( w,w )3 Ellw((2, 2

(1.1.12)

which shows that u is a solution of problem (1.1.1). We have J ( u ) - J ( u ) > 0 unless u = u so that we see once again the solution is unique. Arguing as in the proof of Theorem 1.1.2, it is an easy matter to verify that inequalities (1.1.10) are equivalent to the relations t l v E U, J’(u)v 3 0 and

J ’ ( u ) u = 0,

(1.1.13)

when U is a convex cone with vertex 0, alternately, vvE

u,

J’(u)v = 0,

( 1.1.14)

when U is a subspace. Notice that relations (1.1.13) coincide with relations (1.1.5), while (1.1.14) coincide with (1.1.6). When U = V, relations (1.1.14) reduce to the familiar condition that the first variation of the functional J, i.e., the first order term J f ( u ) w in the Taylor expansion (1.1.12), vanishes for all w € V when the point u is a minimum of the function J : V + R , this condition being also sufficient if the function J is convex, as is the case here. Therefore the various relations (1.1.4), (1.1.5) and (1.1.6), through the equivalent relations (l.l.lO), (1.1.13) and (1.1.14), appear as generalizations of the previous condition, the expression a(u, u - u ) - f ( v - u ) = J’(u)(v- u ) , v E U , playing in the present situation the role of the first variation of the functional J relative to the conuex set U. It is in this sense that the 0 formulations of Theorem 1.1.2 are called “variational”.

The nonsymmetric case. The Lax-Milgram lemma Without making explicit reference to the functional J, we now define an abstract variational problem : Find an element u such that uEU

and Vv E U, a(u, u - u ) P f ( u - u ) ,

(1.1.15)

8

ELLIPTIC BOUNDARY V A L U E PROBLEMS

[Ch. 1, 8 1.1.

or, find an element u such that ( 1.1.16)

if U is a cone with vertex 0, or, finally, find an element u such that

uEU

and Vu E V,

a(u, u ) = f ( u ) ,

(1.1.17)

if U is a subspace. By Theorem 1.1.1, each such problem has one and only one solution if the space V is complete, the subset U of V is closed and convex, and the bilinear form is V-elliptic, continuous, and symmetric. If the assumption of symmetry of the bilinear form is dropped, the above variational problem still has one and only one solution (LIONS & STAMPACCHIA (1967)) if the space V is a Hilbert space, but there is no longer an associated minimization problem. Here we shall confine ourselves to the case where U = V.

Theorem 1.1.3 (Lax-Milgram lemma). Let V be a Hilbert space, let a(., .): V X V + R be a continuous V-elliptic bilinear form, and let f : V + R be a continuous linear form. Then the abstract variational problem: Find an element u such that uEV

and

V u E V,

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

( 1.1.18)

has one and only one solution. Proof. Let M be a constant such that VU,E ~ V,

u)l C Mllull (lull.

( 1.1.19)

For each u E V, the linear form u E V + a(u, u ) is continuous and thus there exists a unique element A u E V’ (V’ is the dual space of V) such that Vv E V, a(u, u ) = Au(u).

Denoting by

11.11*

( 1.1.20)

the norm in the space V’, we have

Consequently, the linear mapping A: V + V’ is continuous, with IlAIlrecv;v’) =sM.

Let

T:

(1.1.21)

V’+ V denote the Riesz mapping which is such that, by

Ch. 1, 5 1.1.1

ABSTRACT PROBLEMS

9

definition,

vf E v',

vu

v,

f ( u ) = ((Tf, u ) ) ,

( I . 1.22)

denoting the inner product in the space V. Then solving the variational problem (1.1.18) is equivalent to solving the equation TAU= ~ fWe . will show that this equation has one and only one solution by showing that, for appropriate values of a parameter p > O , the affine mapping

));s((

uE V+u-p(~Av-~f)€V

(1.1.23)

is a contraction. To see this, we observe that IIv

- p ~ A u 1 1=~llulr - 2p((~Av,v ) ) + p'11~Au11~ Q

( I - 2pff

+ pZMZ)llu(lZ,

since, using inequalities (1.1.3) and ( I . 1.21), ((TAU,0))

= Au(v) = a(v, u ) z-aIlu11*,

11TA4l = IIA4* s IlAll ll4l s

Mll4

Therefore the mapping defined in (1.1.23) is a contraction whenever the number p belongs to the interval ]0,2a/M2[ and the proof is complete.

Remark 1.1.3. It follows from the previous proof that the mapping A : V + V' is onto. Since

ffll~I12= a ( u , u ) = f(u) c IVII*IIUII. the mapping A has a continuous inverse A-I, 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). 0 Exercises 1.1.1. Show that if ui,i = 1,2, are the solutions of minimization problems ( I . 1. I ) corresponding to linear form f i E V', i = I , 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). 2.2.2. 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 1 = T . A: V + V is continuous with 11~411d M,and that aIJu(IG llS4ull for all u E V. It remains to show that

1(V )=

v.

(i) Show that 1 ( V ) is a closed subspace of V. (ii) Show that the orthogonal complement of 1 ( V )in the space V is reduced to (0). 1.2. Examples of elliptic boundary value problems

The Sobolev spaces H " ( R ) . 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 u : A + Y,where A is a subset of X. If the function is k times differentiable at a point a E A, we shall denote D k u ( a ) ,or simply Du(a) if k = 1, its k-th (Fre'chet) derivative. It is a symmetric element of the space 2k(x;Y),whose norm is given by

We shall also use the alternate notations Du(a) = u'(a) and D 2 u (a )= u"(a). In the special case where X = R" and Y = R, let ei, 1 d i S n, denote the canonical basis vectors of R".Then the usual partial derivatives will be denoted by, and are given by, the following:

aiu(a)= Du(a)ei, aiju(a)= D2u(a)(ei,ei), aijku(a)= D3v(a)(ei,ei, ek),etc. . . Occasionally, we shall use the notation V u ( a ) , or grad u ( a ) ,to denote the gradient of the function u at the point a, i.e., the vector in R" whose components are the partial derivatives aiu(a), 1 s i d n.

Ch. I ,

B

1.2.1

11

EXAMPLES

We shall also use the multi-index notation: Given a multi-index

a = ( a 1a , 2 , .. . , a,,) E N " , we let la'(= X;='=, ai. Then the partial derivative aau(a) is the result of the application of the IaI-th derivative D%(a) to

any [at-vector of (R")lalwhere each vector ei occurs ai times, 1 S i G n. For instance, if n = 3, we have a l u ( a )= a(l*O.o)u(a), aluu(a) = a".'*"u(a), alllu(a)= a ( 3 * 0 * 0 ) ~ etc. ( a ).,. There exist constants C ( m , n ) such that for any partial derivative aau(a) with la1 = m and any function u, la"v(a)l

IID"'v(a)ll

C(m,n)

1s:

la"u(a)l,

where it is understood that the space R" is equipped with the Euclidean norm. As a rule, we shall represent by symbols such as D k u , u", a i U , 8% etc.. . , the functions associated with any derivative or partial derivative. When h l = hz = = hk = h, we shall simply write

- -

D ' ~ ( a ) ( h ih2,. , . . ,hk) = D k v ( a ) h k .

Thus, given a real-valued function u, Taylor's formula of order k is written as k 1 u(a + h ) = u ( a ) + 2 7 D'u(a)h' + -~ k + l u ( + a eh)hk+', /=I I . (k + I)! for some 8 E 10, I[ (whenever such a formula applies). Given a bounded open subset J2 in R",the space 9(0)consists of all indefinitely differentiable functions u : J2 +R with compact support. For each integer m a0,the Soboleu space H"'(f2) consists of those functions u E L'(J2) for which all partial derivatives a% (in the distribution sense), with ( a l b m , belong to the space L2(J2),i.e., for each multi-index a with la1 b m, there exists a function a"u E L2(J2)which satisfies

v#, E g(J2),

I

R

8%

4 dx = (-

1)la1

(1.2.1)

Equipped with the norm

the space H"'(J2) is a Hilbert space. We shall also make frequent use of the semi-norm

12

ELLIPTIC BOUNDARY V A L U E PROBLEMS

[Ch. I , 8 1.2.

We define the Sobolev space = (9(0))-,

the closure being understood in the sense of the norm II.llm,a. When the set 0 is bounded, there exists a constant C ( 0 ) such that

vv E Hdcn),

( u l 0 . n C(0)lvIl.n. ~

(1.2.2)

this inequality being known as the Poincark-Friedrichs inequality. Therefore, when the set 0 is bounded, the semi-norm I-Irnn is a norm over the space Ho"(R), equivalent to the norm lI.llm.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 N E ~ A S(1%7), we say that an open set R has a Lipschitzcontinuous boundary r i f the following conditions are fulfilled: There exist constants a > 0 and i3 > 0, and a finite number of local coordinate systems and local maps a,, 1 c r c R, which are Lipschitz-continuous on their respective domains of definitions {arE R"-';1.3.1 d a}, such that (Fig. 1.2.1): R

r = U {(x:, a r ) ;x:= a r ( f r ) , lirl< a}, < a}C 0, 1 6 r c R , {(x:, a'); ar(ar)< x i < a r ( i r )+ B; a,(ir) - p < x l < a,(ar); lirl 0 a.e. on R to get existence).

r r,,

The elasticity problem We now come to the fourth example which is by far the most significant. Let R be a bounded open connected subset of R’ with a Lipschitzcontinuous boundary. We define the space

v = u = {u = (UI.

v2. u3) E (H1(R))’;

vi = 0 on

r,,

1sis3},

(1.2.30)

r,

where To is a dy-measurable subset of with a strictly positive dy-measure. The space V is equipped with the product norm

For any u

= ( v l , v2, v3) E (H’(R))3, we

let

eii(u)= e i i ( u )= :(i+q + aivi), 1 s i, j s 3,

(1.2.31)

and

where aii is the Kronecker’s symbol, and A and p are two constants which are assumed to satisfy A > 0, p > 0. We define the bilinear form a(u,u ) =

=

I c’ fl i.j=l

In{*

aii(u)cii(u)dx

c 3

div u div u + 2 p

i.i = I

q(u)q(u)

24

ELLIPTIC BOUNDARY V A L U E PROBLEMS

[Ch. I , 8 1.2.

and the linear form

where f = ( f ~fi, , f3) E (L2(n)Pand g = (gl,gz, gd E (L2(rl))3,with r~= r - To are given functions. It is clear that these bilinear and linear forms are continuous over the space V. T o prove the V-ellipticity of the bilinear form (see Exercise 1.2.4), one needs Korn's inequality: There exists a constant C ( 0 ) such that, for all u = ( u I , u2, u3)E (H1(f2))3, (1.2.35)

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 = (ulr u2, u3)+

1.

3

=

( 2 lEij(u)li.n)

1/2

i,j= 1

is a norm, equivalent t o the product norm, as long as the d y-measure of Tois 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 > O , p > 0, since by (1.2.33) a ( u , u ) z 2p(uJ2.

We conclude that there exists a unique function u E V which minimizes the functional

over the space V, or equivalently, which is such that

Ch. 1, 0 1.2.1

25

EXAMPLES

Since relations (1.2.37) are satisfied by all functions u E (9(f2))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 (H2(f2))3and all 5 ) E (H’(R))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 u E (H2(f2))3and u E (H1(f2))3. Arguing as in the previous examples, we find that we are formally solving the equations 3

-CaJcjj(u)=fi, j=1

(1.2.39)

1sia3.

It is customary to write these equations in vector form: - p Au - (A

+ p ) grad div u = f

in

0,

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 u = o on To= r - r,. To sum up, we have formally solved the following associated boundary value problem :

I

- p Au - (A u =O

on

+ p ) grad div u = f

ro,

in

0, (1.2.40)

26

ELLIPTIC BOUNDARY V A L U E PROBLEMS

ICh. I , B 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- o r 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 d in the absence of forces, u denoting the displacement of the points of d under the influence of given forces (as usual, the scale for the displacements is distorted in the figure). The body d cannot move along r,,,and along r,,surface forces of density g are given. In addition, a volumic force, of density f, is prescribed inside the body 6. Then we recognize in ( e i j ( u ) ) the strain tensor while (uii(u)) 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 p are the Lamt coeficients of the material of which the body is composed.

Ch. 1, 5 1.2.1

EXAMPLES

21

The variational equations (1.2.37) represent the principle of virtual work, valid for all kinematically admissible displacements u, i.e., which satisfy the boundary condition u = 0 on To. The functional J 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: - ( J n f * u dx + * 0 dr). 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 f o r the invention of the finite element method by engineers.

Ir, g

Remark 1.2.2. It is interesting to notice that the strict positiveness of the dy-measure of r, has a physical interpretation: It is intuitively clear that in case the dy-measure rowould vanish, the body would be free and therefore there could not exist an equilibrium position in general. 0 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. 0 Remark 1.2.4. Since problem (1.2.40) is called system of linear elasticity, the linearity being of course that of the mapping (1. 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 1 eii(v)= -(aiui 2

3

+ aiui + k=l C aivkaivk),

1 =z i, j c 3.

Actually, it suffices that for at least one pair ( i t j ) , the above expression be considered. This is the case for instance of the uon 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. 0

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 ’ ( O ) , we consider in the last examples Sobolev spaces which involve second-order derivatives. We begin with the following data:

I I

v = u = Hi(O), ,-

(1.2.41)

Au Au dx,

a(u, u ) = In

Since the mapping u + l A ~ ( is ~ .a ~norm over the space Hi(O), as we showed in (1.2.81, the bilinear form is H:(O)-elliptic. Thus there exists a unique function u E H&O) which minimizes the functional 1

IAv 1’

J : u +J ( v ) = 2

dx -

f v dx

(1.2.42)

over the space H i ( 0 ) or, equivalently, which satisfies the variational equations Vv E H&O),

In

Au Au dx = I n f v dx.

(1.2.43)

Ch. 1, 8 1.2.1

EXAMPLES

29

Using Green’s formula (1.2.7): ~ o A u A u d x = ~ o A ’ ~u d x - ~ ~ ~ ~ A u u d y + ~dy, ~ A u d , u we find that we have formally solved the following homogeneous Dirichlet problem f o r the biharmonic operator A’: A z u = f in R, u = a,u = 0 on

(1.2.44)

r.

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,

v = u = H;(n), a(u, u ) =

I,

{Au A u

+ (1 - u ) ( 2 ~ 1 2 u ~-l zdllua22u u

-az@iiv))

dx

(1.2.45)

{UAU A V + (1 - u ) ( ~ I I u ~ I+ I u&ZU&~V +2aizuaizU))dx,

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 = (Ee3/12(l- u2))fper unit area. The constants E = p(3A + 2 p ) / ( A + p ) and u = A/2(A + p ) are respectively the Young’s modulus and the Poisson’s coefficient of the plate, A and p being the LamC’s coefficients of the plate material. When f = 0 , the plate is in the plane of coordinates ( X Ix, d (Fig. 1.2.4). The condition u E H i ( R ) 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, 5 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 u satisfying the inequalities 0 < u < i, the bilinear form is H:(fl)-elliptic, since we have V u EH2(fl), a ( v , u ) = uIAu(i.n+(l - U ) I U I : . ~ .

Thus, there exists a unique function u E H:(fl) which minimizes the total potential energy of the plate: J ( ~ ) = -1~ ~ { l A v J ’ + 2- ~( 1) ( ( a ~ 2 u ) ~~- l l u ~ ~ 2 u ) } d x -dx, ~nfu 2 (1.2.46) over the space H;(fl) or, equivalently, which is solution of the variational equations {Au A V + ( 1 - ~ ) ( 2 d l 2 ~ 8 1 2~ ~I

=In

-C32zudllv)}dx

Using Green’s formulas (1.2.7) and (1.2.9):

f u dx.

I u ~ ~ Z U

(1.2.47)

Ch. 1, § 1.2.1

EXAMPLES

31

we find that we have again solved, at least formally, the homogeneous Dirichlet problem for the biharmonic operator A':

A 2 u = f in 0, u = a,u = 0 on I'.

(1.2.48)

Therefore, in spite of a different bilinear form, we eventually find the same problem as in the previous example. This is so because, in view of the second Green's formula which we used, the contribution of the integral (1

-

a ) { 2 d l ~ u a l-~ vd l , u d z z v- d z z u a l l v dx }

is zero when the functions v are in the space 9 ( R ) , and consequently in its closure H , ? ( 0 ) .Thus, the partial differential equation is still A 2 u = f in R. However different boundary conditions might result from another choice for the space V . See Exercise 1.2.7. To distinguish the two problems, we shall refer to a fourth-order problem corresponding to the functional of (1.2.42) as a biharmonic problem, while we shall refer to a fourth-order problem corresponding to the functional of (1.2.46) as a plate problem. In this section, we have examined various minimization or variational problems with each of which is associated a boundary value problem for which the partial differential operator is elliptic (incidentally, this correspondence is not one-to-one, as the last two examples show). This is why, by extension, these minimization or variational problems are themselves called elliptic boundary value problems. For the same reasons, such problems are said to be second-order problems, or fourthorder problems, when the associated partial differential equation is of order two or four, respectively. Finally, one should recall that even though the association between the two formulations may be formal, it is possible to prove, under appropriate smoothness assumptions on the data, that a solution of any of the variational problems considered here is also a solution in the classical sense of the associated boundary value problem.

Remark 1.2.5.

In this book, one could conceivably omit all reference to the associated classical boundary value problems, inasmuch as the finite element method is based only on the variational formulations. By

32

ELLIPTIC BOUNDARY VALUE PROBLEMS

[Ch. 1, 0 1.2.

contrast, finite difference methods are most often derived from the 0 classical formulations. Exercises 1.2.1. Prove Green's formula (1.2.9). The reader should keep in mind that the derivative a,u generally differs from the second derivative of the function u, considered as a function of the curvilinear abcissa along the boundary.) 1.2.2. Let the space V = U and the bilinear form be as in (1.2.10), and let the linear form be defined by

where the functions f and a satisfy assumptions (1.2.11) and uo is a given function in the space H ' ( 0 ) .Show that these data correspond to the formal solution of the nonhomogeneous Din'chlet problem for the operator u + - Au + au, i.e.,

-Au+au=f in 0, u = uo on I'. .Fdx

/ Fig. 1.2.5

Ch. 1, 8 1.2.1

33

EXAMPLES

Is it equivalent to minimizing the functional (1.2.12) over the subset

u = { u E H W ) ; ( u - uo) E Hd(R)} of the space V = H'(R)? With a = O and n = 2, this is another membrane problem. See Fig. 1.2.5, which is self-explanatory. 1.2.3. Find a variational problem which amounts to solving the nonhomogeneous Neumann problem for the operator -A, i.e., problem (1.2.22) when the function a vanishes identically, and when the equality J n f d x + J r g d y = O holds. [Hint: Use the fact that over the quotient space H ' ( R ) / P o ( R ) Po(R): , space of constant functions over R, the semi-norm I*II,n is a norm, equivalent to the quotient norm. See Theorem 3.1.1 for a proof.] 1.2.4. Let R be a connected open subset of R",with n = 2 or 3, and let r o b e a dy-measurable subset of its boundary r,assumed to be Lipschitzcontinuous. Let

v = {u

= ( u i ) E ( H ' ( R ) ) " ; ui =

o

on

ro, 1s i s n }

(i) Show that V is a closed subspace of the space ( H ' ( 0 ) ) " . (ii) Show that the mapping

is a norm over the space V, if the dy-measure of rois strictly positive. [Hint: Show that a function u E (H'(R))"which satisfies (ul = 0 is of the form u ( x ) = a x x + b for some constant vectors a and b, i.e., the displacement u is a rigid body motion. Such a result is proved for example in HLAVACEK& NECAS (1970, Lemma 11.1). See also Section 8.1 for analogous ideas.] (iii) Using Korn's inequality (1.2.35), show that the norm (-1 is equivalent to the norm Il.lll,n. [Hint: Argue as in Theorem 1.2.1.1 1.2.5. Let

v = H1(R),

a(u, u ) =

I {f:

0 i=l

diudiv

+ auu} dx,

f (U ) =

I,fu dx,

and let

U = { u EH'(R); u L O a.e. on r}. Show that U is a closed convex cone with vertex 0. Using charac-

34

[Ch. I , 5 1.2.

ELLIPTIC BOUNDARY V A L U E PROBLEMS

terizations ( 1 . I . 5 ) of Theorem 1.1.2 show that the associated variational problem amounts to formally solving the boundary value problem -Au+au=f

uso,

a,u

in

so,

R, ua,u

=0

on

r.

This type of nonlinear problem is a model problem for Signorini problems, i.e., problems in elasticity for which the boundary conditions are unilateral constraints such as the above ones. For extensive discussions of such problems, see DUVAUT & LIONS (l972), FICHERA (1972). 1.2.6. Extend the analysis made for the data (1.2.23) to the case where the bilinear form is given by

the functions ai being in the space L"(R).In particular, find sufficient conditions for the V-ellipticity of the bilinear form. 1.2.7. Let the bilinear form and the linear form be as in (1.2.45),and let

v = u = ~ ' (nH:,(n) 0) ={V

E ~'(0) u =;o

on

r}.

This is a mathematical model for a simply supported plate. Using the fact that u + ~ A U ( ~is, O again a norm over the space V, equivalent to the norm ~ ~ * ~ analyze ~ 2 , ~ , the associated variational problem. What is the associated boundary value problem? 1.2.8. Let

I

a(u,u ) =

[ f ( ~= )

10

Au A u dx,

f u dx - I r A d z , u dy, f E L'(R), A E L'(r).

Using as in Exercise 1.2.7 the fact that u + ( A U ( is ~ . a~ norm over the space V, equivalent to the norm ll*l12, R , analyze the associated variational problem. In particular, show that it can be decomposed into two second order problems. What is the associated boundary value problem? Does it share the same property?

Ch. I]

BIBLIOGRAPHY A N D COMMENTS

35

Bibliography and comments 1.1. The original reference of the Lax-Milgram lemma is LAX & MILGRAM(1954). Our proof follows the method of LIONS& STAMPACCHIA (1967), where it is applied to the general variational problem (1.1.15), and where the case of semi-positive definite bilinear forms is also considered. STAMPACCHIA (1964) had the original proof. For constructive existence proofs and additional references, see also GLOWINSKI, LIONS& TREMOLIERES (1976a). I. BABUSKA(BABUSKA& AZIZ(1972, Theorem 5.2.1)) has extended the Lax-Milgram lemma to the case of bilinear forms defined on a product of two distinct Hilbert spaces. This extension turns out to be a useful tool for the analysis of some finite element methods (BABUSKA(1971b)). 1.2. For treatments of Differential Calculus with FrCchet derivatives, the reader may consult CARTAN(1967), DIEUDONNE(1967), SCHWARTZ (1967). For the theory of distributions and its applications to partial differential equations, see SCHWARTZ (1966). Other references are (1968), VO-KHACKHOAN(1972a, 1972b). The TRBVES(1967), SHILOV Sobolev spaces are extensively studied in LIONS (1962) and N E ~ A S (1967). See also ADAMS(1975). The original reference is SOBOLEV (1950). Thorough treatments of the variational formulations of elliptic boundary value problems are given in LIONS(1962), AGMON(1965), N E ~ A S (1967), LIONS& MAGENES(1968), VO-KHACKHOAN(1972b). Shorter & AZIZ(1972), ODEN& accounts are given in AUBIN(1972), BABUSKA REDDY ( 1976a). More specialized treatments, particularly for nonlinear problems, are LADY~ENSKAJA & URALTEVA (1968), LIONS(1969), & TEMAM(1974). For regularity results, see GRISVARD (1976), EKELAND KONDRAT’EV(1967). For more classically oriented treatments, see for example BERS, JOHN & SCHECHTER (1964), COURANT & HILBERT(1953. 1962), MIRANDA (1970), STAKGOLD (1968). As an introduction to classical elasticity theory, notably for the elasticity problem, the clamped plate problem, the membrane & LIFSCHITZ (1967). For the variaproblem, see for example LANDAU tional formulations of problems in elasticity along the lines followed here, consult DUVAUT & LIONS(l972), FICHERA (1972), ODEN& REDDY (1976b).

CHAPTER

2

INTRODUCTION TO THE FINITE ELEMENT METHOD

Introduction The basic scope of this chapter is to introduce the finite element method and to give a thorough description of the use of this method for approximating the solutions of second-order or fourth-order problems posed in variational form over a space V . A well-known approach for approximating such problems is Galerkin's method, which consists in defining similar problems, called discrete problems, over finite-dimensional subspaces v h of the space V . Then the finite element method in its simplest form is a Galerkin's method characterized by three basic aspects in the construction of the space v h : First, a triangulation y h is established over the set d, i.e., the set d is written as a finite union of finite elements K E y h . Secondly, the function u h E v h are piecewise polynomials, in the sense that for each K E Y h , the spaces P K = { V h I R ; uh E v h } consist of polynomials. Thirdly, there should exist a basis in the space v h whose functions have small supports. These three basic aspects are discussed in Section 2.1, where we also give simple criteria which insure the validity of inclusions such as v h c H'(R), v h c H ~ ( L ? ) , etc. . . (Theorems 2.1.1 and 2.1.2). We also briefly indicate how the three basic aspects are still present in the more general finite element methods to be subsequently described. In this respect, we shall reserve the terminology conforming finite element method for the simplest such method (as described in this chapter). In Section 2.2, we describe various examples of finite elements, which are either n-simplices (simplicia1 finite elements) or n-rectangles (rectangular finite elements), in which either all degrees offreedom are point values (Lagrange finite elements) or some degrees of freedom are directional deriuatiues (Hermite finite elements), which yield either the inclusion X h c H ' ( R ) (finite elements of class go) or the inclusion 36

Ch. 2, 5 2.1.1

BASIC ASPECTS OF THE FINITE ELEMENT METHOD

37

x h C HZ(n)(finite elements of class %') when they are assembled in a finite element space x h . Then in Section 2.3, finite elements and finite element spaces are given general definitions, and we proceed to discuss their various properties. Of particular importance are the notion of an afine family of finite elements (where all the finite elements of the family can be obtained as images through affine mappings of a single reference finite element) and the notion of the PK-interpolation operator (a basic relationship between these two notions is proved in Theorem 2.3.1). The PK-interpolation operator and its global counterpart, the Xh-interpolation operator both play a fundamental role in the interpolation theory in Sobolev spaces that will be developed in the next chapter. We also show how to impose boundary conditions on functions in finite element spaces. We conclude Section 2.3 by briefly indicating some reasons for which a particular finite element should be preferred to another one in practical computations. In Section 2.4, we define the convergence and the order o f convergence for a family of discrete problems. In this respect, Ce'a's lemma (Theorem 2.4.1) is crucial: The error I I K - U h l l , i.e., the distance (measured in the norm of the space V ) between the solution K of the original problem and the solution uh of the discrete problem, is (up to a constant independent of the space v h ) bounded above by the distance inf u h E V h Ilu - tlhll between the function u and the subspace vh. Indeed, all subsequent convergence results will be variations on this theme !

2.1. Basic aspects of the finite element method

The Galerkin and Ritz methods Consider the linear abstract variational problem: Find

VV E v, a(K, V ) = f ( U ) ,

K

E V such that

(2.1.1)

where the space V , the bilinear form a ( . , .), and the linear form f are assumed to satisfy the assumptions of the Lax-Milgram lemma (Theorem 1.1.3). Then the Galerkin method for approximating the solution of such a problem consists in defining similar problems in finitedimensional subspaces of the space V . More specifically, with any finite-dimensional subspace vh of V , we associate the discrete problem :

38

INTRODUCTION TO THE FINITE ELEMENT METHOD

Find

Uh

[Ch. 2, 5 2.1.

E vh such that vvh

E

vh,

a(uh, v h ) = f ( V h ) .

(2.1.2)

Applying the Lax-Milgram lemma, we infer that such a problem has one and only one solution u h , which we shall call a discrete solution. Remark 2.1.1. In case the bilinear form is symmetric, the discrete solution is also characterized by the property (Theorem 1.1.2)

(2.1.3) where the functional J is given by J ( v ) = ta(v, v ) - f ( v ) . This alternate 0 definition of the discrete solution is known as the Ritz method.

The three basic aspects of the finite element method. Conforming finite elemen t met hods Let us henceforth assume that the abstract variational problem (2.1.1) corresponds to a second-order or to a fourth-order elliptic boundary value problem posed over an open subset R of R",with a Lipschitz-continuous boundary r. Typical examples of such problems have been studied in Section 1.2. In order to apply Galerkin method, we face, by definition, the problem of constructing finite-dimensional subspaces Vh of spaces V such as H ~ ( R H'(R), ), H;(O),etc.. . The finite element method, in its simplest form, is a specific process of constructing subspaces vh, which shall be called finite element spaces. This construction is characterized by three basic aspects, which for convenience shall be recorded as (FEM I ) , (FEM 2) and (FEM 3), respectively, and which shall be described in this section. (FEM I ) The first aspect, and certainly the most characteristic, is that a triangulation T h is established over the set 6, i.e., the set 6 is subdivided into a finite number of subsets K , called finite elements, in such a way that the following properties are satisfied:

d = U ~ c Ky. ~ (Th2) For each K E F,,,the set K is closed and the interior is non empty. (Yh3) For each distinct K I , K 2E Fh, one has Rl f l k2= 4. (Fh4) For each K E F,,, the boundary aK is Lipschitz-continuous. (yhhl)

Ch. 2, 5 2.1.1

BASIC ASPECTS O F THE FINITE ELEMENT METHOD

39

Remark 2.1.2. A fifth condition ( 9 h 5 ) relating “adjacent” finite elements, will be introduced in the next section. 0

Once such a triangulation 9 h is established over the set 6, one defines a finite element space x h through a specific process, which will be illustrated by many examples in the next section and subsequently. We shall simply retain for the moment that x h is a finite-dimensional space of functions defined over the set 6 (we shall deliberately ignore at this stage instances of finite element spaces whose “functions” may have two definitions across “adjacent” finite elements; see Section 2.3). Given a finite element space x h , we define the (finite-dimensional) spaces P K

= {UhlK;

vh

Ex

h }

spanned by the restrictions 1)hlK of the functions v h E x h to the finite elements K E y h . Without specific assumptions concerning the spaces P K , K E Y h , there is no reason for an inclusion such as x h C H ’ ( 0 )- let alone an inclusion such as x h c ~ ’ (-to 0 )hold. In order to obtain such inclusions, we need additional conditions of a particularly simple nature, as we show in the next theorems (converses of these results hold, as we shall show in Theorems 4.2.1 and 6.2.1). Remark 2.1.3. Here and subsequently, we shall comply with the use of the notation H ” ( K ) , in lieu of H ” ( R ) . 0 Theorem 2.1.1. Assume that the inclusions PK C H ’ ( K ) for all K E 9 and x h C %‘(6)hold. Then the inclusions x h

h

cH ’ ( o ) ,

x o h = { V h E X h ;

v h = o

on

r)cHd(n),

hold.

Proof. Let a function u E x h be given. We already know that it is in the space L’(0). Therefore, by definition of the space H ’ ( R ) ,we must find for each i = 1,. . . ,n, a function ui E L 2 ( 0 )such that

For each i, a natural candidate is the function whose restriction to

40

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2, 5 2.1.

each finite element K is the function d i ( u J K ) .Since each finite element K has a Lipschitz-continuous boundary dK, we may apply Green's formula (1.2.4): For each K E y h ,

where vLKis the i-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 Z K ~ T , , . vIK4vi.i.K ~ ~ K dy vanishes: Either a portion of dK is a portion of the boundary r of R in which case 4 = 0 along this portion, or the contribution of adjacent elements is zero. The boundary r being Lipschitz-continuous by assumption, the second inclusion follows from the characterization

H ~ ( R ) = { u E H ' ( O ) ,V = O

on

which was mentioned in Section 1.2.

0, 0

Assuming Theorem 2.1.1 applies, we shall therefore use the finite element space v h = x,h if we are solving a second-order homogeneous Dirichlet problem, or v h = 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. and XI,C

Assume that the inclusions PK C H * ( K ) f o r all K E T,,

% ' ( dhold. ) Then the inclusions

Thus if we are to solve a simply supported plate problem, or a clamped plate problem, we shall use the finite element space vh = Xoh, or

Ch. 2 . 8 2.1.1

BASIC ASPECTS OF THE FINITE ELEMENT METHOD

41

the finite element space v h = Xooh, respectively, as given in the previous theorem. Let us return to the description of the finite element method. (FEM 2) The second basic aspect of the finite element method is that the spaces PK,K E Y h , contain polynomials, or, at least, contain f u n c tions which are “close to” polynomials. At this stage, we cannot be too specific about the underlying reasons for this aspect of the method but at least, we can say that (i) it is the key to all convergence results as we shall see, and (ii) it yields simple computations of the coefficients of the resulting linear system (see (2.1.4) below). Let us now briefly examine how the discrete problem (2.1.2) is solved in practice. Let ( w k ) E l be a basis in the space V,,. Then the solution uh = ZEI U k W k of problem (2.1.2) is such that the coeficients uk are solutions of the linear system

(2.1.4)

whose matrix is always invertible, since the bilinear form, being assumed to be V-elliptic, is a fortiori Vh-elliptic. By reference to the elasticity problem, the matrix ( a ( w k ,w,)) and the vector (f(w1))are often called the stiffness matrix and the load vector, respectively. In the choice of the basis (wk.F-1, it is of paramount importance, f r o m a numerical standpoint, that the resulting matrix possess as many zeros as possible. For all the examples which were considered in Section 1.2 the coefficients a ( w k ,w,) are integrals of a specific form: For instance, in the case of the first examples, one has a(wk,w,) =

I (i R

[=I

aiwkdiwI+ awkwl

so that a coefficient a ( w k ,w I ) vanishes whenever the dx-measure of the intersection of the supports of the basis functions Wk and w1 is zero. (FEM 3) As a consequence, we shall consider as the third basic aspect of the finite element method that there exists at least one “canonical” basis in the space vh whose corresponding basis functions have supports which are as “small” as possible, it being implicitly understood that these basis functions can be easily described.

42

INTRODUCTION TO T H E FINITE ELEMENT METHOD

[Ch. 2. 5 2.1.

Remark 2.1.4. When the bilinear form is symmetric, the matrix ( a ( w k ,w , ) ) 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,*),”=,which is orthonormal with respect to the inner product a ( . , -). This is indeed an efficient way of getting a sparse matrix since the corresponding matrix ( a ( w f ,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. 0 It was mentioned earlier that the three basic aspects were characteristic of the finite element method in its simplest f o r m . Indeed, there are more general finite element methods:

(i) One may start out with more general oariational 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 Vhr 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 R 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 R,, (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, 52.2.1

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 Prove Theorem 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 I(uhlld C (as usual, there is a simpler proof when the bilinear form is symmetric). (ii) The space V being separable, there exists a nested sequence ( V”)”€N of finite-dimensional subspaces such that Vv)-= V. Let ( U ~ ) ” € Nbe the sequence of associated discrete solutions. Show that there exists a subsequence of the sequence ( u ” ) ” , =which ~ weakly converges to a solution u of the original variational problem. (iii) Show that the whole sequence converges in the norm of V to the solution u. (iv) Show that the Sobolev spaces H ” ( R ) are separable. 2.1.1. 21.2.

(u

2.2. Examples of finite elements and finite element spaces

Requirements f o r finite element spaces Throughout this section, we assume that we are using a conforming finite element method for solving a second-order or a fourth-order boundary value problem. Let us first summarize the various requirements that a finite element space X h must satisfy, according to the discussion made in the previous section. Such a space is associated with a triangulation TI, of a set d = KE9,, K (FEM I), and for each finite element K E T h , we define the space

u

P K

= {UhlK;

uh

E

xh}.

Then the requirements are the following:

(2.2.1)

44

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2 . 8 2.2.

(i) For each K € y h , the space PK should consist of functions which are polynomials or "nearly polynomials" (FEM 2). (ii) By Theorems 2.1.1 and 2.1.2, inclusions such as x h C %'(d) or x h C %'(d) should hold, depending upon whether we are solving a second-order or a fourth-order problem. For the time being, we shall ignore boundary conditions, which we shall take into account in the next section only. (iii) Finally, we must check that there exists one canonical basis in the space X h , whose functions have "small" supports and are easy to describe (FEM 3). In this section, we shall describe various finite elements K which are all polyedra in R",and which are sometimes called straight finite elements. As a consequence, we have to restrict ourselves to problems which are posed over sets d which are themselves polyedra, in which case we shall say that the set 0 is polygonal.

First examples of finite elements f o r second order problems: n-Simplices of type ( k ) , (3') We begin by examining examples f o r which the inclusion x h C %'(d) holds, and which are the most commonly used by engineers for solving second-order problems with conforming finite element methods. Inasmuch as such problems are most often found in mechanics of continua, it is clear that the value to be assigned in practice to the dimension n in the forthcoming examples is either 2 or 3 (see the examples given in Section 1.2). We equip the space R" with its canonical basis (ej)yZl. For each integer k 2 0, we shall denote by Pk the space of all polynomials of degree S k in the variables xI, x2,. . . ,x,, i.e., a polynomial p E Pk is of the form p : x = (xi, ~

. . . , x,)

2 ,

E R"+ P ( X )

for appropriate coefficients y a l a 2 . a. n , or using the multi-index notation,

The dimension of the space Pk is given by dim Pk

=

(n + k ).

(2.2.2)

Ch. 2 , § 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

45

If @ is a space of functions defined over R", and if A is any subset of R", we shall denote by @ ( A )the space formed by the restrictions to the set A of the functions in the space @. Thus, for instance, we shall let pk(A) = { p IA;

P E pk).

(2.2.3)

Notice that the dimension of the space Pk(A) is the same as that of the space p k = Pk(R") as long as the interior of the set A is not empty. In R", a (nondegenerate) n-simplex is the convex hull K of (n + 1) points aj = (aii)y=lE R", which are called the vertices of the n-simplex, and which are such that the matrix

(2.2.4)

is regular (equivalently, the ( n + 1) points uj are not contained in a hyperplane). Thus, one has

(2.2.5) Notice that a 2-simplex is a triangle and that a 3-simplex is a tetrahedron. For any integer m with 0 6 m d n, an m-face of the n-simplex K is any m-simplex whose (m + 1) vertices are also vertices of K. In particular, any ( n - 1)-face is simply called a face, any 1-face is called an edge, or a side. The barycentric coordinates Aj = A j ( x ) , 1 d j d n + 1, of any point x E R", with respect to the ( n + 1) points aj,are the (unique) solutions of the linear system

1%

(2.2.6) Aj

= 1,

j= I

whose matrix is precisely the matrix A of (2.2.4). By inspection of the linear system (2.2.6), one sees that the barycentric coordinates are afine

46

FINITE ELEMENTS A N D FINITE ELEMENT SPACES

functions o f

X I , x2,.

Ai =

[Ch. 2, 5 2.2.

. . , x, (i.e., they belong to the space PI):

2

J=I

+ bin+l,

1 s i zz n

+ 1,

(2.2.7)

where the matrix B = (b,) is the inverse of the matrix A. The barycenter, or center o f gravity, of an n-simplex K is the point of K whose all barycentric coordinates are equal to l / ( n + 1). To describe the first finite element, we need to prove that a polynomial p : x + E,rr141 yaxa of degree 1 is uniquely determined by its values at the ( n + 1) vertices ai of any n-simplex in R".T o see this, it suffices to show that the linear system

131ya(ai)"

= pi, 1 ~j

G II

+ 1,

has one and only one solution (yo,la1 C 1 ) f o r all right-hand sides pi, 1 C j C n + 1 . Since dim P = card { a j } )= n + 1, the matrix of this linear system is square, and therefore it suffices to prove either uniqueness or existence. In this case, the existence is clear: The barycentric coordinates verify Ai(ai)= aii, 1 4 i, j S n + 1 , and thus the polynomial

(uyz:

has the desired property. As a consequence, we have the identity n+l

V P E PI,P

=

2 P(ai)Ai.

(2.2.8)

r=l

Although we shall not repeat this argument in the sequel, it will be often implicitly used. A polynomial p E PI being completely determined by its values p ( a i ) , 1 S i G n + 1 , we can now define the simplest finite element, which we shall call n-simplex o f type ( 1 ) : The set K is an n-simplex with vertices ai, 1 6 i C n + 1, the space PK is the space P l ( K ) , and the degrees o f freedom of the finite element, i.e., those parameters which uniquely define a function in the space PK. consist of the values at the vertices. Denoting by .ZK the corresponding set of degrees o f freedom, we shall write symbolically

ZK = { p ( u l ) , 1 s i s n + 1 ) .

Ch. 2 . 5 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

47

In Fig. 2.2.1, we have recorded the main characteristics of this finite element for arbitrary n, along with the figures in the special cases n = 2 and 3. In case n = 2, this element is also known as Courant’s triangle (see the section “Bibliography and Comments”).

Courant‘s triangle. or triangle of type ( I )

tetrahedron of type ( 1 )

dim PK = 3

I

n-simplex of type ( I ) PK

2,

=

P , ( K ) ; dim PK = ( n + 1):

= { p ( a i ) , 1 s is n

+ I}.

Fig. 2.2.1

Let us call aij = ;(ai + ai), 1 s i < j s n + 1, the mid-points of the edges of the n-simplex K . Since Ak(aij>= t ( & i + & j ) , 1 s i < j s n + 1, 1 s k s n + 1, we obtain the identity (where, here and subsequently, indices i ,j , k , . . . , are always assumed to take all possible values in the set {1,2,. . . , n} whenever this fact is not specified) Ai(2Ai - l)p(ai) + 2 4AiAj~(aij),

V P E P z ~P = 1

(2.2.9)

i<j

which yields the definition of a finite element, called the n-simplex of type (2): the space PK is P z ( K ) ,and the set ZKconsists of the values at the vertices and at the mid-points of the edges (Fig. 2.2.2).

48

INTRODUCTION TO THE FINITE ELEMENT METHOD

triangle of type (2) dim PK = 6

[Ch. 2 , s 2.2.

tetrahedron of type (2) dim PK = 10

n-simplex of type (2)

( n + 1" PK = P , ( K ) ; dim PK = S K = {p(ai),

1G i 6n

+ I;

+ 2).

p(aii), I s i < j s n + i}.

Let aiii= f(2ai+ a i ) for i# j , and aiik= ;(ai + ai + a k )for i < j < k. From the identity

(2.2.10) we deduce the definition of the n-simplex of type ( 3 ) (Fig. 2.2.3). One may define analogous finite elements with polynomials of arbitrary degree, but they are not often used. In this respect, we leave to the reader the proof of the following theorem (Exercise 2.2.2), from which for any integer k, the definition of the n-simplex of type ( k ) can be easily derived.

Ch. 2 . 5 2.2.1

triangle of type ( 3 ) dim PK = 10

I

1

tetrahedron of type (3)

dim PK = 20

n-simplex

P,=b(aj),

49

FINITE ELEMENTS AND FINITE ELEMENT SPACES

of

type (3)

I

I s i ~ n + l ;p ( a i j i ) , l s i , j ~ n + l , i # j ; < j < k s n + 1).

p(ajik), 1 s i

Fig. 2.2.3

Theorem 2.2.1. Let K be an n-simplex with vertices aj, 1 j S n + 1 . Then for a given integer k 3 1 , any polynomial p E Pk is uniquely determined by its values on the set

Let us now examine a modification of the n-simplex of type (3), in ) no longer present, and which is which the degrees of freedom p ( a i j k are often preferred by the engineers to the previous element. To describe the corresponding finite element, we need the following result.

50

INTRODUCTION TO THE FINITE ELEMENT METHOD

Theorem 2.2.2.

[Ch. 2 , 8 2.2.

For each triple (i,j , k ) with i < j < k , let I i#m

Then any polynomial in the space

Pi = { p E P 3 ; &k(p)

1 a i <j

= 0,

< k a n + I}

is uniquely determined by its values at the vertices ai, 1 d i d n the points aiij,1 d i, j d n + 1 , i f j . In addition, the inclusion

(2.2.13)

+ 1, and at

P2C P;

(2.2.14)

holds. Proof. The ("f')degrees of freedom '$ijk are linearly independent (since ' $ i j k ( p ) = 12p(aijk)+ . . .) and thus, the dimension of the space Pi is dim Pi = dim P3-

(" ' )

=(n

+ 112,

i.e., precisely the number of degrees of freedom. Using the identity (2.2.10), and arguing as before, we obtain the identity

- 1)

+ T27A i A j k#i,j C, Ak)p(aiij), (2.2.15)

which proves the first part of the theorem. To prove that the inclusion (2.2.14) holds, let p be a polynomial of degree a 2 and let A E 2$(R"; R) be its second derivative (which is constant). From the expansions p ( a l ) = p(aijk)

valid for any triple Z

Dp(aijk)(ai - aijk) + iA(al - aijk)*, 1 E 1,

= {i,j ,

k } with i < j < k, we deduce

Ch.2, 5 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

51

l,mEI, IZm, we deduce

taking into account that a;jk = i(a;;j-k & k , ) Because A is a linear mapping, and because

= i(ajjk f G

k )

= f(akki i- aiii).

and the proof is complete.

a

From Theorem 2.2.2 we deduce the definition of the n-simplex o f type (3‘) (Fig. 2.2.4).

Assembly in triangulations. The associated finite element spaces Next we examine the question of constructing triangulations, using anyone of the finite elements previously described. Being non degenerate n-simplices, these have non empty interiors and Lipschitzcontinuous boundaries, and therefore properties (Yh2) and (9h4) are satisfied. To construct triangulations in the sense understood in Section 2.2, we shall write d = KGFh K in such a way that the n-simplices have piecewise disjoint interiors (cf. properties (.Th1 ) and (Yh3)).In order to satisfy inclusions such as x h c ~ ‘ ( f i(and ) X , c %‘(ji) later on), we shall impose a fifth condition on a triangulation made up of n-simplices, namely: (Yh5) Any f a c e of any n-simplex K , in the triangulation is either a subset of the boundary r, or a face of another n-simplex Kz in the triangulation. In the second case, the n-simplices K , and K 2 are said to be adjacenr. An example of a triangulation for n = 2 is given in Fig. 2.2.5, while

u

52

INTRODUCTION TO THE FINITE ELEMENT METHOD

triangle of type (3')

tetrahedron of type (3')

n-simplex of type (3')

PK = P : ( K ) (cf. (2.2.13)); dim PK = ( n + I)*; ZK = { p ( a , ) ,1 s i s n + I , p(aiii), 1 s i , j 6 n + 1, i+ j } . . Fig. 2.2.4

Fig. 2.2.5

[Ch. 2 . 8 2.2.

Ch. 2 , § 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

53

Fig. 2.2.6.

Fig. 2.2.6 shows an example of a “forbidden situation” since the intersection of K , and K 2 is not an edge of K2. Given a triangulation T h , we associate in a natural way a finite element space x h of functions u h : 6 + R with each type of finite element: With n-simplices of type ( l ) , a function uh E x h (i) is such that each restriction V h ( K is in the space PK = P , ( K )for each K E y h , and (ii) is completely determined by its values at all the vertices of the triangulation. Likewise, with n-simplices of type (2), a function of x h (i) is in the space PK = P , ( K ) for each K E y h h rand (ii) is completely determined by its values at all the vertices and all the mid-points of the edges of the triangulation. Similar constructions hold for n-simplices of type (3) or (3‘). In all cases, a function v h in the space x h is seen to be determined by degrees of freedom which make up a set of the form s h

={Uh(b);

b EN h ) ,

(2.2.16)

where N h is a finite subset of 6.The set &,is the set of degrees of freedom of the finite element space x h . One should observe that if there i s n o ambiguity in the definition of the degrees of freedom across adjacent finite elements, it is precisely because we have satisfied requirement ( Y h 5 ) . This requirement also plays a crucial role in the proof of the following result.

54

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2.5 2.2.

Theorem 2.2.3. Let x h be the finite element space associated with n-simplices of type ( k ) for any integer k a1 or with n-simplices of type (3'). Then the inclusion x h

c %'(h) n H ' ( 0 )

holds.

Proof. We shall give the proof in case n = 2 and for triangles of type ( 2 ) , leaving the other cases as a problem (Exercise 2.2.3). Given a function uh in the space X,, consider the two functions UhlK, and uLlK2 along the common side K' = [bi,b j ] of two adjacent triangles K 1and KZ (Fig. 2.2.7). Let t denote an abscissa along the axis containing the segment K'. Considered as functions of t , the two functions V h l K , and UhlK2 are quadratic polynomials along K', whose values coincide at the three points bi, bj, bij = (bi+ b j ) / 2 . Therefore these polynomials are identical, and the inclusion x h C Ceo(d)holds. Finally the inclusion x h c H'(R) is a consequence of Theorem 2.1.1. 0 It remains to verify requirement (FEM 3 ) , i.e., that there is indeed a canonical choice for basis functions with small supports. In each case,

it,

Fig. 2.2.7

Ch. 2. Fi 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

55

the set 2,, of degrees of freedom of the space is of the form s h

If we define functions Wk

1S k S M}.

= {(bk);

E

xh

Wk,

and

1 6k

S

(2.2.17)

M, by the conditions

W k ( b r ) = &I,

1 6 k,l s M ,

(2.2.18)

it is seen that (i) such functions form a basis of the space X,, and that (ii) they have "small" supports. In Fig. 2.2.8, we have represented the three types of supports which are encountered when triangles of type (3) are employed, for instance. n-Rectangles of type ( k ) . Rectangles of type (29, (3'). Assembly in triangulations

Before we turn to a second category of finite elements, we need a few definitions. For each integer k 3 0, we shall denote by Q k the space of all polynomials which are of degree s k with respect to each one of the n variables xI,x2,. . . , x,, i.e., a polynomial p E Qk is of the form p : x = ( X I , ~ 2 , ... ,x , ) €

R"+ P ( x )

h J P(4) Fig. 2.2.8

56

[Ch. 2 . 8 2.2.

INTRODUCTION TO THE FINITE ELEMENT METHOD

for appropriate coefficients y a l a 2 . . The dimension of the space Qk is given by .(In.

dim

Q k

= (k

+ 1)"

(2.2.19)

and the inclusions p k

c Qk c p n k

(2.2.20)

hold. Notice that the dimension of the space Qk(A) is the same as that of the space Q k = @(itn) as long as the interior of the set A C R" is not empty.

Theorem 2.2.4. A polynomial p E Qk is uniquely determined by its values on the set

, . . . , ;)€R'';

&=(x=(i,;

i i E { O , l, . . . , k } , l c j s n

I.

(2.2.21) Proof. It suffices to use the identity

In R", an n-rectangle, or simply a rectangle if n form

= 2,

is a set of the

n

K

[a;,bi] = {x

= r=l

= (xI,x2,. . . ,x n ) ;

ai zs xi c bi, 1

d

i d n}, (2.2.23)

with - m < ai < b i < + m for each i, i.e., it is a product of compact intervals with non-empty interiors. A f a c e of K is any one of the sets

i#i

i#i

while an edge of K, also called a side, is any one of the sets

n n

[aj, bjI x

i=I

i#i

{ci}.

Ch. 2 . 8 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

51

with ci = ai or bi, 1 G i d n, i# j , 1 s j d n. A vertex of K is any point x = (x,, XP,. . . ,x,) of K with xi = aior bi, 1 s i =sn. Observe that the set M k of (2.2.21) is a subset of a particular n-rectangle, namely the unit hypercube [0, 13". Then, given any nrectangle K, we infer that a polynomial p E Qk is uniquely determined by its values on the subset

Mk(K)= F K ( M ~ )

(2.2.24)

of the n-rectangle K , where FK is a diagonal afine mapping, i.e., of the form F K :x E R"+ F K ( x )= BKx + bK, with bK a vector in R" and BK an n x n diagonal matrix, such that K = FK([O,13"). From this, we deduce the definition of finite elements, called n-rectangles of type ( k ) . Just as in the case of n-simplices, the values k = 1 , 2 or 3 are the most commonly encountered. In Fig. 2.2.9, 2.2.10 and 2.2.1 1 , the corresponding elements are represented for n = 2 and 3, and the numbering of the points occurring in the sets of degrees of freedom is also indicated for n = 2. a 4,

,=3

rectangle of type ( I )

3-rectangle of type ( 1 ) =8

dim P ,

n-rectangle of type ( 1 )

P,

H,

=

Q , ( K ) : dim PK = 2":

={p(a);

LI

E M , ( K ) }(cf. (2.2.24))

Fig. 2.2.9

58

INTRODUCTION TO THE FINITE ELEMENT METHOD

I

[Ch. 2,

2.2.

/

r

el

{

%rectangle of type (2) dim PK = 21

rectangle of type (2) dim P , = 9

n-rectangle of type

(2)

P, = Q 2 ( K ) ; dim PK = 3 " ; a E M , ( K ) } (cf. (2.2.24)).

P, = { p ( a ) .

-

For the numbering of the nodes when n = 2, we have ..Alowed t..is rule: Assuming, without loss of generality, that the set K is the unit square [ O , 112, four points are consecutively numbered if they are the vertices of a square centered at the point (i.4). This rule allows for particularly simple expressions of the corresponding functions p; appearing in identities of the form VP E

P =

Qkr

P(ai)Pi9 I

which are special cases (for k = 1 ,2 ,3 and n = 2) of the identity (2.2.22). Notice that the coordinates of a given point with respect to the four vertices ai, 1 d i d 4, of the unit square are ( X I , XZ),

(x2, 1 - X I ) ,

(1 - XI. 1 - XZ),

(1 - x2, XI),

respectively. Then, if we introduce the variables x3

= 1 -XI,

xq

= 1 - x2,

(2.2.25)

Ch. 2 , 8 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

rectangle of type ( 3 )

I

59

3-rectangle of type ( 3 ) dim PK = 64

I

n-rectangle of type (3)

PK = Q , ( K ) ; dim PK = 4": a E M 3 ( K ) } (cf. (2.2.24)).

f K ={p(a);

Fig. 2.2. I 1

the four functions pi are obtained through circular permutations of the variables xI, x2, x3,x4 (such permutations correspond to rotations of + 7r/2 around the point (i, i)). Corresponding to the unit square of type (1) (recall that K = [ O , I]'), we have the identity

with Pl=(I-xl)(l-x?). P4 = ( 1 - X h ' .

pz=x,(l-x?),

p3=xlx?,

We may thus condense these expressions in PI = x3x4,.

...

(2.2.26)

Likewise, corresponding to the unit square of type (2). we have the

60

[Ch. 2 . 8 2.2.

INTRODUCTION TO THE FINITE ELEMENT METHOD

(2.2.27)

using the above rule. Finally, corresponding to the unit square of type (3), we have 1

Pl=qX3(3X3-

9

ps=-qX3(3X3-

1)(3X3-2)Xq(3Xql)(X3-

9

p9=-X3(3X3-2)(X34 81

p13=qX3(3X3-

1)(X3-

1)(3Xq-2),.

1)Xq(3Xq- 1)(3Xq-2), 1)Xq(3Xq- 1)(3Xq-2), 1)X4(3X4-

1)(Xq- I),

.. ...

I

(2.2.28)

... ....

Remark 2.2.1. The inconsistency for the notations a;, 5 4 i 4 9, between the rectangles of type (2) and (3), avoids the introduction of a new letter.

0 In analogy with the n-simplices of type (3’), one can derive two finite elements, in which the “internal” values of the rectangle of type (2) or (3) are no longer degrees of freedom (for simplicity, we shall restrict ourselves to the case n = 2). The existence of these finite elements is a consequence of the following two theorems.

Theorem 2.2.5. Let the points a;, 1 4 i 4 9, b e as in Fig.2.2.10. Then any polynomial in the space

is uniquely determined b y its values at the points a;, 1 s i s 8. In addition,

Ch. 2, P 2.2.1

61

FINITE ELEMENTS AND FINITE ELEMENT SPACES

the inclusion

P2C

QS

(2.2.30)

holds.

Proof. The first part of the proof is similar to the first part of the proof of Theorem 2.2.2. In particular, we have the identity

with (2.2.31)

To prove the inclusion (2.2.30), let p be a polynomial of degree 2, and let A denote its (constant) second derivative. From the expansions 1 p(ai) = p(a9) + Dp(a9)(ai- as)+-A(ai - ad2, 1 d i d 8, 2

we deduce

since

Because the mapping A is bilinear, and because as= ( a , + a2)/2,. . . ,we obtain

Combining the previous relations, we deduce that

and the proof is complete.

62

INTRODUCTION TO THE FINITE ELEMENT METHOD

Theorem 2.2.6. the space

[Ch. 2 , s 2.2.

Let the points ai, 1 d i =s 16, be as in Fig. 2.2.1 1. Define

Q ; = { p E Q3; & ( p ) = 0 , 1 C i C 4},

(2.2.32)

where

M P )=

+ 4 p ( a 1 )+ 2 p ( a ~+) p(a3)+ 2p(a4) -6p(a~) 3P(a6)- ~ P ( ~ -I ~I P) ( ~ I Z ) ,

(2.2.3 3)

and &(PI, $3(p), and t , ! ~ ~ (are p ) derived by circular permutations in the sets U ~ = , { u i }Uy=s{ai}, , U1$,{ai} and u:$13{ai}.Then any polynomial in the space Q ; is uniquely determined by its values at the points ai, 1 6 i d 12. In addition, the inclusion P3C Q;

(2.2.34)

holds. Proof. The proof is left as a problem (Exercise 2.2.5). We shall only record the identity 12

V PE Q L

P =

2 P(ai)pi, 1=I

with

(2.2.35)

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). I f it happens that the set b 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 n-rectangles: The fifth condition (Th5) on a triangulation now reads: (Fh5)Any f a c e of any n-rectangle K I in the triangulation is either a subset of the boundary I‘, o r a f a c e o f another n-rectangle K2 in the triangulation.

Ch.2 . 5 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

2, = { p ( a i ) , 1 < i S 8). Fig. 2.2.12

I

rectangle of type (3')

PK = Q ; ( K ) (cf. (2.2.32)): dim PK = 12;

& = (p(ai);

1 s is 12).

Fig. 2.2.13

I

63

64

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2 . 8 2.2.

In the second case, the n-rectangles K , and K 2 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 m a y 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 ) f o r any integer k zz 1 or with rectangles of type (2’) or (3’). Then the inclusion x h

C go(&)f-I H ’ ( R )

(2.2.36)

0

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 o f 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 E 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, § 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

65

expressions such as D p ( a ) b , D * p ( a ) ( b ,c ) , e t c . . . , where b, c are vectors in R”. 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 ai, 1 c i c n + 1, and let aijk = ;(ai + uj + ak), 1 d i < j < k d n + 1 . Then any polynomial in the space P3 is uniquely determined by its values and the values of its n first partial derivatives at the vertices ai, 1 s i d n + 1 , and its values at the points aiik.1 d i < j < k d n + 1 .

Proof. It suffices to argue as usual so as to obtain the following identity : Vp EP3, p

= C (-2A;+3A:-7Ai

2

I

{ j #j:i#

AjAk)p(ai) i

The only novelty is that one needs to use the derivatives of the barycentric coordinates in order to show that D p ( a i )= D p ( a i ) , 1 S i d n + 1, denoting momentarily by p’ the right-hand side of (2.2.37). By differentiating the polynomial p’, we obtain

Dp’(ai)=

2 {Dp(ai)(ai- ai)}DAi. j#i

T o show that the above expression is equal to D p ( a i ) ,it is equivalent to show that Dp(ai)(uk- ai) = Dp(ai)(ak- a i ) , 1 d k d n

+ 1,

k # i.

These last relations are in turn consequences of the relations Dhj(ak - ai) =

- Ai(ai), 1 S k S

n

+ 1,

k f i,

which we now establish. Denoting by B the inverse matrix of the matrix A of (2.2.4),we obtain

&Ai=b,, l S j S n + l ,

ldlsn

66

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2, $ 2.2.

(cf. (2.2.7)). Therefore we have n

fl

DAj(ak - X ) =

bjlalk

-

b j ~ x=~ s j k - Aj(x)

for any x E R",and in particular for x

=

ai.

0

From this theorem, we deduce the definition of a finite element, which is called the Hermite n-simplex of type (3) (Fig. 2.2.15). where the directional derivatives Dp(ai)(ai- ai) are degrees of freedom. Of course, the knowledge of these n directional derivatives at a vertex ai is equivalent to the knowledge of the first derivative D p ( a i ) . Such a knowledge is indicated graphically by one small circle, or sphere, centered at the point ai. Since the first derivative @ ( a i ) is equally well determined by the partial derivatives ajp(ai), 1 6 j 6 n, another possible set of degrees of freedom for this element is the set 2; indicated in Fig. 2.2.15.

Hermite tetrahedron of type (3)

Hermite triangle of type (3) dim PK = 10

I

Hermite n-simplex of type (3)

Fig. 2.2.15

Ch. 2, P 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

67

The derivation of a related element without the degrees of freedom p(aijk), i C j < k, is based on the following theorem, whose proof is left to the reader (Exercise 2.2.6).

Theorem 2.2.9.

F o r each triple (i, j , k ) with i C j c k, let

Then any polynomial in the space

P'; = {p E P3; $ijk(p) = 0,

1 c i <j < k d n

+ 1)

(2.2.39)

is uniquely determined by its values and the values of its n first partial derivatives a t the vertices ai, 1 S i c n + 1. In addition, the inclusion

PZC P ; holds.

(2.2.40)

0

From this theorem, one deduces the definition of the Hermite nsimplex of type (37, which, in case n = 2, is also called the Zienkiewicz triangle (Fig. 2.2.16). Given a triangulation made up of n-simplices, we associate in a natural way a finite element space X h with either type of finite elements. To be specific, assume we are using Hermite n-simplices of type (3), the case of Hermite n-simplices of type (3') being quite similar. Then a function vh is in the space X,, if (i) each restriction vhlK is in the space PK = P 3 ( K )for each K E T,,,and (ii) it is defined by its values at all the vertices of the triangulation, its values at the centers of gravity of all triangles found as 2-faces of the n-simplices K E T,,,and the values of its n first partial derivatives at all the vertices of the triangulation. The corresponding set of degrees of freedom of the space X,, is thus of the form

where Nu denotes the set of all the vertices of the n-simplices of the triangulation and Nc denotes the set of all centers of gravity of all 2-faces of the n-simplices found in the triangulation. When a finite element space is constructed with n-simplices of type ( 3 ) or (37, the sets Zk are preferred to the sets X K (cf. Figs. 2.2.15 and

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[Ch. 2 . 5 2.2.

n

Hermite tetrahedron of type (3') dim P , = 16

triangle, o r Hermite triangle of type (3')

Hermite n-simplex of type (3')

2, = ( p ( a , ) ,

1 c i S n + 1; Dp(a,)(a,- a , ) , 1 < i , j G n + 1, i f j } . S ; = { p ( a i ) , ~ ~ i ~ n aip(a,), + t ; I G ~ S ~ + I I G, ~ s ~ ) .

Fig. 2.2.16

2.2.16) inasmuch a s they directly correspond to the set &, but this observation is of a purely practical nature. Again, requirement ( 9 h 5 ) insures that the degrees of freedom are unambiguously defined across adjacent finite elements, and it is also the basis for the following theorem.

Theorem 2.2.10. Let xh be the finite element space associated with Hermite n-simplices of type (3), or with Hermite n-simplices of type (3'). Then the inclusion

x,,c u0@)n ~

' ( 0 )

(2.2.41)

holds. Proof.

Arguing as in Theorem 2.2.3, it suffices to derive the inclusion

xh C qo(d):Along any side common to two adjacent triangles, there is a unique polynomial of degree 3 in one variable which takes on prescribed

Ch. 2, 8 2.2.1

69

FINITE ELEMENTS AND FINITE ELEMENT SPACES

values and prescribed first derivatives at the end points of the side. This argument easily extends to the n-dimensional case.

0

To verify requirement (FEM 3), let us assume for definiteness that we are considering Hermite triangles of type (3), so that the associated set of degrees of freedom of the space is of the form

&I,

1 s k,l

L, a I W k ( b 1 ) = & W k ( b , ) = 0, ’ l s k s L , lslsJ, w:(bl) = 0 , 1 d k S J, 1 C 1 d L, alw:(bl) = &I, &w:(bi) = 0, 1 s k, 1 s J, Wi(b1)= 0 , 1 k J,. 1 d 1 d L, alw:(bj) = 0 , azw:(bi) = &, 1 d k, 1 G J,

Wk(bl)=

(2.2.43)

First examples of finite elements f o r fourth-order problems: The Argyris and Bell triangles, the Bogner-Fox-Schmit rectangle. Assembly in triangulations Finally, we examine some examples of finite elements which yield the inclusion xh C and which may therefore be used for solving fourth-order problems. It is legitimate to restrict ourselves to the case where n = 2, in view of the examples given in Section 1.2. Our first example is based on the following result.

%‘(a),

Theorem 2.2.11. Let K be a triangle with uertices ai, 1 s i s 3, and let aii = ;(ai + ai), 1 d i < j 4 3, denote the mid-points of the sides. Then any polynomial p of degree 5 is uniquely determined by the following set o f 2 1 degrees of freedom : Z K = {aap(ai), JaI4 2,

1 s i s 3;

a,,p(aii), I s i < j d 3}, (2.2.44)

where a,, denotes the normal derivative operator along the boundary of K .

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Proof. Given a set of degrees of freedom, finding the corresponding polynomial of degree 5 amounts to solving a linear system with a square matrix, for which existence and uniqueness for all right-hand sides are equivalent properties, as we already observed. We shall prove the latter property, i.e., that any polynomial p E P5such that a”p(ai)=O, 1 ~ ~ 1 ~1 2~ , i s 3 ,a v p ( a i j ) = O , l d i C j ~ 3 ,

is identically zero. Let t denote an abscissa along the axis which contains the side K‘ = [al,a 2 ] .Then the restriction pIK,,considered as a function q of t, is a polynomial of degree 5 which satisfies q ( a i ) = q’(ai)= q“(ai)= q(a2) = q’(a2)= q’’(a2)= 0,

since, if

T

is a unit vector on the axis containing the side K’, we have q’(al) = a d a d ,

q ” ( a l )= a,p(al),

etc. . . ,

and thus q = 0. Likewise, considered as a function r of t , the normal derivative a y p along K’is a polynomial of degree 4 which satisfies

r(aJ = r‘(a1)= r(a12)= r(a2)= r’(a2) = 0 , since

d a d = ayp(a1), rYad = a,p(al>,

r(a12)= a V p ( a d etc.. , .,

and, thus, r = 0. Since we have alp = 0 along K’ (p = 0 along K’), we have proved that p and its first derivative Dp vanish identically along K’. This implies that the polynomial A: is a factor of p, as we now show: After using an appropriate affine mapping if necessary, we may assume without loss of generality that A3(xl,x2) = xI.We can write

where p i , 0 d i d 5, are polynomials of degree (5 - i) in the variable Theref ore Vx2 E R, ~ ( 0~ , 2 =) po(x2) = 0, VXZ E R, aip(0, ~ 2 =) P I ( X ~ ) 0, 9

which proves our assertion.

x2.

Ch. 2, 4 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

71

Similar arguments hold for the other sides, and we find that the polynomial (A:AiA:) is a factor of p. Since the Ai are polynomials of degree 1 which do not reduce to constants, it necessarily follows that p

=o.

With Theorem 2.2.11, we can define a finite element, the 21-degree of freedom triangle, also known as Argyris triangle (Fig. 2.2.17). Fig. 2.2.17 is self-explanatory as regards the graphical symbols used for representing the various degrees of freedom. We observe that at each vertex ai, the first and second derivatives D p ( a i ) and D 2 p ( a i )are known. With this observation in mind, we see that other possible definitions for the set of degrees of freedom are the sets 2k and 2;

Fig. 2.2.17

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[Ch. 2 . 8 2.2.

indicated in Fig. 2.2.17. In the expression of the set Xi, the indices are numbered modulo 3, and each vector v;, 1 d i d 3 , is the height originating at the point ai. It may be desirable to dispose of the degrees of freedom avp(aij), 1 d i < j d 3. This reduction will be a consequence of the following result.

Theorem 2.2.12. Any polynomial in the space

P ; ( K )= { p E P , ( K ) ; aVpE P3(K') for each side K' of K} (2.2.45)

is uniquely determined by the following set of 18 degrees of freedom:

ZK = { a a p ( a i ) ,(a(6 2,

1 6 i 6 3).

(2.2.46)

The space P X K ) satisfies the inclusion P4(K) c P ; ( K ) .

(2.2.47)

Proof. By writing aVpE P 3 ( K ' ) in definition (2.2.45), it is of course meant that, considered as a function of an abscissa along an axis containing the side K', the normal derivative &p is a polynomial of degree 3. The inclusion (2.2.47) being obvious, it remains to prove the first part of the theorem. To begin with, we prove a preliminary result: Let K' = [ai, ail be a segment in R", with mid-point aji, and let v be a function such that uIK.E P4(K'). Then we have v I K .E P3(K') i f and only if xii(v)= 0, where xij(V)

+

=4 ( ~ ( ~ i~ ) ( a j )-) 8v(aij)

+ Du(aj)(a;- U j ) .

+ D ~ ( a i ) (-~a;) j (2.2.48)

To see this, let, for any x E K', a4= D 4 v ( x ) ~ 4where , along K', so that a4is a constant. Then we have

+

1 2

-

T

v ( a ; )= v ( a j j ) Dv(aii)(ai aij)+ - D 2 v ( a j j ) ( a-i

+ -61 ~ ~ v ( a ; -~ aij)3 ) ( a+-I(a; ~24 a 4

- aiil14,

is a unit vector

Ch. 2.9 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

73

from which we deduce (ai - aij = -(aj - aij)):

Likewise,

1 Dv(aj)(aj- aij) = D2v(aij)(aj - aij)'+ -D3v(aii)(aj- aij)' 2 a 4 + xllaj - aij(I4v

and therefore, D2Y(a;j)(a;- aij)2+ D2v(a;j)(aj- a;$ = Dv(ai)(ai- aij) + Du(aj)(aj- aij) ff4

--{llai 6

- Uij1I4 + llaj - Uij114}.

Combining our previous relations, we get 1

2u(aij)= ~ ( a i+) ~ ( a j+)-{Du(u~)(u~ - aj) + Dv(aj)(ai- aj)} 4 ff4 + -1lai 96

-

and the assertion is proved. As a consequence of this preliminary result, the space P ; ( K ) may be also defined as

P X K ) = { p E P d K ) ; ,yii(dvp) = 0,

1G

i < j s 3},

(2.2.49)

i.e., in view of relations (2.2.48), we have characterized the space P ; ( K ) by the property that each normal derivative avp(aij)is expressed as a linear combination of the parameters a"p(ai), a"p(aj), la( = 1,2. Then the proof is completed by combining the usual argument with the result of Theorem 2.2.1 1. 0

14

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2. 8 2.2.

From Theorem 2.2.12, we deduce the definition of a finite element, called the 18-degree of freedom triangle, or, preferably, Bell’s triangle. See Fig. 2.2.18, where we have indicated three possible sets of degrees of freedom which parallel those of the Argyris triangle. Given a triangulation made up of triangles, we associate a finite dement space xh with either type of finite elements. We leave it to the reader to derive the associated set of degrees of freedom of the space X , and to check that the canonical basis is again composed of functions with “small” support. We shall only prove the following result. Theorem 2.2.13. Let xh be the finite element space associated with Argyris triangles o r Bell’s triangles. Then the inclusion

x,, c ui(d)n ~ * ( n ) holds.

Bell’s triangle or 18-degree of freedom triangle

Fig. 2.2.18

(2.2.50)

Ch. 2, 8 2.2.1

Proof.

FINITE ELEMENTS AND FINITE ELEMENT SPACES

75

By Theorem 2.1.2, it suffices to show that the inclusion XhC

Cge'(d) holds.

Let K1 and K 2 be two adjacent triangles with a common side K ' = [bi,bi] (Fig. 2.2.19) and let u h be a function in the space Xh constructed with Argyris triangles. Considered as functions of an abscissa t along an axis containing the side K ' , the functions Vh(K, and UhlK2 are, along K', polynomials of degree 5 in the variable t. Call these polynomials q1 and q2. Since, by definition of the space xh,we have q(bi)= q'(bi)= q"(bi)= q(bj)= q'(bj)= q"(bj)= 0 ,

with q = q1- q2, it follows that q = 0 and hence the inclusion V,,C Cg0(6) holds. Likewise, call rl and r2, the restrictions to the side K' of the functions & u , , ( ~ ,and -avuhlK2. Then both rl and r2 are polynomials of degree 4 in the variable t and, again by definition of the space X,,, we have

r( b i )= r'(b i )= r( bii)= r( bi) = r'( bi) = 0, with r = rl - r2, so that r = 0. We have thus proved the continuity of the normal derivative which, combined with the continuity of the tangential derivative ( q = 0 along K' implies q' = 0 along K'), shows that the first derivatives are also continuous on 6. If the space Xh is constructed with Bell's triang1es;the argument is

Fig. 2.2.19

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INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2 , s 2.2.

identical for the difference q = q 1- q2.The difference r = rl - r2 vanishes because it is a polynomial of degree 3 in the variable t which is such that

0

r( bi) = r’(bi) = r( b j )= r’(b j )= 0 .

To conclude, we give one instance of a rectangular finite element which may be used for solving fourth-order problems posed over rectangular domains. Its existence depends upon the following theorem, whose proof is left to the reader (Exercise 2.2.8).

Theorem 2.2.14. Let K denote a rectangle with vertices ai, 1 S i C 4 . Then a polynomial p E Q3 is uniquely determined by the following set of degrees of freedom:

ZK = {p(ai)y

aiP(ai), azP(ai), alzP(ai), 1

i

4).

0 (2.2.51)

The resulting finite element is the Bogner-Fox-Schmit rectangle. See Fig. 2.2.20, which is again self-explanatory for the graphical symbols. The proof of the next result is also left to the reader (Exercise 2.2.8):

I

Bogner-Fox-Schmit rectangle

Ch. 2, 5 2.2.1

FINITE ELEMENTS AND FINITE ELEMENT SPACES

77

Theorem 2.2.15. Let xh be the finite element space associated with Bogner- Fox- S c hm it rectangles. Then the inclusion

xh c %'(fi) n H 2 ( n )

(2.2.52)

0

holds.

Finally, the reader should.check, using the standard construction, that a finite element space constructed with any one of the last three finite elements indeed possesses canonical bases whose functions have "small" supports (FEM 3). Exercises 2.2.1. (i) Prove that the dimension of the space Pk, resp. Qk, is ("ik),. resp. ( k + I)". (ii) Prove that dim Pk(A) = dim p k . resp. dim Qk(A)= dim Qk, if the interior of the set A C R" is not empty. 2.2.2. Let K be an n-simplex with vertices aj, 1 s j s n + 1. For a given integer k 2 1, show that a polynomial of degree s k is uniquely defined by its values on the set Lk(K) defined in (2.2.1 1) (NICOLAIDES (1972)). The set L k ( K ) is called the principal lattice of order k of the n-simplex K. 2i2.3. Complete the proof of Theorem 2.2.3 so as to cover all cases. 2.2.4. Give another proof of Theorem 2.2.4 (i.e., without recurring to identjty (2.2.22)), by showing that if a polynomial of Qk vanishes on the set hfk defined in (2.2.21), then it is identically zero. 2.2.5. Prove Theorem 2.2.6. 2.2.6. Prove Theorem 2.2.9. Are the spaces P ; and P i (cf. (2.2.13) and (2.2.39), respectively) identical? 2.2.7. Given a triangle with vertices ai, 1 c i C 3, and mid-points aij = !(ai + a j ) , 1 d i < j s 3 , show that a polynomial p E P4 is completely determined by the following degrees of freedom ( ~ E N ~ S E(1974)): K

p ( a i ) , JiiP(ai), J u p ( a i ) , J z ~ ( a i ) ,1 C i s 3, p ( a i i ) , 1 6 i < j c 3. Does this element yield the inclusion

xh c %'(a)?

xh C %'(fin),

I

resp. the inclusion

2.2.8. (i) Given a rectangle with vertices ai, 1s i s 4 , show that a polynomial p E Q3 is completely determined by the following degrees of

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INTRODUCTION TO T H E F I N I T E E L E M E N T M E T H O D

[Ch. 2 . 5 2.3.

freedom:

p ( a A aiP(ai), 4 P ( a i ) , aizP(ai), 1 s i s 4. (ii) Show that the corresponding space x h

x h

satisfies the inclusion

c c e l ( i - 3 ) n~ ~ ( 0 ) .

Consider the finite element space xh constructed with (Hermite) triangles of type (3) and let w:, 1 C k C MI,be the basis functions of the space xh associated with the values at the barycenters of all the triangles of the triangulation, so that the discrete solution takes the form

2.2.9.

M

Show that in this case the solution of the linear system (2.1.4) amounts, in fact, to solving a smaller linear system, in the unknowns u;, 1 C i S M2, only. This process, known as the static condensation of the degrees o f freedom, is of course to be distinguished from the use of (Hermite) triangles of type (3'). 2.3. General properties of finite elements and finite element spaces

Finite elements as triples ( K , P , 2).Basic definitions. The P-interpolation operator Let us begin by giving the general definition of a finite element. A finite element in R" is a triple ( K , P , 2 ) where: (i) K is a closed subset of R" with a non empty interior and a Lipschitz-continuous boundary, (ii) P is a space of real-valued functions defined over the set K , (iii) Z is a finite set of linearly independent linear forms 4i, 1 6 i S N, defined over the space P (in order to avoid ambiguities, the forms 4; need to be defined over a larger space; we shall examine this point later; cf. Remark 2.3.3). By definition, it is assumed that the set 2 is P unisofvent in the following sense: given any real scalars a;,I S i =z N, there exists a unique function p E P which satisfies + , ( p ) = a,, 1

s i S N.

(2.3.1)

Ch. 2 . 8 2.3.1

79

PROPERTIES OF FINITE ELEMENTS A N D SPACES

Consequently, there exist functions pi E P, 1 S i

4 j ( p i )= aij,

1sj s

N.

S

N , which satisfy

(2.3.2)

Since we have

(2.3.3) Of course this implies that the space P is finite-dimensional and that dim P = N. The linear forms &, 1 S i s N, are called the degrees of freedom o f the finite element, and the functions p i , 1 s i c N , are called the basis functions of the finite element. Whenever we find it convenient, we shall use the notations P K , Z K , &K and piK in lieu of P, 2,+i and pi. Remark 2.3.1. The set K itself is often called a finite element, as we did in the previous section, and as we shall occasionally do in the sequel. 0 Remark 2.3.2. The P-unisolvence of the set 2 is equivalent to the fact that the N linear forms 4i form a basis in the dual space of P. As a consequence, one may view the bases (4i)Lland (pi)EIas being dual bases, in the algebraic sense.

In the light of the definition of a finite element, let us briefly review the examples given in the previous section. We have seen examples for which the set K is either an n-simplex, in which case the finite element is said to be simplicial, or triangular if n = 2, or tetrahedral if n = 3, or an n-rectangle in R”,in which case the finite element is said to be rectangular. As we already mentioned, these are all special cases of straight finite elements, i.e., for which the set K is a polyhedron in R“.Other polygonal shapes are found in practice, such as quadrilaterals (see Section 4.3 and Section 6.1) or “prismatic” finite elements (see Remark 2.3.6). We shall also describe (Section 4.3) “curved” finite elements, i.e., whose boundaries are composed of “curved” faces. The main characteristic of the various spaces P encountered in the examples is that they all contain a “full” polynomial space Pk(K) f o r

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[Ch. 2, 5 2.3.

some integer k 2 1, a property that will be shown in subsequent chapters to be crucial as far as convergence properties are concerned. In all the examples described previously, the degrees of freedom were of some of the following forms:

(2.3.4)

where the points a:, r = 0, 1,2, belong to the finite element, and the (non zero) vectors .$iIk, ti, Si: are either constructed from the geometry of the finite element (e.g., D p ( a j ) ( a j- ai), a v p ( a j j ) ,etc. . .) or fixed vectors of R" (e.g., a i p ( a j ) , ajjp(ak)). The points a:, r = 0, 1,2, are called the nodes of the finite element and make up a set which shall be denoted JVK in general. Whereas only directional derivatives of order 1 or 2 occurred in the examples, one could conceivably consider degrees of freedom which would be partial derivatives of arbitrarily high order, but these are seldom used in practice. As we shall see later, however, (Section 4.2 and Section 6.2) there are practical instances of degrees of freedom which are not attached to nodes: They are instead averages (over the finite element or over one of its faces) of some partial derivative. When all the degrees of freedom of a finite element are of the form p + p ( a ; ) , we shall say that the associated finite element is a Lagrange finite element while if at least one directional derivative occurs as a degree of freedom, the associated finite element is said to be a Hermite finite element. As the examples in the previous section have shown, there are essentially two methods for proving that a given set 2 of degrees of freedom is P-unisolvent: A f t e r it has been checked that dim P = card(Z)), one either (i) exhibits the basis functions, or (ii) shows that if all the degrees of freedom are set equal to zero, then the only corresponding function in the space P is identically zero. We have used method (i) for all the examples, except for the Argyris triangle where we used method (ii). Given a finite element (K, P,2),and given a function u = K - R , sufficiently smooth so that the degrees of freedom di(u), 1 S i C N , are

Ch. 2 . 8 2.3.1

PROPERTIES OF FINITE ELEMENTS AND SPACES

81

well defined, we let

(2.3.5) denote the P-interpolant of the function u, which is unambiguously defined since the set Z is P-unisolvent. Indeed, the P-interpolant, also denoted &u, is equivalently characterized by the conditions

n u E P, and

di(17u) = di(u), 1 6 i Q N .

(2.3.6)

Whenever the degrees of freedom are of the form (2.3.4), let s denote the maximal order of derivatives occurring in the definition of the set 2. Then, for all finite elements of this type described here, the inclusion P c % " ( K ) holds. Consequently, we shall usually consider that the domain dom 17 of the P-interpolation operator 17 is the space dom I7 = FeS(K).

(2.3.7)

This being the case, it follows that ouer the space P C dom 17, the interpolation operator reduces t o the identity, i.e., VpEP, n p = p .

(2.3.8)

Remark 2.3.3. In order that the P-interpolation operator be unambiguously defined, it is necessary that the forms di be also defined on the space % ' ( K ) ,for the following reason. Assume again that the space P is contained in the space %'(K). Then if the domain of the operator 17 were only the space P, infinitely many extensions to the space %'(K) would exist. Let us give one simple example of such a phenomenon: Let K be an n-simplex with barycenter a. Then the linear form p E % O ( K ) + l/(meas ( K ) )J K p dx is one possible extension of the form p E Pl(K)+ p(a). Of course, these considerations are usually omitted inasmuch as when one considers a degree of freedom such as aip(aj) for instance, it is implicitly understood that this form is the usual one, i.e., defined over the space %'(K),not any one of its possible extensions from the space P to the space % ' ( K ) .For another illustration of this circumstance, see the description of Wilson's brick, in Section 4.2. 0 Whereas for a Lagrange finite element, the set of degrees of freedom is unambiguously defined - indeed, it can be conveniently identified with

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INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2 . 4 2.3.

the set of nodes - there are always several possible definitions for the degrees of freedom of a Hermite finite element which correspond to the “same” finite element. More precisely, we shall say that two finite elements ( K , P , 2 ) and ( L , Q. 8)are equal if we have K = L,

P = Q and

n~=n~.

(2.3.9)

As an example, let us consider the Hermite n-simplex of type (3’) with the two sets of degrees of freedom (cf. Fig. 2.2.16):

2 = { p ( a i ) , 1 s i s n + 1; Dp(ai)(ai- ai), 1si. j s n + l ,

i#j},

2’ = { p ( a i ) , 1 s i s n + 1; &p(ai), 1 d i s n + 1, 1c k c n}. Let us denote by n and n’ the corresponding P,(K)-interpolation operators. Then, for any function u E % ‘ ( K )= dom = dom nl, we have, with obvious notations, =

2 u(ai)pi + Ci.i fi(ai)(aj - ai)pij,

x i

nlv =

I

u(ai)p:+

c.

aku(ai)p:k.

i.k

One has, for each pair ( i , j ) , Du(ai)(aj- ai) = X L ,piikaku(ai)for appropriate coefficients pijk.To conclude that IZ = n’, it suffices to observe that for each polynomial p E P K , one also has D p ( a i ) ( a , - a i ) = Xi=,pi&p(ai) with the same coefficients pijk. Afine families of finite elements

We now come to an essential idea, which we shall first illustrate by an example. Suppose we are given a family (K, PK, Z K ) of triangles of type (2). Then our aim is to describe such a family as simply us possible. Let k be a triangle with vertices di, and mid-points of the sides ciij = (cii + h j ) / 2 , 1 6 i < j s 3, and let

2 = {p(cii). 1 s i s 3; p(cii,), 1 s i < j c 31, so that the triple (k,P, 2) with fi = PI(&) is also a triangle of

type (2).

Ch. 2,B 2.3.1

PROPERTIES OF FINITE ELEMENTS AND SPACES

8

83

Given any finite element K in the family (Fig. 2.3.1), there exists a unique invertible afine mapping

F K : 2 E R2+ F K ( ~= )BKi

+ bK,

i.e., with BK an invertible 2 x 2 matrix and bK a vector of R2, such that FK(cij)= ai, 1 d i c 3 . Then it automatically follows that FK(Bij)

= aii,

I s i < j c 3,

since the property for a point to be the mid-point of a segment is preserved by an affine mapping (likewise, the points which we called aiii or ajjk keep their geometrical definition through an affine mapping). Once we have established a bijection 2 E k + x = F K ( i E ) K between the points of the sets l? and K, it is natural to associate the space Pk=(p: K+R;

p=b-

Fi' j=Pl

Fig. 2.3. I

9

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INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2, § 2.3.

with the space P. Then it follows that

PW

= P , ( K ) = PK,

because the mapping FK is affine. In other words, rather than prescribing such a family by the data K , PK and Z K , it sufices to give one reference finite element (K.2, @ ) and the afine mappings FK.Then the generic finite element (K, P K ,Z K ) in the family is such that

K = FK(K), P K = ( P : K + R ; p = C * F - ' , CEP},

Z K = (P{FK(hi)),

1 s i s 3;

p(FK(&i,)), 1 =Z i < j s 3).

I

With this example in mind, we are in a position to give the general definition: Two finite elements ( k , @ , $ and ) (K, P , Z ) , with degrees of freedom of the form (2.3.4), are said to be afine-equivalent if there exists an invertible afine mapping:

F : 2 E R" + F ( i ) = B i + b E R",

(2.3.10)

such that the following relations hold:

K = F(K), P = { p : K + R ; p = f i * F - ' , CEP}, a:= F(S9, r = 0,1,2, (i'k = BiA, 8; = B& 6; = B&, whenever the nodes a:, resp. S:, and vectors occur in the definition of the set Z, resp. 2.

(ilk,

(2.3.1 1 ) (2.3.12) (2.3.13) (2.3.14) (;,

ti, resp. jilk,li, &,

Remark 2.3.4. The justification of the relations (2.3.14) will become apparent in the proof of Theorem 2.3.1. 0 With this definition of affine-equivalence, let us return to the examples given in Section 2.2 (the reader should check for oneself the various statements to come). T o begin with, it is clear that two n-simplices of type k for a given integer k 2 1 , are affine equivalent, and that this is also the case for n-simplices of type ( 3 ' ) , in view of the definition (2.2.13) of the associated space PK. Likewise, two n-rectangles of type ( k ) for a given

Ch. 2 , 8 2.3.1

85

PROPERTIES OF FINITE ELEMENTS A N D SPACES

integer k L 1, or two rectangles of type (2') or (3') are affine equivalent through diagonal affine mappings. In other words, any two identical Lagrange finite elements that we considered are afine-equivalent. When we come to Hermite finite elements, the situation is less simple. Consider for example two Hermite n-simplices of type (3) with sets of degrees of freedom in the form ,ZK (Fig. 2.2.15). Then it is clear that they are affine-equivalent because the relations ai- ai = F ( h j )- F(hi)= B(hj - hi), 1 S i, j =sn + I ,

j # i,

hold, among other things. However, had we taken the sets of degrees of freedom in the form Zk, it would not have been clear to decide whether the two finite elements were affine-equivalent, and yet these two sets of degrees of freedom correspond to the same finite element, as we already pointed out. The same analysis and conclusion apply to the Hermite n-simplex of type (3') or to the Bogner-Fox-Schmit rectangle. In this last case, it suffices to observe that this finite element can also be defined by the following set of degrees of freedom (the index i is counted modulo 4)

-Zk = { P ( a i ) , Dp(aiI(ai-1- ai), Dp(ai)(ai+l- ai), D*p(ai)(ai-,- ai, ai+,- a i ) , 1 s i s 4},

(2.3.15)

for which relations (2.3.14) hold. There are counter-examples. For instance, consider a finite element where some degrees of freedom are normal derivatives at some nodes. Then two such finite elements are not in general affine equivalent: The property for a vector to be normal to a hyperplane is not in general preserved through an affine mapping. Thus two Argyris triangles are not affine equivalent in general, except for instance if they happen to be both equilateral triangles. The case of Bell's triangles is left as a problem (Exercise 2.3.4). Let us return to the general case. We shall constantly use the correspondences (2.3.16) (2.3.17) between the points 2 E K and x E K, and the functions p^ E and p E P corresponding to two affine-equivalent finite elements. As a consequence

86

INTRODUCTION TO THE FINITE ELEMENT METHOD

[Ch. 2, 8 2.3.

of the correspondences (2.3.16) and (2.3.17), notice that we have $(.t)=p(x)

for all

~EKD , EP.

(2.3.18)

We next prove a crucial relationship between the P-interpolation operator fi and the P-interpolation operator ll associated with affineequivalent finite elements. This relationship will be itself a consequence of the fact that the basis functions are also in the correspondence (2.3.17). Theorem 2.3.1. Let ( & , p , s ) and ( K ,P , S ) be two afine-equivalent finite elements with degrees o f freedom in the form (2.3.4). Then if Bi, 1 d i d N , are the basis functions of the finite element K,the functions pi, 1 d i d N , are the basis functions of the finite element K. The interpolation operators l 7 and fi are such that

(nu)^ = dd

(2.3.19)

for any functions v^ E dom d and u +dom ll associated in the correspondence

9 Edom d-+v= 6 .F - ' E d o m

T.

(2.3.20)

Proof. The P-interpolation operator ll is of the form (with obvious notations):

Using the derivation of composition of functions, we obtain

and, taking also into account that D2F = 0,

Thus we also have

Ch. 2 . 8 2.3.1

PROPERTIES OF FINITE ELEMENTS AND SPACES

87

from which we deduce, using the correspondence (2.3.17),

If we apply the previous identity to a function u E P, we infer that the functions fi!, fii, fii are the basis functions of the finite element ( k ,$, $), by virtue of identity (2.3.8). Using this result, we conclude that the function (nu)*is equal to the function fi;, by definition of the $-interpolation operator fi. 0

Remark 2.3.5. To obtain the conclusion of the previous theorem when the sets of degrees of freedom are in the general form f = {&; 1 < i S N} and 2 = {&; 1 S i d N}, it is necessary and sufficient that the degrees of freedom be such that Vv^E dom fi, &(;)

= &(u),

I O (O(hk+l)

if k = 1, if k 3 2 ,

where I ~ ~ O . m and ,R ~ ~ ~ ~ stand ~ l , for m , the ~ norms of the spaces L"(O) and W',"(O),respectively. Restricting ourselves for brevity to the case k = 1, the corresponding error estimates are obtained in Theorem 3.3.7. It is worth pointing out that all the error estimates found in Section 3.2 and 3.3 are optimal in the sense that, with the same regularity assumptions on the function u, one gets the same asymptotic estimates (or "almost" the same for the norms I.lO,m,R and Il-l l.m,n when k = 1) when the discrete solution u h E v h is replaced by the Xh-interpolant n h u E vh.

3.1. Interpolation theory in Sobolev spaces

The Sobolev spaces W m s p ( 0The ) . quotient space W k + l ' P ( 0 ) / P k ( f l ) We shall consider the Soboleu space W m . p ( 0which, ) for any integer m 3 0 , and any number p satisfying 1 S p S m , consists of those func-

Ch. 3, 5 3.1.1

INTERPOLATION THEORY IN SOBOLEV SPACES

113

tions u E L p ( R )for which all partial derivatives 8% (in the distribution sense) with IaI d m belong to the space L P ( R ) Equipped . with the norm

1 1

(IUJJ,,,,~,~ =(

laaulp dx)'",

if 1

P < 00,

(3.1.1)

lalsrn

IIUllrn.m,n = max {ess.sup laau(x)l} XER

if p = Q),

laldm

the space W m * p ( R is)a Banach space. We shall also use the semi-norms

(,F

jRlilLIuIpdx)'",

JVJ,,,~.~=

LI

if 1 S p

(Q),

=m

lUlm.m.n=

(3.1.2) {esze%upI ~ Q U ( ~ ) I ] if P

= 00.

The Soboleu space WCp(R)is the closure of the space 9 ( R ) in the space W m V p ( R ) . Given a subset A of R" and given a function u E %"(A), the notation I ~ ~ ~ l m . mand ,A 1Vlrn.m.A will also denote the norm maxl,lsm supxEA laau(x)l and the semi-norm maxlol=m supxEA la"u(x)l, respectively. Notice that

wm*2(n) = H"(R), ll-llm.~.R = ll*llm.~,

W,m**(R)= H,m(R),

1.lm.Z.R

= I*lm.R-

As usual, the open sets R that will be considered in this section will be assumed to have a Lipschitz-continuous boundary. In addition they will be assumed to be connected when needed (this assumption is used in the proof of Theorem 3.1.1). In view of future needs, we shall record here some basic properties of the Sobolev spaces that will be often used. In what follows, the notation X 4 Y indicates that the normed linear space X is contained in the no;med linear space Y with a continuous injection, and the notation x C Y indicates in addition the compactness of the injection. Finally, for any integer m 2 0 and any number a €10. I], %"'.u(fi) denotes the space of all functions in cern(fi)whose m-th derivatives satisfy a Holder's condition with exponent a. Equipped with the norm

where 11.11 denotes the Euclidean norm in R", the space (ern,"(fi)is a Banach space.

I I4

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 5 3.1.

By the Soboleu's imbedding theorems. the following inclusions hold, for all integers m 2 0 and all I c p c 03,

W r n @ ( RL) p* 4 (0) with

-P1_* -

1 rn --P n'

n if m ( n l p ) . In the case of n-simplices of type ( k ) , show that we may choose for vectors ti,1 ~ i a n any , n vectors out of the n(n + 1)/2 vectors of the form ( a j - ail, 1 s i < j s n + 1. As a consequence show that in the case of triangles for instance, we may have IIuIIK U I I ~ . ~ . O = O ( h k )even though Zlfimal's condition is violated. (v) Apply the previous analysis to rectangles of type (k).

3.2. Application to second-order problems over polygonal domains

Estimate of the error IIu

- Uh1II.R

Let there be given a second-order boundary value problem, posed over a space V which satisfies the usual inclusions H d ( 0 ) C V = C H1(R). One basic hypothesis will be that the set d is polygonal, essentially because such an assumption allows us to exactly cover the set d with polygonal finite elements. Then with any such finite element, we associate a finite element space xh. Next, we define an appropriate subspace vh of xh (this takes into account the boundary conditions contained in the definition of the space V) which is included in the space V, so that we are using a conforming finite element method. One main property that we shall assume is that the space v h contains the Xh-interpolant of the solution u : See Section 2.3 where the special cases v h = xh C V = H ' ( R ) and vh = XOh C V = Hd(R) have been thoroughly discussed. This would also be true of a problem where V = { u E H ' ( R ) ; u = O on r0} provided the subset of can be written exactly as a union of faces of some finite elements. By contrast, this is not true in general of a

v

r,, r

132

[Ch. 3, 0 3.2.

CONFORMING FTNITE ELEMENT METHODS

nonhomogeneous Dirichlet problem. In this direction, see Exercises 3.2.1 and 3.2.2. Throughout this section, we shall make the following assumptions, denoted (H l), (H 2) and (H 3), whose statements will not be repeated. (H 1) We consider a regular family of triangulations T h in the following sense: (i) There exists a constant u such that (3.2.1) (ii) The quantity (3.2.2) approaches zero. In other words, the family formed by the finite elements ( K , PK,Z K ) , K E uh fi-h, is a regular family of finite elements, in the sense of Section 3.1. Remark 3.2.1. There is of course an ambiguity in the meaning of h, which was first considered as a defining parameter of both families ( T h ) and (&), and which was next specifically defined in (3.2.2). We have nevertheless conformed to this often followed usage. 0

uh

(H 2) All the finite elements (K, PK,Z K ) , K E r h , are affineIn other words, equivalent to a single reference finite element (k,fi, the family (K, P K , ~ ~K )E, T h f o r all h, is an afine family of finite elements, in the sense of Section 2.3. (H 3) All the finite elements ( K , PK,Z K ) , K E uhrh, are of class Ce0. We first prove an approximation property of the family (vh)(Theorem 3.2.1), from which we derive an estimate for the error in the norm (l*Ill,a (Theorem 3.2.2). In the sequel, C stands for a constant independent of h and of the various functions involved (not necessarily the same at its various occurences).

e).

Theorem 3.2.1. In addition to (H l), (H 2) and (H 3), assume that there exist integers k 3 0 and 1 3 0 with 1 S k, such that the following inclusions

Ch. 3, 5 3.2.1

I33

APPLICATION TO SECOND-ORDER PROBLEMS

are satisfied:

(3.2.3) (3.2.4) where s is the maximal order o f partial derivatives occurring in the definitions o f the set 2. Then there exists a constant C independent of h such that, f o r any function u E ~ ~ " n ( V0, ) I(U

-n h Ullm,ns Chk+'-"l ~ l k + ~ . R ,0 s m s

(c

minil, 0,

I12

IIU

-n

h Ull'm,~)

s Chk+'-"( ~ l k + ~ . R 2, G m

d

KEYh

(3.2.5)

min{k + 1, I } (3.2.6)

where n h u E vh is the Xh-interpo/antof the function u. Proof. Applying Theorem 3.1.6 with p

11

-n K

uJI,.K

G

= q = 2,

we obtain

O G m s min{k Chf;+l-mlvlk+l,K,

+ 1, I}.

Using the relations (nhu ) I K = nKv, K E y h (cf. (2.3.32)) and the inequalities hK s h, K E y h (cf. (3.2.2)), we get

= Chk+l-mlvlk+l.n, OSmGmin{k+

I,/}.

Thus inequalities (3.2.6) are proved and inequalities (3.2.5) follow by observing that

for m

X, c

=0

and for rn

%'(a)implies

=1 x h

(when 13 1) since the inclusions

c ~ ' ((Theorem 0) 2.1.1).

C

H ' ( k ) and

0

Remark 3.2.2. Analogous interpolation error estimates hold if the function u is "only" in the spaces ( % " ( a ) n n ~ , , N " " ( K )V).nIt suffices to replace the semi-norm I U ~ ~ + ~ . R by the semi-norm ( E K ~I UTI : ~ + I , K ) ~ I ~ in the right-hand sides of inequalities (3.2.5) and (3.2.6). 0 Such more general estimates are seldom needed.

134

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 5 3.2.

Theorem 3.2.2. In addition to ( H l), ( H 2) and (H 3), assume that there exists an integer k 3 1 such that the following inclusions are satisfied:

p k ( k )c P c H ’ ( R ) , H k + ’ ( k )4 %“k),

(3.2.7) (3.2.8)

where s is the maximal order of partial derivatives occurring in the definition of the set %. Then if the solution u E V of the variational problem is also in the space Hk+’(L?),there exists a constant C independent o f h such that IIU

- uhlll,ns

ChklUlk+l.a,

(3.2.9)

where uh E vh is the discrete solution. Proof. It suffices to use inequality (3.2.5) with u = u and m conjunction with CCa’s lemma (Theorem 2.4. I), which yields

=

1, in

Suficient conditions f o r limh4 IIu - UhI(1.R = 0 The previous result has been established under the assumptions that the solution u is suficiently smooth (in Hk+’(L?) for some k 3 1) and that the Xh-interpolant n h u exists (cf. the inclusion Hk’’(k)b%’(k) which is satisfied if k > (n/2)- 1 + s). If these hypotheses are not valid, it is still possible to prove the convergence of the method if the solution u belongs to the space V n H ’ ( 0 ) and if the “minimal” assumptions (3.2.10) below hold, using a “density argument” as we now show (one should notice that the assumption s =s1 in the next theorem is not a restriction in practice for second-order problems). For a different approach, see Exercise 3.2.3. Theorem 3.2.3. In addition to ( H l), ( H 2) and (H 3), assume that the inclusions (3.2.10) P * ( I z )c P c H ’ ( k )

are satisfied, and that there are no directional derivatives of order 2 2 in the set %. Then we have

qg Ilu - Uh(I1.R = 0.

(3.2.11)

Ch. 3, 8 3.2.1

135

APPLICATION TO SECOND-ORDER PROBLEMS

Proof. Define the space Y=

w2qn)n v.

(3.2.12)

Since the inclusions (3.2.10) and

w2*”(IZ)b %’(IZ),

s=O

or

I,

W2’(R) 4 H’(IZ), hold, we may apply Theorem 3.1.6 with k = 1 , p = 00, rn = 1 , q = 2: There exists a constant C such that Vu E Y9

llu - n K ~ l 1 1 . K

~(mea~(K))’”hKl~)2.~,K,

from which we deduce that

and thus we have proved that

h uI(I.o= 0. lim I(u - n

(3.2.13)

h 4

For all h and all u E Y, we can write IlU

-

nh Ulll,o

1IU

- U1II.R -t

IIu

-n h UIII.R.

(3.2.14)

Given the solution u E V and any number E > 0, we first determine a function u, E Y which satisfies the inequality I(u - u,II I,R s 4 2 . This is possible because fhe space Y is dense in the space V . Then by (3.2.13), there exists an h , , ( ~ such ) that IIu, - n h u , l l , , ~S ~ / for 2 all h S h d ~ ) In . view of inequality (3.2.14), we have therefore shown that

and the conclusion follows from CCa’s lemma (Theorem 2.4.1).

0

A close look at the above proof shows that the choice (3.2.12) of the space Y is the result of the following requirements: On the one hand it had to be dense in the space V and on the other hand the value k = 1 was needed in order to apply Theorem 3.1.6 so as to obtain property (3.2.13) with the assumption PI(& C @.Therefore the space Y had to contain derivatives of order s 2 (this condition limits in turn the admissible values of s to 0 and 1 ) and consequently, one is naturally led to the space of the form (3.2.12). In fact, any space of the form ‘If=

136

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 6 3.2.

W2,p(12) r l V with p sufficiently large, would have also been acceptable, as one may verify. Estimate of the error Iu - UhlO.0. The Aubin-Nitsche lemma In theorem 3.2.2, we have given assumptions which insure that IIu i.e., the quantity - U h l ( l . 0 = O ( h k ) so that the error in the norm I-lo.O, lu - UhIO.0. is at least of the same order. Our next objective is to show that, under mild additional assumptions, one has in fact Iu - uhlO,O= O(hk+l). Let us first define an abstract setting which is well adapted to this type of improved error estimates: In addition to the space V , with norm 11.11, we are given a Hilbert space H , with norm 1.1 and inner product (.;), such that V = H with a continuous injection (in the present case, we shall have typically V = Hd(l2), H ' ( O ) , or any closed space contained in between these two spaces, and H = L 2 ( 0 ) ) . Then we shall identify the space H with its dual, so that the space H may be identified with a subspace o f the dual space V' of V , as we now show: Let f € H be given. Since V C H with a continuous injection L, we have

VvE v ,

I(f9

v)lG

If1 bl ll4 VI IbII,

(3.2.15)

and therefore the mapping u E V - + ( f u, ) defines an element f E V ' . The mapping f E H + f € V' is an injection for if (f, u ) = 0 for all u E V , then ( f , u ) = 0 for all u E H since V is dense in H,and thus f = 0. We shall henceforth identify f and f,i.e., we shall write

V fE H,

vu E v,

cf, u ) = f(u).

(3.2.16)

We next prove an abstract error estimate. With the same assumptions as for the Lax-Milgram lemma (Theorem 1.1.3), we let as usual u E V and uh E vh denote the solutions of the variational problems

V u E V , a(u, u ) = f ( u ) , vuh

E

vh,

a(uh. V h ) = f(uh),

(3.2.17) (3.2.18)

respectively. We recall that M denotes an upper bound for the norm of the bilinear form a ( - ,-) (cf. (1.1.19)).

Ch. 3, 8 3.2.1

APPLICATION TO SECOND-ORDER PROBLEMS

137

Theorem 3.2.4 (Aubin-Nitsche lemma). Let H be a Hilbert space, with norm 1.1 and inner product (., such that , ) s

V=H

and

V 4 H .

(3.2.19)

Then one has (3.2.20)

where, f o r any g E H, rp, E V is the unique solution of the variational problem : V v E V , a ( v , Cp,) = (g, u ) . Proof.

(3.2.21)

To estimate lu - uhl, we shall use the characterization (3.2.22)

Using the identification (3.2.16), we can solve problem (3.2.21) for all g E H (the proof is exactly the same as that of the Lax-Milgram lemma). Since ( u - uh) is an element of the space V, we have in

particular

a ( u - Uhr (pB) = (8, u - uh), on the one hand, and we have vph

E

vh,

a ( u - uh, ( P h ) = 0,

on the other, which we obtain by subtracting (3.2.17) and (3.2.18). Using the above relations, we obtain vph

E

vh,

(8,u - uh) = a ( u - uhy pg - (Ph)r

and therefore, ( ( g ,u - uh)l

- Uhll

$ch 11%

- (Phil.

(3.2.23)

Inequality (3.2.20) is therefore a consequence of the characterization (3.2.22) and inequality (3.2.23). 0 A look at the above proof shows that pR had to be the solution of problem (3.2.21), i.e., where the arguments are interchanged in the bilinear form. Problem (3.2.21) is a special case of the following varia-

138

CONFORMING FINITE ELEMENT METHODS

[Ch. 3. 5 3.2.

tional problem: Given any element g E V', find an element rp E V such that vu E

v,

a ( u , rp) = g ( u ) .

Such a problem is called the adjoint problem of problem (3.2.17). Of course the two problems coincide if the bilinear form is symmetric. It is easily verified that when the variational problem (3.2.17) corresponds to a second-order boundary value problem (cf. the examples given in Section 1.2), the same is true for its adjoint problem. As we shall see, the abstract error estimate of Theorem 3.2.4 yields an improvement in the order of convergence for a restricted class of second-order problems, which we now define: A second-order boundary value problem whose variational formulation is (3.2.17), resp. (3.2.21), is said to be regular if the following two conditions are satisfied:

(9 For any f E L2(fl), resp. any g E L2(fl),the corresponding solution uf,resp. uB,is in the space H2(fl)f l V. (ii) There exists a constant C such that (3.2.24) (3.2.25) Remark 3.2.3. Consider for instance problem (3.2.17). Then without the assumption of regularity, we simply know that (use Remark 1.1.3 and the identification (3.2.16)):

Indeed, this regularity is not too restrictive a condition: For example the homogeneous Dirichlet problem and homogeneous Neumann problem associated with the data of (1.2.23) (with g = 0) are regular if 0 is convex and if the functions aii and a are sufficiently smooth. However, this would not be the case for the homogeneous mixed problem of (1.2.28). We are now in a position to estimate the error in the norm l.IO.R. Theorem 3.2.5. In addition to (H I ) , (H 2) and (H3). assume that s = 0 . that the dimension n is s 3. and that there exists an integer k 3 I such

[Ch. 3. 8 3.2.

I39

APPLICATION TO SECOND-ORDER PROBLEMS

that the solution u is in the space Hk"(f2) and such that the inclusions

Pk(R)c r; c H ' ( R )

(3.2.26)

hold. Then if the adjoint problem is regular, there exists a constant C independent o f h such that (3.2.27)

Iu - Uh(0.ns Chk+'IUIk+l.n.

Proof. Since n s 3, the inclusion H 2 ( & 4 Y o ( @ holds (if s = 1, the inclusion H ' ( k ) 4 %'(I?) holds only if n = 1; this is why we have restricted ourselves to the case s = 0). Applying Theorem 3.2.1 and inequality (3.2.25), we obtain, for each g E H = L 2 ( 0 ) , inf

Ch 6 vh

II% - cphl1i.n s IIPg- nh(P&i.ns

ChIpgIz.n Chlglo,n.

Combining the above inequality with inequality (3.2.20) yields IU

- UhIo.a

Chllu - Uhlli.n,

and it remains to use inequality (3.2.9) of Theorem 3.2.2.

0

Concluding remarks. Inverse inequalities Although we restricted ourselves to the case of a single partial differential equation, it should be clear that the analysis of this section includes the systems o f equations of plane and three-dimensional elasticity (cf. (1.2.40)) posed over polygonal domains. In this case, the space vh is a product of two or three identical finite element spaces vh: With each degree of freedom of the space vh, one associates two or three unknowns which are the corresponding components of the approximate displacement. The asymptotic estimates obtained in Theorems 3.2.2 and 3.2.5 are the best one could hope for, inasmuch as the orders of conoergence are the same a s if the discrete solution uh were replaced by the Xh-interpolant o f the function u : Compare (3.2.9) and (3.2.5) with m = 1 , and (3.2.27) and (3.2.5) with m = 0. Consequently, the table in Fig. 3.1.2 is also useful for getting a practical appraisal of the upper bounds of Theorems 3.2.2 and 3.2.5. For instance, one gets IIu - ~ ~ =l O(h2-"), l ~ . m~ = 0 , 1, with n-simplices or rectangles of type ( I ) , or IIu - Uhllm.0 = 0(h3-'"), m = 0 , 1 , with n-simplices of type (2) or (3') or rectangles of type (2) or (3'), e t c . . . Neverthe-

140

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 5 3.2.

less, the higher the order of convergence, the higher the assumed regularity of the solution, and this observation limits considerably the practical value of such estimates. For example, let us assume that we are using n-simplices of type (3) while the solution is “only” in the space H’(f2): Then the application of Theorems 3.2.2 and 3.2.5 with k = 1 shows that one gets only ((u- U , , ( ( ~ , R= O ( h 2 - m ) ,m = 0 , l . Therefore, unless the solution is very smooth, the use of polynomials of high degree does not improve the quality of the approximation. Interestingly, the same conclusion was also drawn through purely practical considerations, by the engineers who seldom use polynomials of degree 3 4 for approximating the solution of second-order boundary value problems. To conclude this section, we shall define a simple property of a family of triangulations (of the type described at the beginning of this section), whose value lies essentially (as usual) in the consequences which we shall derive (cf. Theorem 3.2.6). Although we had no immediate need for this property in the present section, it shall be used subsequently at various places, beginning in the next section, so that it seemed appropriate to record it here. We shall say that a family of triangulations satisfies an inverse assumption, in view of the inverse inequalities to be established in the next theorem, if there exists a constant Y such that h hK

--d

(3.2.28)

Y.

Notice that this is by no means a restrictive condition in practice. For such families, we are able to estimate the equivalence constants between familiar semi-norms (we remind the reader that u is the constant which appears in the regularity assumption; cf. (3.2.1)). Theorem 3.2.6. Let there be given a family of triangulations which satisfies hypotheses (H I), (H 2) and an inverse assumption, and let there be given t w o pairs ( I , r ) and ( m , q ) with 1, m 3 0 and ( r , q ) E [ I , 001 such that

I

S

m

and

@ C W ’ - r ( Kn) W m . q ( k ) .

Then there exists a constant C

=

(3.2.29)

C(u, v, I, r, m , q ) such that

(3.2.30)

Ch. 3, 8 3.2.1

141

APPLICATION TO SECOND-ORDER PROBLEMS

if p , < 00, with

Proof. Given a function by Theorem 3.1.2,

uh

Ex

h

and a finite element K E Yh, we have

(3.2.31) where the function GK is in the standard correspondence with the function UhIK. Define the space

Since 1 s m by assumption, the implication

B E fi

+ 1PlIrn.q.k

=0

holds and therefore the mapping

8 E p/fi+ 118IIm.q.k = fin!€ N IB

- 2lm.q.t

is a norm over the quotient space p/fi.Since this quotient space is finite-dimensional, this norm is equivalent to the quotient norm ll-ll/.r.k and therefore there exists a constant = r, m,q ) such that

c c(1,

B li E p* IB Im,q,k = II8IIm.q.R c cII8II/,r,k= ~ I I l.r,k.

( 3.2.32)

Taking into account the regularity hypothesis and the inverse assumption, we obtain from inequalities (3.2.31) and (3.2.32) and Theorem 3.1.3,

Assume first that q = m, so that there exists a finite element K OE y such that, using (3.2.33),

h

142

[Ch. 3, I 3.2.

CONFORMING FINITE ELEMENT METHODS

Assume next that q < 03. We deduce from inequality (3.2.33) that

Then we distinguish three cases: Either r s q, so that

and inequality (3.2.30) is proved in all cases.

0

Inequalities of the form (3.2.30) are of course immediately converted into inequalities involving the semi-norms I . l m , q . ~ or I . I I . r . ~ if it so happens that the inclusions x h c ~ “ - ~ ( ( . or n ) x h c w’.‘((.n) holds. For example, let us assume that hypothesis (H 3) is satisfied and that the inclusion P C HI(& holds so that the inclusion x h C r l H’(R) holds. Then we have

%‘(a)

C vuh

xh,

vvh E X h ,

(3.2.34)

bh(o.-.RGj;;;Ti(~h(o.R~

1Uhll.R s ~ l ~ h l O , ~ e, t c . .

.

If now hypothesis (H3) is satisfied and if the inclusion holds, then we get similarly C vuh E xh, ( ~ h ~ l , m . ~ 2 ~ h ( ~ h ( 0etc. . c n ,.R. l

(3.2.35)

@ c W’.yk) (3.2.36)

Another observation is that similar inequalities between izorms can be directly derived from these inequalities. For instance, we obtain

Ch. 3, 5 3.2.)

APPLICATION TO SECOND-ORDER PROBLEMS

143

Remark 3.2.4. Inverse inequalities can be likewise established between the above semi-norms and other semi-norms or norms, such as II*IIL~(n. In this direction, see Exercise 3.2.4. 0

Exercises 3.2.1. The object of this problem is to indicate a way of approximating the solution of a nonhomogeneous Dirichlet problem (see also the next problem) whose solution u E H ' ( R )satisfies (cf. Exercise 1.2.2)

i

( u - uo) E Hd(R),

vu E Hd(W,

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

where uo is a given function in the space H ' ( R ) , and the forms a(., .) and f ( - ) satisfy the usual assumptions of the Lax-Milgram lemma, the bilinear form being assumed to be symmetric in addition. Given a finite element space x h , we let as usual XOh ={vh

E X h ;

(i) Given a function Uh

Ex

h

r

Uoh

uh

=o

on

r}.

E x h , show that the discrete problem: Find

such that U h E UOh+XOh={VhEXh; v u h

EXOh,

(Uh-UOh)EXOh}

a(Uh9 u h ) = f ( u h ) .

has a unique solution. (ii) Show that (STRANG & FIX(1973, p. 200)):

(iii) Assume that the spaces x h are made up of n-simplices of type ( k ) . Indicate how should one choose the function U 0 h in order that

IIu - U h l I I . R =

o(hk),

assuming the functions u and uo are sufficiently smooth. 3.2.2. This problem describes a penalty method for approximating the solution of a nonhomogeneous Dirichlet problem, whose solution u E H'(R) satisfies (cf. Exercise 1.2.2):

t

( u - uo) E H ; ( o ) , vu E

Hd(R), a ( u , u ) = f ( u ) ,

144

[Ch. 3, 8 3.2.

CONFORMING FINITE ELEMENT METHODS

where uo is a given function in the space H ' ( R ) and where for simplicity, we shall assume that a(u, u ) =

I

&u&u dx,

R i=l

f(u)

= fRfu

dx, f € L2(R).

In what follows, we consider a family of finite element spaces x h . We are also given a family of real numbers E ( h ) > 0 with limh4 e ( h ) = 0. (i) Show that, for each h, the discrete problem: Find uh E x h such that Vvh E x h ,

I

a(uh9 uh)+(uh - u0)Uh d y 4h) r

= f(Uh).

has a unique solution. (ii) Assume that the solution u is in the space H Z ( R )Show . that, for all v h E xh, 1 Iu-uhI:.R+-(Iuh-u0112L2(r)=

4h)

a(uh-u, uh-u)

+Lj (Uh-fd)(Uh-U)dy 4h)

-Ir

r

& u ( u ~, u )dy a#(u - V h ) dy.

-

Using this identity and the inequality ab d q a 2 + ( 1 / q ) b 2 valid for all q > O , derive the following abstract error estimate: There exists a constant C independent of h such that

Iu - uhll.Rs

c

inf

'JhsXh

Iu - UhI:.R+-1IUO-

1

4h)

2

uhII~2(n

(iii) Assume that the spaces x h are made up of n-simplices of type (k) and that the solution u is sufficiently smooth. Show that, for some constant C independent of h,

and thus deduce the optimal choice for E(h), so as to maximize the order

Ch. 3. 5 3.2.1

APPLICATION TO SECOND-ORDER PROBLEMS

145

of convergence (therefore, as far as the order of convergence is concerned, the method proposed in the previous exercise is preferable). 3.2.3. The purpose of this problem is to analyze a procedure of CLEMENT (1975) for defining an operator whose approximation properties are similar to those of the standard interpolation operator but which can be defined in more general situations. For simplicity, we shall consider finite element spaces xh made up of triangles of type (I), but the analysis can be extended to triangular finite elements with polynomials of higher degree. With each vertex bi, 1 d i d M , of the triangulation, we associate a basis function w iE X h in the usual manner, i.e., one has

w i ( b j )= aij, 1 d i, j S M.

For each i, we set

si = supp w;. Given a function u E L 2 ( R )we , let Piu denote, for each i = 1 , . . . , M , the projection of the function u in the space L2(Si)over the subspace P l ( S i ) , i.e., one has

PiuE P l ( S i ) and Vp E Pl(Si), and we set

I,

( u - Piu)p dx = 0,

M

Piu(bi)wi.

rhu = i=l

In this fashion, we have defined a mapping In the sequel, we consider a family of spaces xh associated with a regular family of triangulations. (i) Show that there exists a constant C independent of h such that

Vi E [ 1 , MI, K

C

S; 3 diam Si s ChK,

and that there exists an integer v independent of h such that

Vi E [ I , MI, card{K

€ y h ;

K

c S i } s v.

(ii) Show that there exists a constant C independent of h such that

Vi E [ I . MI, Vu E H ' ( S i ) ,

146

[Ch. 3, 8 3.2.

CONFORMING FINITE EL E M ENT METHODS

OC ( u - Piulm,siCC(diam Si)’-m~u~,.si,

msI c ~ .

(iii) Show that there exists a constant C independent of h such that

V K E y h , vp E Pl(K), Ip I m,a.K c C(meas(K))-”2him(pJO.K, m

= 0,

I.

(iv) Show that there exists a constant C independent of h such that

Vi E [ I , MI, I w ; ~ , , , ~ s C(meas(K))”*hg”, m

= 0,

I.

(v) Show that

vu E L’(R),

{

Iu - r h U I O . R s CIUIO.R,

lim Iu - rhulo.n= 0, h 4

v u E H’(R),

I/?

where C denotes as usual various constants independent of h. [Hint: Let K E F,, be a triangle with vertices bi, 1 s i s 3 . Prove the identity 3

(rhu - u ) l K = (plu - u ) l K

+

i=2

(piu(bi) --

PIu(b;))WilK

and use the previous questions to estimate appropriate semi-norms of the functions (Plu- u ) l ~and 23=2 (Piu(bi)- P , u ( ~ ~ ) ) w ~ J K . I (vi) If the function t’ belongs to t h e space Hh(R),is the function rhu in the space X,, = { D,, E X , , ; vh = 0 on r)? (vii) Apply the results of question (v) to the approximation of a second-order boundary value problem. Compare with Theorems 3.2.2 and 3.2.5. 3.2.4. Let there be given a family of triangulations which satisfies hypothesis (H 2) and an inverse assumption. It is also assumed that P C %“(K).Then show that for each p E [ I , m], there exists a constant C = C ( p ) independent of h such that V u h E Xh.

llt’hllL”
C p(UhlO.p.0.

Ch. 3.

B

UNIFORM CONVERGENCE

3.3.1

147

3.3. Uniform convergence A model problem. Weighted semi-norms J.lb:m.n

For ease of exposition, we shall simply consider the homogeneous Dirichlet problem for the operator -A, which corresponds to the following data:

(3.3.1)

Assuming that d is a convex polygonal subset of R2, we shall restrict ourselves to finite element spaces xh which are made up of triangles of type ( l ) , so that the corresponding discrete problems are posed in the (results concerning the use of trianspaces v h = { U h E xh; uh = 0 on gles of type (k) and higher dimensions are indicated at the end of this section and in the section “Bibliography and Comments”). We shall assume once and for all that we are given a family of triangulations of the set d which (i) is regular and (ii) satisfies an inverse assumption, i.e., there exist two constants u and v such that

r}

(3.3.2)

Our main tool in the study of the error in the norms ( * l O , m , ~ and J(.Jll.m,~ will be the consideration of appropriate weighted norms and semi-norms. Accordingly, the first part of this section will be devoted to the study of those properties of such semi-norms which are of interest for our subsequent analysis (cf. Theorems 3.3.1 to 3.3.4). Given a weight-function 4, i.e., a function which satisfies

4 E L ” ( R ) and 4 a O a . e . we define, for each integer m

3 0,

on 0,

(3.3.3)

the weighted semi-norms (3.3.4)

I48

CONFORMING FINITE ELEMENT METHODS

[Ch. 3.

B 3.3.

To begin with, we observe that, if the function 4-I exists and is also in the space L " ( 0 ) ,an application of Cauchy-Schwarz inequality gives V a E R , V u , u E H 1 ( 0 ) , a ( u , u ) S lul+~:l.nluJ+-a:l.~.(3.3.5)

Departing from the general case, we shall in fact concentrate our subsequent study on weighted semi-norms of the particular type (Y E R, where the function 4 is of the form (3.3.7) below. Our first task is to extend to such weighted semi-norms the property that there exists a constant cI,solely dependent upon the set 0,such that

I.ld~:m.R,

V W E f f d ( 0 ) n H2(a),IwIz.RS cilAwlo,n.

(3.3.6)

Theorem 3.3.1. There exists a constant CI = C1(a)such that, f o r all functions 4 of the f o r m

4:xEcL4(x)=

1

IIx - zl12+ e2'

8 > 0,

2 = (ZI, 22) E R2,

(3.3.7)

(3.3.8) Proof. Let u be an arbitrary function in the space H d ( 0 ) n H W ) . Then the function w = (XI - 2 l ) U

also belongs to the space H A ( 0 ) n H 2 ( 0 ) ,and

a1,w - 2 a l q - 2l)al2u = a12w - azu,

(XI - 2 , ) a , , u= (XI

= azzw, AW = ( X I - .TI)Au + 2 8 1 ~ .

( X I - Pl)a22u

Using these relations and inequality (3.3.6), we find a constant c2 such that

Ch. 3,

P 3.3.1

UNIFORM CONVERGENCE

149

Since we have likewise

we eventually obtain

As exemplified by the above computations, we shall depart in this section from our practice of letting the same letter C denote various constants, not necessarily the same in their various occurrences. This is due not only to the unusually large number of such constants which we shall come across, but also - and essentially - to their sometimes intricate interdependence. Therefore, constants will be numbered and, in addition, their dependence on other quantities will be made explicit when necessary. However the possible dependence upon the set R and the constants u and Y of (3.3.2) will be systematically omitted. While we shall use capital letters Ci,i 2 1 , for constants occurring in important inequalities, small letters ci, i L 1, will rather be reserved for intermediate computations. In the next two theorems, we examine the relationships between the weighted semi-norms I*Jd=:m,a (the function 4 being as in (3.3.7)) and the standard semi-norms I . l m , m . n . Such relationships will play a crucial role in the derivation of the eventual error estimates.

Theorem 3.3.2. For each number a > 1 and each integer m 3 0, there exists a constant C2(a,m ) such that, f o r all functions 4 of the f o r m

4 :x € 6 + ( x )

=

1 IJx - -f1I2 + e*' e > o , n ~ 6 ,

(3.3.9)

we have 1 Vu E WmTm(fl),1 U J ~ " ; m . RG~ ( a ,m ) F l u l m , - , n .

(3.3.10)

For each number p €10, 1 [ , and each integer m 3 0 , there exists a

150

CONFORMING FINITE ELEMENT METHODS

[Ch.3. 5 3.3.

constant C3(p,m) such that f o r all functions 4 of the form (3.3.9), we have

ve s p,

vv E

wrn-”(n), Ivl+;,.,,n =sc,(P, m)lln 011/21Ulrn.m.n. (3.3.1 1)

Proof.

Clearly, one has

Next let 6 = diam(O), so that

If a > 1 , write

and inequality (3.3.10) is proved with C 2 ( a ,m ) = c 3 ( m ) ( d ( a- l))”2. If a=l,wehavefor 8 S p < 1 ,

with

and inequality (3.3.1 1) is proved with C3(/3,m) = ~ ~ ( m ) ( 2 r c @ ) ) ” ~ . 0 We next obtain inequalities in the opposite direction. In order that they be useful for our subsequent purposes, however, we shall establish these inequalities only for functions in the finite element space Xhrand further, we shall restrict ourselves to weight-functions of the form 4 or d 2 , with 4 as in (3.3.9), for which (i) the parameter 8 cannot approach zero too rapidly when h approaches zero (cf. (3.3.13)), and for which (ii) the points 3 depend upon the particular function uh E X h under consideration (cf. (3.3.13 and (3.3.17)).

Ch. 3, 0 3.3.1

UNIFORM CONVERGENCE

151

Theorem 3.3.3. For each number y > 0 , there exist constants C d y ) and C,(y) such that, i f for each h, we are given a function c b h of the form

in such a way that (3.3.13)

37 > 0, V h , 6 h 2 yh, then (i) we have V u h

Ex

h ,

if, f o r each function such a way that Ivh(xh)(

IvhlO.m,flC

l)h

6,’

C4(Y’)~I~hI&;O,fl

E x h , the point

xh

(3.3.14)

E d in (3.3.12) is chosen in (3.3.15)

= (vh(O.m.fl,

and (ii) we have v u h

Exh,

if, for each function such a way that

l ~ h I l p , f l C CS(Y)h o lhv h l & : I . O

uh

maxi181v h ( x h ) l ,

E x h , the point

xh

1 8 2 u h ( x h ) ( } = 1 vh I

(3.3.16)

E d in (3.3.12) is chosen in

I.m.fl.

Proof. (i) Let u h be an arbitrary function in the space point x h be chosen as in (3.3.15). We can write

(3.3.17)

xh,and k t the

for some constant c5 (in the last inequality we have used the fact that the family of triangulations satisfies an inverse assumption; cf. Theorem 3.2.6). In other words,

152

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 8 3.3.

and consequently ( B ( a ; r ) = { x E R2;Ilx - all C r}),

The set

d being polygonal,

there exists a constant

c6

such that

We also have

and

by assumption (3.3.13). Combining the previous inequalities, we obtain an inequality of the form (3.3.14) with

(ii) Let v h be an arbitrary function in the space xh, let the point x h be chosen as in (3.3.17), and let K h E y h denote a triangle which contains the point xh. Since the gradient v v h is constant over the set Khrwe deduce dx lUhl$h:l.n5

lUhli.m.OfKh

1lx - xh112+ 4;.

With this inequality and the inequalities meas Kh3 c d u , u ) h 2 ,

we obtain an inequality of the form (3.3.16) with C 4 Y )=

Jm.

0

Ch. 3, 0 3.3.1

UNIFORM CONVERGENCE

153

T o conclude this analysis of weighted semi-norms, we examine in the next theorem the interpolation error estimates in the semi-norms (-I+g:,,,.n where, for each h, the function +h is of the form (3.3.12). The conclusion (cf. (3.3.20)) is that the error estimates are exactly the same as in the case of the usual semi-norms I.lrnTn provided the parameter 6 h does not approach zero too rapidly with h (cf. (3.3.19)). Notice, however, that if the behavior of the function 6h can be “at best” linear a s in the previous theorem, the constant which appears in inequality (3.3.19) is not arbitrary, by contrast with the constant y which appeared in inequality (3.3.13). Finally, observe that no restriction will be imposed upon the points xh.

Theorem 3.3.4. There exists a constant c 6 and, f o r each a E R - {o}, there exist constants C,((Y)> 0 and Cs(a) such that, if for each h, we are given a function +h of the form (3.3.18)

in such a way that Vh,

6h 3

(3.3.19)

C,(a)h,

then (i) we have

Proof. (i) There exists a constant cs such that

V u E H Z ( 0 ) , V K E Y,,, ( u - nKu ( , . ~s C & ~ - ~ ( U ) Z . K , rn = 0, 1. Next, we have IV

-n K Ul4R:m.K

Iu(2.K

(4t(XK))'/21u- n

K u(rn,Kv

(4t(XK))-'/Z)U16R:2.K~

where, for each K E Y,,, the points x K E K and X K E K are chosen in

154

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 5 3.3.

we obtain

and therefore

Consequently, if we let

the conjunction of the above inequalities yields inequality (3.3.20) with

ca= f i e * .

%'(a)

(ii) Since the function 4tuh is in the space = dom n h and since the restrictions 4 E u h l K belong to the space H 2 ( K )for all K E Yh, the same argument as in (i) shows that

Ch. 3. § 3.3.1

UNIFORM CONVERGENCE

155

Using the inequalities

we deduce that

we conclude that there exists a constant

c9(a) such

that

and thus,

Therefore we have proved inequality (3.3.21), with C,(a) = C , d / c s o .

0

Uniform boundedness of the mapping u + u h with respect to appropriate weighted norms After the above preliminaries, we now come to the central object of this section, i.e., the estimate of the errors Iu - uhlo.m,n and ( u - u h l l . m . f l via the method of weighted norms of J.A. Nitsche. The analysis will comprise three stages. In the first stage (cf. the next theorem), we consider for each h the projection operator Ph: u E Hd(n)*Phu E v h

(3.3.22)

associated with the inner product a ( . , .) of (3.3.1), and which is therefore

I56

CONFORMING FINITE ELEMENT METHODS

[Ch. 3.

B

3.3.

defined for each u E Hd(0 by the relations

phv E

v h

vwh E

and

vh,

a(U - P h V , wh) = 0.

(3.3.23)

Thus we have in particular uh = phu, where uh is the discrete solution found in the space v h and u is the solution of the problem defined in (3.3.1). We shall then show that for an appropriate choice of the parameters Oh in the functions 4 h (cf. (3.3.25) and (3.3.26) below), the mappings Ph are also bounded independently of h when both spaces Hd(0) and v h are equipped with the weighted norm

(3.3.24)

Theorem 3.3.5. There exist three constants h o E ] O ,1[, C9> 0 and Clo such that, if f o r each h, we are given a function +h of the form (3.3.25)

in such a way that Vh,

Oh

(3.3.26)

= C9hIln h ) 1 / 2 ,

Proof. For convenience, the proof will be divided in four steps. (i) There exist two constants C I Iand CI2such that, if

V h , & 3 Clih,

(3.3.28)

then

v v Hd(0), For brevity, let = phu.

Since

lPhui$h:I.nd

c12(IphulifXl.R+

(‘1ihh:l.d.

(3’3.29)

Ch. 3, 5 3.3.1

157

UNIFORM CONVERGENCE

and

we deduce that

1 V h I2&,h:I.R

a ( V h r '$huh)

(3.3.30)

+ 2)Vh l:$:O.t2-

Using relations (3.3.23) we can write a(Uh9 4 h V h )

=

a(uh

- u, &huh

-nh(4huh))+

An application of inequality (3.3.5) with a

=

(3.3.31)

a(u, 4hUh).

1 shows that

By Theorem 3.3.4, we have that, if

Vh,

then (cf. inequality (3.3.21) with a

I+huh

(3.3.32)

Oh b cloh, with clo= C7(1),

-n h(

h u h

)I&61:I,R

=

1)

CI I

-Oh (Ih

I

I I ; I .d?

uh 6f;O.n+ Uh

6h

with cI1= C8(l). Combining the previous inequalities, we find that, for Oh 2 cloh, la(uk

- u, 4 h U h - n h ( 4 h u h ) ) l s

By another application of inequality (3.3.5) with a = 1 , we obtain a(u, &huh)

~~16h;I,Rl&h~h~6h1;l.R.

(3.3.34)

Using the inequality

(3.3.35) (3.3.35)

158

[Ch. 3, 5 3.3.

CONFORMING FINITE ELEMENT METHODS

Combining relations (3.3.30) to (3.3.33, we have found that, for cioh,

O h B

i.e., an inequality of the form

h A'S 2C2+ cI~B(A+ C ) + cll-(A Oh

+ B)(A + C ) .

Assuming

Vh,

(3.3.36)

32 ~ l I h ,

we get A' c 4C2+ (1 + 2clz)BC + A ( ( l + 2clz)B + C ) A2 c 4c2+ - + c12 (B2 + C2) + + (1 + 2c12)2B2+ c2, 2

G

)

and therefore step (i) is proved with (cf. (3.3.32) and (3.3.36)) (3.3.37)

CII= max(clo,2~11) in relation (3.3.28) and C12= max{l1+ 2 ~ 1 2(1. + 2cI2)(3+ 4~12))

in relation (3.3.29). (ii) There exists a constant C13such that, if we assume constant CI1has been determined in step (i)), we have

(3.3.38) Oh

3 Cllh

(the

vu E Hd(R),IPhvI$f;O.fl+

IPhvI$h:*,flc

cl,(lv

where, for each h,

@h

I:f:o.fl

+ I v l $ h i l , f l + h21

@hI$i';2d?

(3.3.39)

= @ , , ( l ) ) is the soh.ition of the variational problem:

(3.3.40) Notice that because the set R is assumed to be convex, the function

Ch. 3, 0 3.3.1

UNIFORM CONVERGENCE

159

$h is in the space H2(f2) and therefore, it is legitimate to consider the semi-norm I* I+;I ;2.nin inequality (3 3.39). Using the definition of the function $h, and letting again u h = p h u , we can write (3.3.41) Il)hI:f:o.n = a ( u h - V , $h - n h $ h ) + 4 ' , u h u dx-

I,

By applying inequality (3.3.5) with (Y = 1 and inequality (3.3.20) with a = - 1 (this is possible because we assume o h a Cllh and CII2 cI0=

C7(- 1); cf. (3.3.32) and (3.3.37)), we obtain

(3.3.42)

so that, by combining relations (3.3.41), (3.3.42) and (3.3.43), we obtain the inequality

I u h I26f:O.R

C 7 M l ~ h l d h : I . R+ l ~ l ~ ~ : l . R ) l J l h I 6 5 ; ~ : 2 , n

1

+3 (I uhl$:o.n + Iul&:o.n), which in turn implies the inequality

(3.3.44) Let then 6 = 1/(3Cl2),where CI2is the constant appearing in inequality (3.3.29). The corresponding inequality (3.3.44) added to inequality (3.3.29) times the factor 2/(3CI2)yields

i.e., an inequality of the form (3.3.39).

160

CONFORMING FINITE ELEMENT METHODS

[Ch. 3, 5 3.3.

(iii) Given any number 8oE]0,1[, there exists a constant Cl4(O0)such

1

Iv h l$i;O.O

Iv h I$i;O.O*

(3.3.47)

To take care of the other term which appears in the right-hand side of inequality (3.3.46), we shall prove that f o r each number O0 E 30, 1[, there exists a constant cl3(O0)such that, f o r all functions 4 of the f o r m

4: x E 6 +(b(X)

=

(Ix

1

+ e2'

- 2112

2 E 6,

o < 8 s eo,

(3.3.48)

we have V + E Hd(0) n H 2 ( 0 ) , l+l?,~s C I ~ ( lln ~ O81) ~ I A I , ~ I (3.3.49) $-~;O.~.

Taking into account that lA+h14i2:o.R

= IVhl42,:o.a.

and applying inequalities (3.3.49) (with I,b = I,bh and 4 = & ) , (3.3.46), and (3.3.47), we then find an inequality of the form (3.3.43, with

cl4(e0) = cl(m 1

+ cl3(eO)).

(3.3S O )

It therefore remains to prove relation (3.3.49) (another method for proving the same relation is suggested in Exercise 3.3.1). Given an arbitrary function E H d ( 0 ) n H 2 ( 0 ) ,we have

+

(3.3.51)

Ch. 3,

P 3.3.1

UNIFORM CONVERGENCE

161

Let then G denote the Green’s function associated with the operator - A in R and the boundary condition u = 0 on R, so that

X

{IR

42(v)G(x, 7 ) ds} dx} d5.

There exists a constant P. 143))

(3.3.52)

such that (cf. for example STAKGOLD (1968,

~ 1 4

Vx, y E 6, x # y, 0 c G(x, y ) S c1411nIIx - ylll.

(3.3.53)

Using this inequality, we proceed to show that for arbitrary points x, 2 E R and for any number 8 with OC 8 C 8,< 1, there exists a constant c15(8,)such that

To see this, write

where

R, = ~ d x0), = (77 E 0 ; 1177 - xII s 81,

n2= R ~ ( x8, ) = {77 E 0 ;8 c 1177 -XI] R3 = R,(x)

= (17

c I},

E 0; 1 s 1177 - xll}.

We then obtain the following inequalities (observe that the last two inequalities make sense only if the sets R2 and R 3 are not empty, and that we have diam R 3 1 if the set 0,is not empty):

162

[Ch. 3, 0 3.3.

CONFORMING FINITE ELEMENT METHODS

Consequently, inequality (3.3.54) is proved, with

The conjunction of inequalities (3.3.5 1) to (3.3.54) then implies inequality (3.3.49) with 1

cl3(e0)= 2 (1 + ~ ? ~ ( e ~ ) ) . (iv) It remains to combine the results of steps (ii) and (iii): W e have determined constants CII,CI3 and Cl4(e0)for each O O E ] O , 1[ such that (cf. inequalities (3.3.39) and (3.3.45))

Cllh s o h

cO< 1

)PhuI$gO,R+ IPhvl$,,;l,f2

[In OhJh2

s C,~(IUI:~:~.~ + l ~ l $ , , : ~+,c13c,4(eo) ~) ___ 1Phu 1 $f.O.R6:

(3.3.55) Let for example O0 = 4 and let

(3.3.56) Then there exists a number ho E 10, 1 [ such that

(3.3.57)

Ch. 3, 8 3.3.1

UNIFORM CONVERGENCE

I63

This being the case, we have found an inequality of type (3.3.27) with ClO =

2

c

.

(3.3.58)

0

Estimates of the errors Iu - Uh(O.m,R and ( u - Uh1l.m.R. Nitsche's method of weighted norms We next develop the second stage of our analysis. Using the inequalities (cf. Theorems 3.3.2 and 3.3.3) between the semi-norms l-lm,m,~, m = 0 , 1, and the weighted semi-norms which appear in inequality (3.3.27), we show in the next theorem that the projection operators Ph of (3.3.22), considered as acting from the subspace Hd(0)fl Wl+"(0) of the space H d ( 0 ) onto the space vh, are bounded independently o f h when the space Hd(0) n W'*"(0)is equipped with the norm 21

+

IuI0.m.R

and the space

v h

Theorem 3.3.6.

(u(i.m.~

(3.3.59)

is equipped with the norm

u + (In h

Remark 3.3.1. norms.

+ hlln hl

1-' 21

v IO,m,R

+ h I 1 I,m,R.

(3.3.60)

Such norms may be viewed as "weighted W'*"(R)-like"

0 There exists a constant CISsuch that

V h c ho, V u E H d ( 0 ) n W'*m(0), lln hl-''21Phu10.m,R+ hlPhull,m,R CdIvI0,m.n + hIln hI IuIl.m.R),

(3.3.61)

where the constant ho> 0 has been determined in Theorem 3.3.5. Proof. Let there be given a function u in the space Hd(0)flW'."(0). For each h c ho, we define the function

(3.3.62)

164

CONFORMING FINITE ELEMENT METHODS

[Ch. 3. 5 3.3.

where ho and C9 are the constants determined in Theorem 3.3.5. Since o h 3 C I I for ~ h s ho (cf. (3.3.57), we may apply inequality (3.3.14)): There exists a constant cl6

(3.3.64)

= c4(cll)

such that lPhu(0.m.R

c16

0;

(3.3.65)

IPhU(&:O,fl-

By inequality (3.3.27), Clo((u(&:o.fl

(PhU(&h:o.fl

+ lu(&h:l.n)7

(3.3.66)

and by inequalities (3.3.10) and (3.3.1 I ) , there exists a constant ( o h C O0 =

4 for h s ho: cf. (3.3.57)) cl7

=

max{c2(2,o),

C& 1))

(3.3.67)

such that Iul&:o.fl

+ IUl&h:l.flS

1

+ Iln eh11/21ull.m.o).

c ~ 7 (o-h ~ ~ ~ o , m . ~

(3.3.68)

Combining inequalities (3.3.65) to (3.3.68), we find that

Using the relation &, = C9h(In (cf. (3.3.26)) and the inequality (In &,I s 21111 h ( (cf. (3.3.57)), we eventually get, for all h c ho,

lln

hl-'/2)Phu10.m.fl

cI8

=

CldlulO.m.fl + hlln

hl lul~.m,d9

(3.3.69)

with cIOc16c17

Likewise, for each h

S

max{C9, d5 c;}.

(3.3.70)

h,, define the function

(3.3.71)

Ch. 3, § 3.3.1

165

UNIFORM CONVERGENCE

Then there exists (cf. inequality (3.3.16)) a constant (3.3.73)

cl9 = c5(cII)

such that IPhuIl.m.0

oh

(3.3.74)

cl9h IPhul,lh:I.O,

and, by inequality (3.3.27), IPhulblh:I.fl

cIo(Iuld:h:o.n

+ Iuldlh:l,a)’

(3.3.75)

Then, arguing as before, we get, for all h s ho, (3.3.76)

. R hl IvI1,m.n) hlPhvll.=.ns C Z O ( ) ~ ~ O+, ~hlln

with cz0=

~

~max{l,~fi GI.c

~

~

c

(3.3.77) ~

The conjunction of inequalities (3.3.69) and (3.3.76) implies inequality (3.3.61) with

0

CIS = CIS + c20.

Remark 3.3.2. In Theorem 3.3.5, the behavior of the function Oh was somehow “bounded below” by a constant times (hlln hI’’*). The key to the success of the present argument was that such a function o h tends nevertheless sufficiently rapidly vers zero with h so as to produce the right factors (as functions of h ) in the inequalities (3.3.69) and (3.3.76).

0 In the third -and final - stage of our study, the uniform boundedness of the projection mappings p h which we just established allows us in turn to easily derive the desired error estimates (recall that the discrete solution uh is nothing but the projection Phu of the solution u ) . Theorem 3.3.7. Assume that the solution u E Hd(0) of the boundary value problem associated with the data (3.3.1) is also in the space W2.”(0). Then there exists a constant C independent of h such that lu - Uh(O.m.0s Ch2/lnh13”Ju12.m.n,

(3.3.78)

Iu - Uhll.=.R

(3.3.79)

Chlln hl

IUIZ.~.~.

~

166

[Ch. 3. B 3.3.

C O N F O R M I N G FINITE ELEMENT METHODS

The norm of the identity mapping acting from the space H,,'(O)n W'*"(O)equipped with the norm of (3.3.59) into the same space, but equipped with the norm of (3.3.60), is bounded above by 11n h01-1'2for all h s hh = min{ho, l/e}. Next we have the identity

Proof.

vuh

E

vh,

u

- uh = u - Phu

= (1 - P h ) ( U - O h ) ,

so that we infer from Theorem 3.3.6 that, for all h Iln h l - I ' * ( U --

uhlO.m.0

S

hh,

+ hlu - uhllp.0

s (IIn h01-1/2 + cis) inf 1%

(lu - UhlO.a.0

E vh

+ h(In hl ( u - uhll.=.n)

Since there exists a constant c21such that

inequalities (3.3.78) and (3.3.79) follow with

C = czl(lln hol-"'

+ cis).

0

In fact, the error estimate of (3.3.78) is not optimal: J.A. NITSCHE (1976b) gets the improved error bound Iu - u h l O . m . 0

s Ch211nhl I ~ J 2 . m , 0 r

(3.330)

at the expense, however, of a technical refinement in the argument, special to triangles of type (1). At any rate, the discrepancy between (3.3.78) and (3.3.80) is somehow insignificant: Both error estimates (3.3.78) and (3.3.80) show an O(h2-')convergence for any E > 0. To conclude, we point out that all the essential features of Nitsche's method of weighted norms have been presented: Indeed, the extension to more general cases proceeds along the same lines. In particular, the use of higher-order polynomial spaces (i.e., Px = P k ( K )for some k Z= 2, n arbitrary) yields a simplification in that the "lln hl" term present for k = 1 disappears in the norms then considered. Thus inequality (3.3.61) is replaced by an inequality of the simpler form (cf. NITSCHE(1975)) IPhUlO.m.0

+ hlPhvIl.m.0

C(lvlO.m.0 + h l U l l . m . 0 ) .

(3.3.8 1)

Such inequalities are obtained after inequalities reminiscent of that of (3.3.27) have been established for appropriate weighted norms of the

Ch. 3, 5 3.3.1

I67

UNIFORM CONVERGENCE

form I - J ~ ~ + ~ :+o ,l*14g~l.~, o (n/2) < a < ( 4 2 ) + I , with functions 4,, again defined as in (3.3.25).

Exercises 3.3.1. Following NITSCHE(1977), the object of this problem is to provide another proof of inequality (3.3.49). i.e., that for any OOE 30,1[, there exists a constant ~ ( 6 , such ) that, for any function 4 of the form fEfi.

0 < e =s 00,

we have

(i) Let

A(O)=

inf

lA+l :-l.o.n

-

IL€Hh(flNlH2(0)

I$l:.fl

and show that A ( 0 ) is the smallest eigenvalue of the eigenvalue problem

[

-Au = A ~ * in u 0, u =Oonr,

so that A ( 0 ) is a strictly positive quantity (references about eigenvalue problems can be found in the section “Additional Bibliography and Comments” at the end of Chapter 4). (ii) Let 6 = B ( f ; diam (0)) and show by a direct computation that

(iii) Conclude, by using the implication

01C 0 2 3 h(Rz) A(O1). 3.3.2. The object of this exercise is to show how an error estimate in the norm I.lo.mn can be quickly derived, once one is willing to accept a poorer order of convergence than that obtained in Theorem 3.3.7. The terminology is the same as in Section 3.2. In addition to ( H l ) , (H2) and (H3), assume that s = 0 , that the

168

CONFORMING FINITE ELEMENT METHODS

[Ch. 3,

P

3.3.

dimension n is G3, that the family of triangulations satisfies an inverse assumption, and that the inclusions

P , ( I z )c F c H’(EZ) hold (notice that by hypothesis (H3), we also have fi c L“(k)). Let then uh be the corresponding discrete solution which approximates the solution u of a general second order boundary value problem of the type considered in Section 3.2. Show that if the adjoint problem is regular, and if u E H2(L!)n V , there exists a constant C independent of h such that Iu

- UhlO.m.0

n = 2, d(uIz.n i if n = 3.

6 Chlu12,n if

Iu - uhl0.rn.n G

~

[Hint: write Iu - UhlO.m,R luh - nhul0.rn.R + J u appropriate inverse inequality for the first term.]

and use an

Bibliography and comments 3.1. The content of this section is essentially based on, and slightly improved upon, CIARLET& RAVIART(1972a). In particular, R. ArcangCli suggested the simpler proof of inequality (3.1.33) given here. For reference about the Sobolev spaces W m . p ( 0and ) their various properties, see ADAMS(1975), LIONS(1962), NECAS (1967), ODEN & REDDY(1976a, chapter 3). Theorem 3.1.1 was originally proved in DENY & LIONS(1953-1954) for open sets which satisfy the “cone property” (such sets are slightly more general than those with Lipschitz-continuous boundaries). An abstract extension of this lemma is indicated in Exercise 3.1.1. There has been considerable interest in interpolation theory and approximation theory in several variables during the past decade, one reason behind this recent interest being the need of such theories for studying convergence properties of finite element methods. Special mention must be made however of the pioneering works of P ~ L Y(1952) A and SYNGE (19571, for what we call here rectangles of type ( 1 ) and triangles of type ( l ) , respectively. The “classical” approach consists in obtaining error estimates in Cm-norms. In this direction, see the contributions of BARNHILL&

Ch. 3.1

BIBLIOGRAPHY AND COMMENTS

169

GREGORY(1976b), BARNHILL& WHITEMAN(1973), BIRKHOFF(1971, 1972), BIRKHOFF,SCHULTZ & VARGA(1968), CARLSON& HALL (1973), CIARLET& RAVIART(1972a), CIARLET& WAGSCHAL(1971), COATMBLEC (1966), LEAF& KAPER (1974), NICOLAIDES (1972, 1973), NIELSON (1973), SCHULTZ (1969b, 1973), STRANG (1971, 1972a), AENfSEK (1970, 1973), ZLAMAL(1968, 1970). Although in most cases a special role is played by the canonical Cartesian coordinates, a more powerful coordinate-free approach, using FrCchet derivatives, can be developed, such as in CIARLET& RAVIART(1972a), CIARLET& WAGSCHAL(1971), where the interpolation error estimates are obtained as corollaries of multi-point Taylor formulas (Exercise 3.1.2). See also COATMBLEC (1966). Another frequently used tool is the kernel Theorem of SARD (1963). Some authors have considered the problem of estimating the constants which appear in the interpolation error estimates. See A R C A N G ~ L I & GOUT (1976) (cf. Exercise 3.1.2), ATTEIA(1977), BARNHILL & WHITEMAN (1973), GOUT (1976), MEINGUET(1979, MEINCUET & DESCLOUX (1977). The approach in Sobolev spaces which has been followed here has been given much attention. In this respect, we quote the fundamental contributions of BRAMBLE& HILBERT(1970,1971), BRAMBLE & ZLAMAL (1970). Other relevant references are AUBIN(1967a, 1967b, 1968a, 1968b, 19721, BABUSKA (1970, 1972b), BIRKHOFF,SCHULTZ & VARGA (1968), BRAMBLE(1970), CIARLET& RAVIART(1972a), FIX& STRANG (1969), D I GUGLIELMO (1970), HEDSTROM& VARGA(1971), K O ~ K A (1973). L NITSC HE (1969, 1970), SCHULTZ (1969b), VARCA(1971). Interesting connections with standard spline theory can be found in ATTEIA (19751, MANSFIELD (1972b), NIELSON(1973) and, especially, DUCHON (1976a, 1976b). The dependence of the interpolation error estimates upon the geometry of the element (through the parameters hK and p K ) generalize Zlamal’s condition, as given in ZLAMAL(1968, 1970), and the “uniformity condition” of STRANC (1972a). JAMET (1976a) has recently shown (cf. Exercise 3.1.4) that, for some finite elements at least, the regularity condition given in (3.1.43) can be replaced by a less stringent one. In a special case, the same condition has been simultaneously and independently found by BABUSKA & Azrz (1976). In essence, it amounts to saying, in case of triangles, that no angle of the triangle should approach T in the limit while by the present analysis no angle should

170

CONFORMING FINITE ELEMENT METHODS

[Ch. 3.

approach 0 in the limit. Incidentally, this was already observed by SYNGE (1957). 3.2. and 3.3. There exists a very large literature on various possible error estimates one can get for conforming finite element method and here we shall merely record several lists of references, depending upon the viewpoints. We shall first observe that almost all the papers previously referred to in Section 3.1 also contributed to the error analysis in the norm ~ ~ - ~ ~ , , n , inasmuch as this simply requires a straightforward application of CCa’s lemma, just as we did in Theorem 3.2.2. Even though the Xh-interpolation operator cannot be defined for lack of regularity of the function to be approximated, an approximation theory can still be developed, as in CLEMENT (1975), HILBERT(1973), P I N I(1974), STRANG (1972a). See Exercise 3.2.3 where we have indicated the approach of Ph. Clement. Historically, the first proof of convergence of a finite element method, albeit in a special case, seems to be due to FRIEDRICHS (1962). Early works on convergence in the engineering literature are JOHNSON & MCLAY (1968), MCLAY (l963), OLIVEIRA (1968, 1969). The reader who wishes to get general introductions to, and surveys on, the various aspects of the convergence of the finite element method may consult BIRKHOFF& FIX( 1 9 7 4 ) , C ~ ~&~ Hs A oL ~ L(~~~I),CIARLET(I~~~ FELIPPA& CLOUCH (1970), KIKUCHI(1975c), ODEN (1975), STRANC (1972a, 1974b), THOMBE(1973a), VEIDINGER (1974), ZLAMAL(19734. Using a priori estimates (in various norms) on the solution (cf. N E ~ A S (1967) and KONDRAT’EV(1967)). it is possible to get error estimates which depend solely on the data of the problem. See BRAMBLE& ZLAMAL(l970), NITSCHE(1970), OGANESJAN & RUKHOVETS(1969). In the case of the equation -Au = f over a rectangle, BARNHILL& GREGORY (1976a) obtain theoretical values for the constants which appear in the error estimate, which are realistic, as shown in BARNHILL, BROWN, MCQUEEN& MITCHELL(1976). “Nonuniform” error estimates are obtained in BABUSKA& KELLOC (1975), HELFRICH(1976). The case of indefinite bilinear forms is considered in C L ~ M E N T (1974), SCHATZ(1974). DOUGLAS,DUPONT& WHEELER(1974a) give estimates for the flux on the boundary. HOPPE (1973) has suggested the use of piecewise harmonic polynomials, and his idea has been justified by RABIER(1977). See also BABUSKA(1974a), ROSE (1975).

Ch. 3.1

BIBLIOGRAPHY A N D C O M M E N T S

171

There are various ways of treating nonhomogeneous Dirichlet boundary conditions. The most straightforward method is suggested in Exercise 3.2.1. See AUBIN(1972), STRANG & FIX(1973, Section 4.4). THOMEE (1973a). Lagrange multipliers may also be used as in BABUSKA(1973a), as well as penalty techniques (cf. Exercise 3.2.2) a s in BABUSKA (1973b). See also the section “Bibliography and Comments”, Section 4.4. For domains with corners or, more generally, for problems where the solution presents singularities, see BARNHILL& WHITEMAN(1973, 1975), BABUSKA(1972a, 1974b, I976), BABUSKA & ROSENZWEIG (1972), BARSOUM(1976), CIARLET,NATTERER& VARGA (1970), CROUZEIX & THOMAS(1973), DAILEY& PIERCE (1972), FIX (1969), FIX, GULATI& WAKOFF (1973), FRIED& YANG (1972), HENNART& MUND (1976), NITSCHE(1976a), SCHATZ & WAHLBIN(1976a), SCOTT(1973b), STRANG & FIX(1973, Chapter 8), THATCHER (1976), VEIDINGER (1972), WAIT& MITCHELL(197 1). Recent references in the engineering literature are HENSHELL& SHAW(1975), YAMAMOTO & SUMI(1976). For further results concerning the error estimates in the norm ll*ll,,~, see BABUSKA& AZIZ (1972, Section 6.4) where it is notably discussed whether they are the best possible, using the theory of n-widths. Many “abstract” finite element methods, or variants thereof, have been considered, by AUBIN(1967b, 1972), BABUSKA (1970, 1971a, 1971b, 1972b), FIX & STRANG (1969), DI GUGLIELMO (1971), MOCK (1976), STRANG (1971), STRANG & FIX(1971). The inverse inequalities established in Section 3.2 are found in many places. See notably DESCLOUX(1973). The technique which yields the error estimate in the norm ( . I O , ~was developed independently by AUBIN(1967b) and NITSCHE(1968), and also by OGANESJAN & RUKHOVETS (1969). See KIKUCHI(1975~)for a generalization. The subject of uniform convergence has a (relatively) long story. In one dimension, we mention NITSCHE(1969), CIARLET(1968), CIARLET & VARGA(1970), and the recent contributions of DOUGLAS & DUPONT(1973, 1976b), DOUGLAS, DUPONT& WAHLBIN (1975b), NATTERER (1977). For special types of triangulations in higher dimensions, see BRAMBLE, NITSCHE & SCHATZ (1975), BRAMBLE& SCHATZ (1976), BRAMBLE & THOMBE(19741, DOUGLAS,DUPONT& WHEELER(1974b), NATTERER (1975b). The first contribution to the general case is that of NITSCHE (1970). Then CIARLET& RAvrARr (1973) improved the analysis of J.A. Nitsche

172

CONFORMING FINITE ELEMENT METHODS

[Ch. 3.

by using a discrete maximum principle introduced in CIARLET(1970). More specifically, CIARLET& RAVIART(1973) have considered finite element approximations of general second-order nonhomogeneous Dirichlet problems posed over polygonal domains in R". Then the discrete problem is said to satisfy a discrete maximum principle if one has f s 0 .$ ma? uh(x) s max (0, max U h ( X ) } . xa-

XER

In the case of the operator ( - A u + a u ) with a 2 0 and n = 2, it is shown that the discrete maximum principle holds for h small enough if there exists E > 0 such that all the angles of all the triangles found in all the triangulations are d [(7r/2)- €1 (in case a = 0, it suffices that the angles of the triangles be s7r/2). Returning to the general case, it is shown that when the discrete problems satisfy a maximum principle, one has lim Iu - Uhl0.rn.R = o if u E U

4

IU

- UhlO.rn,R = O ( h )

w'.p(0) with p > n,

if u E W 2 @ ( 0with ) 2 p > n,

i.e., there was still a loss of one in the expected order of convergence. Recently, NATTERER(1975a), NITSCHE(1975, 1976b, 1977) and SCOTT (1976a) obtained simultaneously optimal (or nearly optimal) orders of convergence. The greatest generality is achieved in the particularly penetrating analysis of J.A. Nitsche, which we have followed in Section 3.3 (the proof of inequality (3.3.49) is that of RANNACHER (1977)). While weighted Sobolev norms are also introduced by F. Natterer, R. Scott's main tool is a careful analysis of the approximation of the Green's function. The uniform boundedness in appropriate norms of particular Hilbertian projections, on which J.A. Nitsche's argument is essentially based, was also noticed by DOUGLAS,DUPONT& WAHLBIN (1975a) who have proved (albeit through a different approach) the boundedness in the norms I - ( O , q , R , 1 d q d m , of the projections, with respect to the inner-product of the space L 2 ( 0 ) ,onto certain finite element spaces. J.A. Nitsche's technique has since then been successfully extended in several directions, notably to more general second-order boundary value problems by RANNACHER (1976b), to the obstacle problem by J.A. Nitsche himself, to the minimal surface problem and other nonlinear problems by J. Frehse and R. Rannacher (cf. Chapter 5 ) , to plates by R. Rannacher (cf. Chapter 6), to mixed methods by R. Scholz (cf. Chapter 7).

Ch. 3.1

BIBLIOGRAPHY A N D COMMENTS

I73

There is currently a wide interest in obtaining various refinements of the error estimates, such as “interior” estimates, superconvergence results, etc.. . . In addition to the previously quoted reference, we mention BRAMBLE& SCHATZ (1974, 1976), BRAMBLE& T H O M E E (1974), DESCLOUX (1975), DESCLOUX & NASSIF(1977), DOUGLAS & DUPONT (1973, 1976a), NITSCHE(1972a), NITSCHE& SCHATZ (1974), SCHATZ & WAHLBIN(1976, 1977). A little explored direction of research is that of the optimal choice of triangulation: For a given number of finite elements of a specific type, the problem consists in finding the “best” triangulation so as to minimize the error in some sense. For references in this direction, see CARROLL & BARKER(1973), MCNEICE& MARCAL(1973), PRACER(1979, RAJAGOPALAN (1976), TURCKE & MCNEICE(1972).

CHAPTER 4

OTHER FINITE ELEMENT METHODS FOR SECOND-ORDER PROBLEMS

Introduction

Up to now, we have considered finite elements methods which are conforming, in the sense that the space vh is a subspace of the space V , and the bilinear form and the linear form which are used in the definition of the discrete problem are identical to those of the original problem. In this chapter, we shall analyze several ways of violating this “conformity”, which are frequently used in everyday computations. In Section 4.1, we assume, as before, that the domain d is polygonal and that the inclusion vh C V still holds, but we consider the use of a quadrature scheme for computing the coefficients of the resulting linear system: each such coefficient being of the form

the integrals

are approximated by finite sums of the form

, ~nodes bl.KE K, which are derived from a single with weights w ~ and quadrature formula defined over a reference finite element. This process results in an approximate bilinear form ah(. ,.) and an approximate linear form fh(.) which are defined over the space vh. Our study of this approximation follows a general pattern that will also be the same for the two other methods to be described in this chapter: I74

Ch. 4.1

INTRODUCTION

I75

First, we prove (the first Strung lemma ; cf. Theorem 4.1. I ) an abstract error estimate (which, as such, is intended to be valid in other situations; cf. Section 8.2). It is established under the critical assumption that the approximate bilinear forms are uniformly Vh-elliptic, i.e., that there exists a constant di > 0, independent of h, such that a h ( v hO. h ) 2 d i ( ( 2 ) h ( ( 2 for all uh E vh. This is why we next examine (Theorem 4.1.2) under which assumptions (on the quadrature scheme over the reference finite element) this property is true. The abstract error estimate of Theorem 4.1.1 generalizes Cia’s lemma: In the right-hand side of the inequality, there appear, in addition to the term inf u h E V h IIu - uhll, two consistency errors which measure the quality of the approximation of the bilinear form and of the linear form, respectively. We are then in a position to study the convergence of such methods. More precisely, we shall essentially concentrate on the following problem: Find suficient conditions which insure that the order of convergence in the absence of numerical integration is unaltered b y the effect of numerical integration. Restricting ourselves for simplicity to the case where PK = P a ( K ) for all K E T h , our main result in this direction (Theorem 4.1.6) is that one still has IIu - UhII1.R

=o(hk),

provided the quadrature formula is exact f o r all polynomials of degree (2k - 2). The proof of this result depends, in particular, on the BrambleHilberf lemma (Theorem 4.1.3), which is a useful tool for handling linear functionals which vanish on polynomial subspaces. In this particular

case, it is repeatedly used in the derivation of the consistency error estimates (Theorems 4.1.4 and 4.1.5). We next consider in Section 4.2 a first type of finite element method for which the spaces vh are not contained in the space V. This violation of the inclusion Vh C V results of the use of finite elements which are not of class go (i.e., which are not continuous across adjacent finite elements), so that the inclusion vh C H ’ ( R ) is not satisfied (Theorem 4.2.1). The terminology “nonconforming finite element method” is specifically reserved for this type of method (likewise, for fourth-order problems, nonconforming methods result from the use of finite elements which are not of class % I ; cf. Section 6.2). For definiteness, we assume through Section 4.2 that we are solving a homogeneous Dirichlet problem posed over a polygonal domain fi. Then

176

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

the discrete problem consists in finding a function uh E vh such that, for all u h E vh, ah(&,,U h ) = f ( U h ) , where the approximate bilinear form ah(. , .) is defined by

the integrand {. . .} being the same as in the bilinear form which is used in the definition of the original problem. The linear form f(.) need not be approximated since the inclusion vh C L2(L2)holds. Assuming that the mapping

is a norm over the space vh, we prove an abstract error estimate (the second Strang lemma; cf. Theorem 4.2.2) where the expected term inf,h,vh IIu - U h ( ( h is added a consistency error. Just as in the case of numerical integration, this result holds under the assumption that the approximate bilinear forms are uniformly Vh-elliptic,in the sense that there exists a constant 6 > 0 independent of h such that ah(Uhr &I(Uhl(Zh for all U h E vh. We then proceed to describe a three-dimensional “nonconforming” finite element, known as Wilson’s brick, which has gained some popularity among engineers for solving the elasticity problem. Apart from being nonconforming, this finite element presents the added theoretical interest that some of its degrees of freedom are of a form not yet encountered. This is why we need to adapt to this finite element the standard interpolation error analysis (Theorem 4.2.3). Next, using a “bilinear lemma” which extends the Bramble-Hilbert lemma to bilinear forms (Theorem 4.2.5), we analyze the consistency error (Theorem 4.2.6). In this fashion we prove that

if the solution u is in the space H2(12). In passing, we establish the connection between the convergence of such nonconforming finite element methods and the patch test of B. Irons. Another violation of the inclusion vh C V occurs in the approximation of a boundary value problem posed over a domain b with a curved boundary (i.e., the set d is no longer assumed to be polygonal). In this

r

Ch. 4.1

INTRODUCTION

177

case, the set h is usually approximated by two types of finite elements: The finite elements of the first type are straight, i.e., they have plane faces, and they are typically used “inside” h.The finite elements of the second type have at least one “curved” face, and they are especially used so as to approximate “as well as possible” the boundary r. In Section 4.3, we consider one way of generating finite elements of the second type, the isoparametric finite elements, which are often used in actual computations. The key idea underlying their conception is the generalization of the notion of affine-equivalence: Let there be given a Lagrange finite element (K,P, {C(bi), 1 S i < N}) in R” and let F : f E K + F(f)= (F,(f))y=, E R” be a mapping such that F, E fi, 1 S i S n. Then the triple

(K = F(IZ), P = {p = 6 * F-I; 6 E fi}, {p(ai= F ( b i ) ) , 1 s i s N } ) is also a Lagrange finite element (Theorem 4.3.1), and two cases can be distinguished: (i) The mapping F is afine (i.e., F, E Pl(k),1 s i s n ) and therefore the finite elements (K, P , 2 ) and (k,P, 2) are affine-equivalent. (ii) Otherwise, the finite element (K, P , 2 ) is said to be isoparametric, and isoparametrically equivalent to the finite element (k,P, 2). If ( K , f i , $ ) is a standard straight finite element, it is easily seen in the second case that the boundary of the set K is curved in general. This fact is illustrated by several examples. We then consider the problem (particularly in view of Section 4.4) of developing an interpolation theory adapted to this type of finite element. In this analysis, however, we shall restrict ourselves to the isoparametric n-simplex of type ( 2 ) , so as to simplify the exposition, yet retaining all the characteristic features of a general analysis. For an isoparametric family (K, P K ,ZK)of n-simplices of type ( 2 ) , we show (Theorem 4.3.4) that the nK-interpolants of a function u satisfy inequalities of the form

1u - nK

s C~~-”’IIUII~.~, 0 s m s 3,

where hK = diam(K). This result, which is the same as in the case of affine families (cf. Section 3.1) is established under the crucial assumption that the “isoparametric” mappings FK do not deviate too much from affine mappings (of course the family is also assumed to be regular, in a sense that generalizes the regularity of affine families).

178

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , 8 4.1.

Even if we use isoparametric finite elements K E Fh to “triangulate” a set d, the boundary of the set .izh = U K is very close to, but not KEYh identical to, the boundary r. Consequently, since the domain of definition of the functions in the resulting finite element space vh is the Set dh, the space vh is not contained in the space V and therefore both the bilinear form and the linear form need to be approximated. In order to be in as realistic a situation as possible we then study in Section 4.4 the simultaneous effects of such an approximation of the domain d and of isoparametric numerical integration. As in Section 4.1, this last approximation amounts to use a quadrature formula over a reference finite element k for computing the integrals of the form J K q ( x ) dx (which appear in the linear system) via the isoparametric mappings F~ : k + K , K E y h . Restricting ourselves again to isoparametric n-simplices of type (2) for simplicity, we show (Theorem 4.4.6) that, if the quadrature formula over the set k is exact for polynomials of degree 2, one has IIi - UhIli.nh= O ( h * ) ,

where 17 is an extension of the solution of the given boundary value problem to the set ah (in general d),and h = maxKEy,hK.This error estimate is obtained through the familiar process: We first prove an abstract error estimate (Theorem 4.4. I), under a uniform Vh-ellipticity assumption of the approximate bilinear forms. Then we use the interpolation theory developed in Section 4.3 for evaluating the term inf Ilii - uhlll.flh (Theorem 4.4.3) and finally, we estimate the two consistency errors (Theorems 4.4.4 and 4.4.5; these results largely depend on related results of Section 4.1). It is precisely in these last estimates that a remarkable conclusion arises: In orderto retain the O ( h 2 )convergence. it is not necessary t o use more sophisticated quadrature schemes f o r approximating the integrals which correspond t o isoparametricfiniteelements than f o r straight finite elements.

f%,c

4.1. The effect of numerical integration

Taking into account numerical integration. Description of the resulting discrete problem.

Throughout this section, we shall assume that we are solving the second-order boundary value problem which corresponds to the follow-

Ch.4, B 4.1.1

THE EFFECT OF NUMERICAL INTEGRATION

179

ing data:

(4.1.1)

where 6 is a polygonal domain in R”,the functions aijEL”(0) and f EL*(O) are assumed to be everywhere defined over 6. We shall assume that the ellipticity condition is satisfied i.e., V x E 6 , Vei, I S i S n ,

3p>O,

(4.1.2)

so that the bilinear form of (4.1.1)is Hd(O)-elliptic. This problem corresponds (cf. (1.2.28)) to the homogeneous Dirichlet problem for the operator

i.e.,

( U = O

on

r.

(4.1.3)

The case of a more general operator of the form n

u

-,- 2 ai(aijaiu)+ au i.i=l

is left as a problem (Exercise 4.1.5). We consider in the sequel a family of finite element spaces x h made up of finite elements ( K ,PK,ZK), K E Y,,, where Yh are triangulations of the set 6 (because the set 6 is assumed to be polygonal, it can be exactly covered by triangulations). Then we define the spaces Vh= {uhEXh;Uh=Oonr}. The assumptions about the triangulations and the finite elements are the same as in Section 3.2. Let us briefly record these assumptions for convenience: (H 1) The associated family of triangulations is regular.

180

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

ICh. 4 . 8 4.1.

( H 2) All the finite elements ( K , PK,&), K E u h Yl,, are affineequivalent to a single reference finite element (R,P, 2). (H3 ) All the finite elements ( K , PK,ZK),K E y h , are of class V".

ul,

As a consequence, the inclusions xh C H ' ( R ) and v h C Hd(R)hold, as long as the inclusion fi C H ' ( K ) (which we will assume) holds. Given a space vh, solving the corresponding discrete problem amounts to finding the coefficients uk, 1 s k S M , of the expansion uh = Xi?=,UkWk of the discrete solution uh over the basis functions wk, 1 s k S M,of the space vh. These coefficients are solutions of the linear system (cf. (2.1.4)) M

a(Wk9 W m ) U k

1

=f(Wm)r

m

M,

(4.1.4)

k=l

where, according to (4.1. I ) , (4.1.5) (4.1.6)

In practice, even if the functions aii, f have simple analytical expressions, the integrals J K . . . dx which appear in (4.1.5) and (4.1.6) are seldom computed exactly. Instead, they are approximated through the process of numerical integration, which we now describe: Consider one of the integrals appearing in (4.1.5) or (4.1.6), let us say J K cp(x) dx, and let FK

:i E

12 -+ FK(.Iz)

= BKf

+ bK

be the invertible affine mapping which maps Z? onto K. Assuming, without loss of generality, that the (constant) Jacobian of the mapping FK is positive, we can write cp(x) dx

= det(BK)

Ik

$(f) df,

(4.1.7)

the functions cp and 4 being in the usual correspondence, i.e., cp(x)= $(a) for all x = FK(f),f E R. In other words, computing the integral IKcp(x) dx amounts to computing the integral Jk $(a)d f . Then a quadrature scheme (over the set k ) consists in replacing the integral JR$(f)d f by a finite sum of the form Xf=l d,4(6,),an approxi-

Ch. 4, 5 4.1.1

T H E EFFECT OF NUMERICAL INTEGRATION

181

mation process which we shall symbolically represent by (4.1.8)

The numbers dl are called the weights and the points gf are called the nodes of the quadrature formula X!:l &l$(6f).For simplicity, we shall consider in the sequel only examples for which the nodes belong t o the set K and the weights are strictly positive (nodes outside the set K and negative weights are not excluded in principle, but, as expected, they generally result in quadrature schemes which behave poorly in actual computations). In view of (4.1.7) and (4.1.8), we see that the quadrature scheme over the set K automatically induces a quadrature scheme over the set K , namely (4.1.9)

with weights w ~ and , ~nodes bf.Kdefined by WI,K

= det(BK)&l

and

b1.K

=

F K ( ~ I )1, C 1

L.

(4.1.10)

Accordingly, we introduce the quadrature error functionals (4.1.11) (4.1.12)

(4.1.13)

Remark 4.1.1. It is realized from the previous description that one needs only a numerical quadrature scheme over the reference finite element. This is again in accordance with the pervading principle that most of the analysis needs to be done on the reference finite element only, just as was the case for the interpolation theory (Section 3.1). 0 Let us now give a few examples of often used quadrature formulas. Notice that each scheme preserves some space of polynomials and it is this polynomial invariance that will subsequently play a crucial role in the problem of estimating the error.

182

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 . 5 4.1.

More precisely, given a space 6 of functions @ defined over the set K, we shall say that the quadrature scheme is exact f o r the space 6,or exact f o r the functions (;I E 6,if &(@)= 0 for all @ E 6. Let k be an n-simplex with barycenter 1

n+l

a=----( n + 1)

Bi.

(Fig. 4.1.1). Then the quadrature scheme

@(a)d i - meas(k)@(d)

(4.1.14)

is exact f o r polynomials of degree < I , i.e., V@E Pl(K),

@(a)d i - meas(k)@(B)= 0.

To see this, let

be any polynomial of degree d 1. Then using the equalities di

= meas(K)/(n

(Exercise 4.1.1), 1 d i d n

+ 1)

+ 1, we obtain

Fig. 4.I . I

(4.1.15)

Ch. 4, $4.1.1

183

THE EFFECT OF NUMERICAL INTEGRATION

Fig. 4.1.2.

Let n

=2

and let

K

be a triangle with mid-points of the sides

dij,

1 Q i < j C 3 (Fig. 4.1.2).

Then the quadrature scheme (4.1.16) is exact f o r polynomials of degree C2 (cf. Exercise 4.1.1), i.e.,

Let n = 2 and let K be a triangle with vertices Cii, 1 < i s 3 , with (Fig. 4.1.3). mid-points of the sides Ci, 1 C i < ’ C 3, and with barycenter ilZ3

a2

Fig. 4.I .3.

184

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , 8 4.1.

Then the quadrature scheme

(4.1.18) 3 (cf. Exercise 4.I.l),i.e., V 4 E

is exact f o r polynomials of degree P,(&.

4(f) d f

(3

=

2 i=l

4 ( h i )+ 8

c

l 0,

vuh

E

Vhr

&lI(Uhll*

d

ah(uhr u h ) ,

(4.1.26)

where the constant h is independent of the subspace vh. Notice that such an assumption implies the existence of the discrete solutions. Theorem 4.1.1 (first Strang lemma). Consider a family o f discrete problems f o r which the associated approximate bilinear forms are uniformly Vh-elliptic. Then there exists a constant C independent of the space vh such that

(4.1.27)

Proof. Let u h be an arbitrary element in the space assumption of uniform Vh-ellipticity, we may write:

hlluh - uh112

ah(Uh - u h , uh - u h ) = a ( u - U h , uh - V h ) +{a(uhrUh-Z)h)-ah(Uh,Uh-Uh)}

+ {fh(uh- u h ) - f (uh - U h ) ) r

so that, using the continuity of the bilinear from a ( . , .),

vh.

With the

Ch. 4, 54.1.1

T H E EFFECT OF NUMERICAL INTEGRATION

187

Combining the above inequality with the triangular inequality and taking the infimum with respect to

uh

E Vh yields inequality (4.1.27).

0

Remark 4.1.3. The abstract error estimate (4.1.27) generalizes the abstract error estimate established in CCa’s lemma (Theorem 2.4.1) in the case of conforming finite element methods, since, in the absence of numerical integration, we would have a h ( * , = a(., -) and f h ( ’ ) = f ( . ) . 0 a )

Suficient conditions for uniform Vh-ellipticity We now give sufficient conditions on a quadrature scheme which insure that the approximate bilinear forms are uniformly Vh-elliptic: Notice in particular that in the next theorem assumptions (i) and (ii) exhibit the relationship which should exist between the reference finite element (K, fi, $) and the quadrature scheme defined on K (for the case of negative weights, see Exercise 4.1.2). Theorem 4.1.2.

Let there be given a quadrature scheme L

ouer the integer k‘

&(a)d2 - &l&(61)with &, > 0 , 1 4 1 s L, reference finite element (K,fi, $), f o r which there 2

exists an

1 such that:

(i) The inclusion P C Pk,(K) holds. (ii) The union {gI} contains a Pk._I(k)-unisolventsubset andlor the quadrature scheme is exact f o r the space P2k,-2(K).

u

Then there exists a constant & > 0 independent o f h such that, f o r all approximate bilinear forms of the form (4.1.24) and all spaces Vh, vuh E

vh,

(4.1.28)

~ l u h 1 ? , O Ga h ( v h , U h ) .

uF=l

Proof. (i) Let us first assume that the union {&} contains a Pp-,(I?)-unisolvent subset. Using the strict positivity of the weights, we find that

6 E fi and lSiSn,

L

I=I

n

GI

2 (a$(&))’= i=l

1siSL.

0 3 aiC(&) = 0 ,

188

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4,

§ 4.1.

For each i E [ I , n], the function a$ is in the space Pks-I(@ by assumption (i), and thus it is identically zero since it vanishes on a Pk,-l(k)unisolvent subset, by assumption (ii). As a consequence, the mapping

defines a norm over the quotient space @/Po(&). Since the mapping ell,^ is also a norm over this space and since this space is finitedimensional, there exists a constant 6 > 0 such that (4. I .29) If the quadrature scheme is exact for the space P 2 k t - 2 ( kthe ) , above inequality becomes an equality with 6 = 1 , since the function ZY==, (ai$)’ belongs to the space P2k,-.2(K) for all 6 E @ and since

is precisely the quadrature formula which corresponds to the integral

(ii) Let us next consider the approximation of one of the integrals

Let U h l K = P K and let $ K E @ be the function associated with pK through the usual correspondence 2 E k -+ F ( 2 ) = BK2 + bK = x E K. We can write, using the ellipticity condition (4.1.2), and the positivity of the weights,

L

3p

2 1=1

n

W/.K

2

(aiPK(bl.K))2.

(4.1.30)

i=l

Observe that ZY=l aipK(bl.K))2 is the square of the Euclidean norm 11.1( of the vector D P K ( ~ I Since . K ) . llDhd61)ll IlB~ll( ( D P K ( ~ (for I . K )all ( ~5 E R“, we have D$(61)S = Dp(bl.K)(BK,f)), we can write, using relations (4.1.10) and (4.1.29),

Ch. 4, $4.1.1

THE EFFECT OF NUMERICAL INTEGRATION

189

2 2( a i f i ~ ( 6 ) ) ~ L

= det(BK)IIBKIl-’ I=1 ;I

a

n

i=l

c det(BK)I(BKII-21fiK(:.R

a ~(IIBKII IIBi111)-21pKI?.K,

(4.1.31)

where we have also used Theorem 3.1.2. Since we are considering a regular family of triangulations, we have

(4.1.32) for some constant C independent of K E yh and h. Combining inequalities (4.1.30), (4.1.31) and (4.1.32), we find that there exists a constant & > 0 independent of K E yh and h such that L

v u h E Vh, 1=1

2 n

w1.K

(aijaivhajuh)(bi.K) 3 ai:IuhI:.K.

(4.1.33)

i,j=l

(iii) It is then easy to conclude: Using inequalities (4.1.33) for all K E y h , we obtain

Remark 4.1.4.

Notice that the expressions

are exactly the approximations we get when we apply the quadrature scheme to the integrals JfiKJ:.k,which in turn correspond to the model problem -Au = f in 0,u = 0 on r. Therefore it is natural to ask for assumptions (ii) which essentially guarantee that the mapping

is a norm over the quotient space l?Po(K).

0

1%

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , p 4.1.

In view of this theorem, let us return to the examples of quadrature schemes given at the beginning of this section. If (K,fi, 2) is an n-simplex of type (1) (fi = P I ( @ and thus k' = l), we may use the quadrature scheme of (4.1.14) since {Ci} is a Po(@-unisolvent set. If (K, B, 2) is a triangle of type (2) (fi = P 2 ( k )and thus k' = 2), we may use the quadrature scheme of (4.1.16) since i<j{Sij}is a PI(&)unisolvent set. Notice that in both cases, the second assumption of (ii) is also satisfied. If (k,P, 2) is a triangle of type (3) or (3') (@ c P 3 ( k )and thus k' = 3), we may use the quadrature scheme of (4.1.18) since the set of numerical integration nodes (strictly) contains the P2(k)-unisolvent subset i{Cii}) U i<j{Ciij}). However the quadrature scheme is not exact for the space P4(k)as the second assumption of (ii) would have required.

u

(u

(u

Consistency error estimates. The Bramble-Hilbert lemma Now that the question of uniform Vh-ellipticity has been taken care of, we can turn to the problem of estimating the various terms appearing in the right-hand side of inequality (4.1.27). For the sake of clarity, we shall essentially concentrate on one special case (which nevertheless displays all the characteristic properties of the general case), namely the case where

P = Pk(K) 1 (the cases where Pk(k)c P c P k , ( k )or where are left as problems; cf. Exercises 4.1.6 and 4.1.7). This being the case, if the solution is smooth enough so that it belongs to the space H k " ( 0 ) , we have

for some integer k

2

p k ( k )c P c

inf IIu - V h l ( l . 0

IIu - nh uI11.a Chklulk+l.n,

UhEVh

assuming the Xh-interpolant of the solution K is well-defined, and thus, in the absence of numerical integration, we would have an O ( h k ) convergence. Then our basic objective is to give suficient conditions on the quadrature scheme which insure that the efect of numerical integration does not decrease this order of convergence. Remark 4.1.5.

This criterion for appraising the required quality of the

Ch. 4, 84.1.1

THE EFFECT OF NUMERICAL INTEGRATION

191

quadrature scheme is perhaps arbitrary, but at least it is well-defined. Surprisingly, the results that shall be obtained in this fashion are nevertheless quite similar to the conclusions usually drawn by engineers 0 through purely empirical criteria. Let us assume that the approximate bilinear forms are uniformly Vh-elliptic so that we may apply the abstract error estimate (4.1.27)of Theorem 4.1.1. Consequently, our aim is to obtain consistency error estimates of the form

(4.1.34) (4.1.35) Notice that, in the usual terminology of numerical analysis, the uniform ellipticity condition appears as a stability condition, while the conditions (implied by the above error estimates)

appear as consistency conditions. This is why we call consistency errors the two terms of the form supwhEvh (. . .) appearing in the left-hand side of inequalities (4.1.34)and (4.1.35).By definition of the quadrature error functionals EK(.) (cf.(4.1.11)),we have, for all wh E vh,

(4.1.37) It turns out that we shall obtain (Theorems 4.1.4 and 4.1.5) ‘‘local’’ quadrature error estimates of the form

192

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4, 8 4.1.

from which the "global" consistency error estimates (4.1.34) and (4.1.35) are deduced by an application of the Cauchy-Schwarz inequality (this is possible only because the constants C(aiilK;&p') and C ( f i K )appearing in the above inequalities are of an oppropriate form). To begin with, we prove a useful preliminary result.

Theorem 4.1.3 (Bramble-Hilbert lemma). Let O be an open subset of R" with a Lipschitz-continuous boundary. For some integer k 2 0 and some number p E [0,m], let f be a continuous linear f o r m on the space W k " s p ( 0with ) the property that VP E PdO),

f ( P ) = 0.

(4.1.40)

Then there exists a constant C ( O ) such that

vu E where

W"".P(O), If(u)l

~~~~~~ is+the l , norm p.R

==

C(~>lVll:+l.p,nl~lk+l.P.Rr

(4.1.41)

in the dual space o f W k + l . p ( 0 ) .

Proof. Let u be any function in the space W k + i . p ( 0Since ) . by assumption, f ( u ) = f ( u + p ) for all p E P k ( 0 ) ,we may write VP E Pk(O), If(v)'l=

If(u + P)I s l l f I l : + i . P . ~ b

+ Pllk+l.P.R>

and thus

The conclusion follows by Theorem 3.1.1.

0

In the sequel, we shall often use the following result: Let the functions W r n . q ( 0and ) w E Wrnq"(0) be given. Then the function cpw belongs to the space W r n . q ( 0and ),

cp E

f o r some constant C solely dependent upon the integers m and n, i.e., it is in particular independent of the set 0. To prove this, use the formula

Ch. 4, 64.1.1

THE EFFECT OF NUMERICAL INTEGRATION

193

in conjunction with inequalities of the form

Theorem 4.1.4.

Assume that, for some integer k

P = Pk(k), v+ E P*k-*(k), I?($)

2

1, (4.1.43)

= 0.

(4.1.44)

Then there exists a constant C independent of K E T,, and h such that V U E W k * m ( K )v, p E pk(K), Vp'E P k ( K ) ,

IEk (aaiP 'dip )I s C haIa IIk.m.KII aiP 'Ilk- I .K IaiP IO.K C h ala l l k . m , K b ' l l k . K b I1.K.

(4.1.45)

Proof. We shall get an error estimate for the expression EK(auw)for a E W'."(K),u E P k - I ( K ) w , E Pk-l.From (4.1.13), we infer that E K ( a u w )= det(BK)&hi%),

(4.1.46)

with a^ E W k . " ( k ) fi, E P k - I ( k ) ,G E P k - l ( k ) .For a given G E P k - d k ) and any 4 E W'."(k),we have ( W k * " ( k )@ 4 ( k since ) k 3 1)

s tI4fiIo.m.k s e l 4 l o . m . k l ~ I o . m . ~ .

e

where, here and subsequently, the letter represents various constants solely dependent upon the reference finite element. Since I$lo.m.~ S I(411k.m.~, and since all norms are equivalent on the finite-dimensional space P k - l ( k ) , we deduce that

IB(4G)I s el1411k.m.kI~lo.k. Thus, for a given B E Pk-I(k)r the linear from

4 E Wk,"(k)+l?(+G) continuous with norm s klBlo,k on

is the one hand, and it vanishes over the space Pk-I(k) on the other hand, by assumption (4.1.44). Therefore, using the Bramble-Hilbert lemma, there exists a constant 6

194

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , 8 4.1.

such that VG E Wk."(R),

v+ E P k - , ( K ) ,

IE(~G)Ie1+lli.m.klGlo.k. let 4 = 48 with ci E Wksm(k), B € Pk-l(k). Using

Next, taking into account that JBlk.m,k= 0, we get

ex k-I

1Glk.m.K

= lav^lk.m.K

s

(4.1.42) and

k-I

la^Ik-j.m,Klv^lj.m,k

I =O

s

ez

lalk-j,m.kIv^lj.k,

j=O

where, in the last inequality, we have again used the equivalence of norms over the finite-dimensional space Pk-I(R).Therefore, we obtain

va E W"."(K),

V 6 E Pk-I(k)r VG E (4.1.47)

Then it suffices to use the inequalities (cf. Theorems 3.1.2 and 3.1.3) 1ilk-jp.g

o s j 6 k - 1,

s &zk-jlalk-j.m.K,

O s j s k - 1, I6(j,R s ~'hk(det(BK))-1'2(uli.K,

IG10.k s & W B K ) -1'21

w10.~,

in conjunction with relations (4.1.46) and (4.1.47). We obtain in this fashion: V a E Wk.m(K),Vu E P k - d K ) , Vw E P k - I ( K ) ,

s ChklIa Ilk.m.KII u Ilk-1.K Iw lo.K, and the conclusion follows by replacing u by &p' and w by dip in the last inequality. 0 Remark 4.1.6. Let us indicate why a direct application of the BrambleHilbert lemma to the quadrature error functionals &(.) (in this direction, see also Exercise 4.1.4) would not have yielded the proper estimate. Let us assume that VG E P , ( K ) , I?($, = 0,

for some integer I 3 0, and let r E [ I ,

001

be such that the inclusion

Ch. 4, 8 4.1.1

THE EFFECT OF NUMERICAL INTEGRATION

I95

W'+'.'(k)&@ ( k holds, ) so that we have

VGE IB(G)I s e I ~ I o . m . 1 2s eIIGIIl+l.r,k. Then assumption (4.1.44), together with the Bramble-Hilbert lemma, implies that VG E W ' + ' J ( k ) , IB(G)I s elGll+l,r,k.

Let us then replace 6 by the product 666, with a sufficiently smooth D E &I(&). Using inequalities of the form function ci, $ E & I ( & ) , (4.1.42) and the equivalence of norms over the space Pk-l(k), we would automatically get all the semi-norms 1 wIi,K,0 s j S min{l + I , k - I}, in the right-hand side of the final inequality, whereas only the semi-norm IwI, , K should appear. 0 The reader should notice that the ideas involved in the proof of the previous theorem are very reminiscent of those involved in the proof of Theorem 3.1.4. In both cases, the central idea is to apply the fundamental result of Theorem 3.1.1 (in the disguised form of the Bramble-Hilbert lemma in the present case) over the reference finite element and then to use the standard inequalities to go from the finite element K to k, and back. The same analogies also hold for our next result. Theorem 4.1.5.

Assume that, f o r some integer k

3

1,

= Pk(k),

VG E PZk-Z(k). and let q E [ I ,

m]

(4.1.48)

B(4) = 0,

(4.1.49)

be any number which satisfies the inequality

n k-->O. 4

(4.1 S O )

Then there exists a constant C independent of K E T,, and h such that

vf E W k ' q ( K ) ,v p E p k ( K ) ,

(4.1.5 1)

IEK(fp)Is Ch~meas(K))"'2'-"'q'~~f~~ k,q.K 11p 11 I . K . Proof. For any f E W k . ' ( K )and any p E P k ( K ) we , have E d f p ) = det(&)&f^ri),

(4.1.52)

196

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4, 5 4.1.

with f E W k . q ( K )I;, E P k ( k ) . Let us write

E(ffi) = E(ffi6) + E(f(6 - TIC)),

(4.1.53)

where fi is the orthogonal projection in the space L 2 ( K ) onto the suhspace PI(&). (i) Let us estimate E(ffi$). For all E W k . q ( k we ) , have

4

I&lj,l s 6lljlO.rn.R

aljllk,q.K9

since inequality (4.1.50) implies that the inclusion Wk.'(K)4 @ ( k holds, ) and, in addition, = O for all &EPk-l(k),by virtue of assumption (4.1.49) (therefore, this assumption is not fully used at this stage, unless k = I ) . Using t h e Bramble-Hilbert lemma, we obtain

B(&)

vlj E W'Jyk), @($)I s 6'lljlk.q.R. In particular, let

4

=

ffic

with f E W'.'(K),

6 E P k ( k ) . Using inequality

(4.1.42), we find:

Iffi~

1k.q.R

s &(IfIk,q,R

Ifi~

+ I f I k - l . q . ~Ifie II . m . g ) r

I0.m.~

since all semi-norms Ififilr,m.~ are zero for 1 2 (fit E Pl(k)). Using the equivalence of norms over the finite-dimensional space P,(k), we get

IffiB h q . K s &IPlk.q.R lfiB l0.R + IfIk-l.q.RlfiB

1I.R).

Further we have

IfiB since

I0.R

s IB lo.K*

fi is a projection operator. and lfiBll.R

IB - fiBl1.R + lBII.Ez.

Applying Theorem 3.1.4 to the operator

fi, which

leaves the space

Po(k)invariant, we find, for some constant 6, lB - fiBll.R s ~ l B 1 l . K .

Thus, upon combining all our previous inequalities, we have found a constant 6 such that V f E W'JyK),

vg E P k ( K ) ,

(4.1.54)

IWfiB)l s ~ ( I f l k . q . K I B l " . R+ Iflk-l.q.R1811.K). (ii) Let us next estimate l?(f(fi - &)).

Observe that if k = 1, the

Ch. 4, 5 4.1.1

THE EFFECT OF NUMERICAL INTEGRATION

I97

difference (6 - Z@) vanishes and therefore, we may henceforth assume that k 3 2. This being the case, there exists a number p E [l, +w] such that the inclusions

Wk*'(k)L W ' @ ( K ) 4P(K) hold. To see this, consider first the case where 1 d q < n, and define a number p by letting ( l l p ) = ( l l q )- ( l l n ) , so that the inclusion W ' . ' ( K ) 4L p ( k )(and consequently the inclusion W " ( k ) L w"-'.p(K)) holds. Then the inclusion W k - ' @ ( k )@ 4 ( R )also holds because we have k - 1 - ( n / p ) = k - ( n / q )> 0 by (4.1.50). Consider next the case where n c q. Then either n < q and the inclusion W ' * ' ( k ) L Lp(& holds for all p E [l,w], or n = q and the same inclusion holds for all (finite) p 3 1, so that in both cases the inclusion W k V q ( K )p4- ' @ ( k holds ) for all p a 1. Since in this part (ii) we assume k 2 2, it suffices to choose p large enough so that k - 1 - ( n l p ) > 0 and then the inclusion W k - ' * p ( K %"(&I )4 holds. Proceeding with the familiar arguments, we eventually find that

v.fE W k - ' * p ( k )@(k), 4 vg E P k ( k ) , 12(.f(~ - fib>>/ e~.f(~^ - f i 6 ) l o . m . ~ s e l f l o . m . ~ l B- fitI0.m.k ~

Thus for a given

~

~

f

~

~

k --

fi6lO.m.k. i . p .

~

~

~

6 E p k ( k ) ,the linear form

f E Wk-'.p(k,+E(.f(6-I?$))

- fifi)O,m,k, and it vanishes over the space assumption (4.1.49)is

is continuous with norm

&2(k)(notice that, contrary to step (i), the "full"

used here). Another application of the Bramble-Hilbert lemma shows that v f E W k - ' @ ( k ) v6 , E Pk(k),

IE(.f(6 - fi6))l e.(flk-l.p.el6- fi6lO.K. Since the operator fi leaves the space P o ( k )invariant

Theorem 3.1.4,

16 - fi6lo.R

C

el6ll.K.

Also, we have

vg E w'.q(kh

1810,p,lz

&l8lo.q.K

+ 181I.q.R)r

we have, again by

198

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 . 5 4.1.

since the inclusion W ' . ' ( k ) bL p ( k )holds, and thus

Combining all our previous inequalities, we obtain:

Remark 4.1.7. Several comments are in order about this proof. (i) First, there always exists a number q which satisfies inequality (4.1.50). In particular, the choice q = 03 is possible in all cases. (ii) Just as in the case of Theorem 4.1.4, a direct application of the Bramble-Hilbert lemma would yield unwanted norms in the right-hand side of the final inequality, which should be of the form ( & ( f p ) ( s * * * llplll.~ (cf. Remark 4.1.6). (iii) Why did we have to introduce the projection fi? otherwise (arguing as in part (ii) of the proof), we would find either

lB(.ffi)l

el.flk-l.p.kIfilO.k?

or

IB O , and let r E [ l , ~ ]be such that the inclusion W'+l.r( K )4go(K ) holds. Using the Bramble-Hilbert lemma, show that there exists a constant C independent of K E .Thand h such that V(P E W f + l ' r ( K ) ,JEK(p)J c C(det B K ) " ' " ' ) h ~ ' l ~ l l + l . r , ~ . 4.1.5. The purpose of this problem is to analyze the effect of numerical integration for the homogeneous Neumann problem corresponding to the following data:

V

=

H'(O),

where, in addition to the assumptions made at the beginning of the section, it is assumed that the function a is defined everywhere over the set d and that 3ao>0,

VxEd,

a(x)2ao>0.

Thus the discrete problem corresponds to the approximate bilinear

204

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , 8 4.1.

form

(i) With the same assumptions as in Theorem 4.1.2, show that there exists a constant & > 0 such that vvh

E vh,

~ ~ ~ ~ h ~ah(Vh, ~ ~ .oh). 1 2 s

(ii) Assume that, for some integer k 2 1,

P = P&), vg E PZk-2(K),

I?($)

= 0.

Show that there exists a constant C independent of K E that

y h

and h such

va E Wk.”(K), v p E Pk(K), Vp’ E Pk(K), I&(aP’P

)I

Ch;lla Ilk.m.KIIP ’llk.KIIP Il1.K.

(iii) State and prove the analogue of Theorem 4.1.6 in this case. 4.1.6.

space

The purpose of this problem is to consider the case where the satisfies the inclusions

@

P&)

c P c P&).

In this case the question of Vh-ellipticity is already settled (cf. Theorem 4.1.2). (i) Show that the analogues of Theorems 4.1.4 and 4.1.5 hold if the quadrature scheme is exact for the space Pk+k,-2(k). (ii) Deduce that the analogue of Theorem 4.1.6 holds if all the weights are positive, the union { g I } contains a Pp-I(k)-unisolvent subset and the quadrature scheme is exact for the space Pk+k,-2(R). (iii) Deduce from this analysis that triangles of type (3‘) may be used in conjunction with the quadrature scheme of (4.1.18). Could the quadrature scheme of (4.1.16) be used?

u

4.1.7.

The purpose of this problem is to consider the case where the

Ch. 4, 64.1.1

space

THE EFFECT OF NUMERICAL INTEGRATION

205

6 satisfies the inclusions pk(k)c 6 c Qk(R),

i.e., essentially the case of rectangular finite elements. (i) Let n = 1 and K = [0, 11. It is well known that for each integer k a O , there exist ( k + 1) points b i E I O , I] and ( k + 1) weights w i > O , 1 s i s k + 1, such that the quadrature scheme

I

10. I 1

k+l

r ~ ( xdx )

- C wip(bi) i=l

is exact for the space P2k+l([0, I]). This particular quadrature formula is known as the Gauss-Legendre formula. Then show that the quadrature scheme

is exact for the space Q2k+l([0, 11"). (ii) Assuming the positivity of the weights, show that the approximate bilinear forms are uniformly Vh-elliptic if the union u f = l { 6 f } contains a Qk(k) n Pnk-l(k)-unisolventsubset. (iii) Show that the analogues of Theorems 4.1.4 and 4.1.5 hold if the quadrature scheme is exact for the space Q2k-I(k). (iv) Deduce that the analogye of Theorem 4.1.6 holds if all the weights are positive, if the union U contains a Qk(k) n Pnk-,(k)-unisolvent subset, and if the quadrature scheme is exact for the space Q2k-dk). As a consequence, and contrary to the case where 6 = Pk(k)(cf. Remark 4.1.8), it is no longer necessary to exactly compute the integrals

aiiaiuhaivhdx when the coefficients aii are constant functions. (v) Show that consequently one may use the Gauss-Legendre formula described in (i). Let d = [ O , Zp] x [0, Jp] where Z and J are integers and p is a Strictly positive number, and let y h be a triangulation of the set d made up of rectangles of type ( 1 ) of the form

4.1.8.

[ip,(i + 1)pl x [ j p , ( j + l)p],

0 c i s I - 1,

0 c j s J - 1.

206

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4, 5 4.1.

Let Uij denote the unknown (usually denoted u k )corresponding to the (k-th) node (ih,j h ) , 1 G i G I - 1 , 1 S j s J - 1 . In what follows, we only consider nodes (ip, j p ) which are at least two squares away from the boundary of the set 0,i.e., for which 2 s is GI-2,2GjGJ-2. Finally, we assume that the bilinear form is of the form a(u, u ) =

[ 2 &u&u dx, fl I=I

i.e., the corresponding partial differential equation is the Poisson equation -Au = f in 0. (i) Show that, in the absence of numerical integration, the expression (usually denoted) Xf=l a ( w k ,wm)uk corresponding to the (m-th) node (ip. j p ) is, up to a constant factor, given by the expression 8Uij

(Ui+l,j+ Ui+l,j+l + Ui.j+l+ Ui-l,j+l+ + Ui-1.j + Ui-l,j-l+ Ui.j-I + Ui+l.j-l).

(ii) Assume that the quadrature scheme over the reference square is

k = LO, 11’

Show that this quadrature scheme is exact for the space Ql(k).Since the set of nodes is QI(l?)-unisolvent, the associated approximate bilinear forms are uniformly V,-elliptic and therefore this scheme preserves the convergence in the norm (J-IJr.n (cf. Exercise 4.1.7). Show that the corresponding equality (usually denoted) Z f ! , a h ( w k wm)uk , = f h ( W n 1 ) is given by 4Uij -(ut+l.j+ Ui.j+I+

Ui-1.j

+ Ui,j-I) = p’f(ip, j p ) ,

which is exactly the standard five-point diference approximation to the equation

-Au

= f.

(iii) Assume that the quadrature scheme over the reference square is

Show that this quadrature scheme is exact for the space

Ql(k).

Ch. 4, 54.2.1

A NONCONFORMING METHOD

207

Show that the associated approximate bilinear forms are not uniformly Vh-elliptic, however. Show that the expression (usually denoted) X,"==, ah(wh,w,)uk is, up to a constant factor, given by the expression 4uij - (ui+l.j+l+ LJi-I,j+l+

LJi-l.i-l+

ui+l.j-l)s

It is interesting to notice that the predictably poor performance of such a method is confirmed by the geometrical structure of the above finite difference scheme, which is subdivided in two distinct schemes !

4.2. A nonconforming method Nonconforming methods f o r second-order problems. Description of the resulting discrete problem

Let us assume for definiteness that we are solving a second-order boundary value problem corresponding to the following data:

[ V = Hd(R), a(u, v ) =

[ 2 aij&uajvdx,

(4.2.1)

R i.j= I

At this essentially descriptive stage, the only assumptions which we need to record are that aijE L"(R), I c i,j c n, f E L*(R),

(4.2.2)

and that the set d is polygonal. Just as in the previous section, this last assumption is made so as to insure that the set fi can be exactly covered with triangulations. Given such a triangulation d = K E y h K, we construct afinite element s p a c e X h whosegenericfiniteelement is not of class %'.Then the space xh will not be contained in the space H ' ( R ) , as we show in the next theorem, which is the converse of Theorem 2.1.1.

u

Theorem 4.2.1.

Assume that the inclusions PK C Ce"(K)f o r all K E Y,,

208

and

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4, 84.2.

xh C H ' ( 0 ) hold. Then the inclusion xh c g0(d)

holds. Proof. Let us assume that the conclusion is false. Then there exists a function u E xh, there exist two adjacent finite elements K , and K 2 ,and there exists a non empty open set 0 c K , U K 2 such that (for example)

-

( U I K ~ UIK,)

> 0 along K' n 0,

(4.2.3)

where K' is the face common to K I and K 2 . Let then cp be a (non zero) C 9(0). Using Green's formula positive function in the space 9(6) (1.2.4), we have (with standard notations)

lo

&ucp dx =

c

A=1,2

&up

dx

KA

and thus we reach a contradiction since the integral along K' should he strictly positive by (4.2.3).

For the time being, we shall simply assume that the inclusions

VK E y h ,

PK C H 1 ( K ) ,

(4.2.4)

hold, so that, in particular, the inclusion

Xh c L 2 ( 0 )

(4.2.5)

holds. Then one defines a subspace X0h of X h which takes as well as possible into account the boundary condition u = 0 along the boundary r of 0. For example, if the generic finite element is a Lagrange element, all degrees of freedom are set equal to zero at the boundary nodes. But, again because the finite element is not of class g o (cf. Remark 2.3.10), the functions in the space XOh will in general vanish only at the boundary nodes. In order to define a discrete problem over the space vh = XOh,we observe that, if the linear form f is still defined over the space vh by

Ch. 4, 44.2.1

209

A NONCONFORMING METHOD

virtue of the inclusion (4.2.5), this is not the case of the bilinear form a(., To obviate this difficulty, we define, in view of (4.2.1) and (4.2.4), the approximate bilinear form a).

(4.2.6) and the discrete problem consists in finding a functior vuh

E vh,

h

E

v h

such that (4.2.7)

ah(Uh. uh)=f(uh)-

We shall say that such a process of constructing a finite element approximation of a second-order boundary value problem is a noncon forming finite element method. By extension, any generic finite element which is used in such method is often called a nonconforming finite element.

Abstract error estimate. The second Strung lemma In view of our subsequent analysis, we need, of course, to equip the space v h with a norm. In analogy with the norm l-l,.o of the space V = Hd(fl), a natural candidate is the mapping

(4.2.8) which is a priori only a semi-norm ouer the space v h . Thus, given a specific nonconforming finite element, the first task is to check that the mapping of (4.2.8) is indeed a norm on the space v h . Once this is done, we shall be interested in showing that, for a family of spaces Vh, the approximate bilinear forms of (4.2.6) are uniformly Vh-elliptic in the sense that 3 G > 0,

vvh, v u h E v h ,

&[l:(lvhllis

ah(Uh, uh).

(4.2.9)

This is the case if the ellipticity condition (cf. (4.1.2))is satisfied. Apart from implying the existence and uniqueness of the solution of the discrete problem, this condition is essential in order to obtain the abstract error estimate of Theorem 4.2.2 below. From now on, we shall consider that the domain of definition of both the approximate bilinear form of (4.2.6) and the semi-norm of (4.2.8) is the space Vh + V. This being the case, notice that Vu E V,

ah(u, u ) =

a ( u , u ) and

llullh

=

JuI~.R.

(4.2.10)

210

[Ch. 4, 44.2.

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

Also, the first assumptions (4.2.2) imply that there exists a constant M independent of the space vh such that v u , E (vh +

v),

[ah(&v ) l

~

~

~

~

~

~

h

~

(4.2.11) ~ u

Theorem 4.2.2 (second Strang lemma). Consider a family of discrete problems f o r which the associated approximate bilinear forms are unif ormly Vh-ellipt ic. Then there exists a constant C independent o f the subspace v h such that

Proof. Let v h be an arbitrary element in the space v h . Then in view of the uniform Vh-ellipticity and continuity of the bilinear forms ah (cf. (4.2.9) and (4.2.1 1)) and of the definition (4.2.7) of the discrete problem, we may write Glluh - vhllzhs

&(Uh

- vh,

uh

-vh)

= ah(U - vh, uh - u h ) + {f(uh- o h ) -

ah(U, Uh - uh)},

from which we deduce

Then inequality (4.2.12) follows from the above inequality and the triangular inequality - Uhllh

IIu - VhIlh

-k Iluh - uhllh-

0

Remark 4.2.1. The error estimate (4.2.12) indeed generalizes the error estimate which was established in Cia's lemma (Theorem 2.4.1) for conforming methods, since the difference f ( W h ) - ah(u,wh) is identically zero for all wh E vh when the space vh is contained in the 0 space V.

~

~

h

-

Ch. 4, 64.2.1

21 1

A NONCONFORMING METHOD

An example of a nonconforming finite element: Wilson’s brick Let us now describe a specific example of a nonconforming finite element known as Wilson’s brick, which is used in particular in the approximation of problems of three-dimensional and two-dimensional elasticity posed over rectangular domains. We shall confine ourselves to the three-dimensional case, leaving the other case as a problem (Exercise 4.2.1). Wilson’s brick is an example of a rectangular finite element in R3,i.e., the set K is a 3-rectangle, whose vertices will be denoted ai, 1 d i d 8 (Fig. 4.2.1). The space PK is the space P 2 ( K ) to which are added linear combinations of the function ( x l x 2 x 3 ) .Equivalently, we can think of the space PK as being the space Q , ( K ) to which have been added linear combinations of the three functions x f , 1 s j s 3. We shall therefore record this definition by writing

PK = P Z ( K ) @ V { X ~ = XQ ~ IX( ~ K }) @ V { x f 1, s j

I

Wilson’s brick; n = 3

Fig. 4.2.1

d

3).

I

(4.2.13)

212

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 . 6 4 . 2 .

Notice that the inclusions

P2(K)c P K ,

Ql(K) c P K

(4.2.14)

hold and that dirn(PK) = 1 1 .

(4.2.15)

It is easily seen that the values p ( a i ) , 1 s i s 8, a t the vertices, together with the values of the (constant) second derivatives aiip, 1 d j =z 3, form a PK-unisolvent set. To see this, it suffices to check the validity of the following identity: For all functions p^ E Pg, with k = [- 1 , + lI3, one has

6 = ~ ( 1 + x l ) ( l + x Z ) { (+x3)p^(cil)+(l-x3)p^(B5)} I - xl)(l + xZ){(l + x3)6(62) + ( 1 - x3)p^(h6)}

+Q4ll'l

{'ply;

so that, using the second inequality of (4.3.28) and the first inequality (4.3.29),

and the second inequality of (4.3.29) is proved. Using (4.3.34), we can write

v i € K,

B = DF(i)(I+ B - ' E ( i ) ) - ' ,

and thus, by (4.3.33),

Therefore, we deduce that

-- 1 lJF'lO.m,K

-

1

9EE IJF-l(X)l = inf J&)

3 (1 - y)"Idet(B)I 5 C

P

meas(R),

and the second inequality of (4.3.30) is proved, which completes the proof. 0

Ch. 4, Ei 4.3.1

ISOPARAMETRIC FINITE ELEMENTS

24 1

Combining Theorems 4.3.2 and 4.3.3, we are in a position to prove our main result (compare with Theorems 3.1.6).

Theorem 4.3.4. Let there be given a regular isoparametnc family of n-simplices K of type (2) and let there be given an integer m 2 0 and two numbers p , q E [ l , 001 such that the following inclusions hold: W3*P(R)4%O(R),

(4.3.35)

W3*P(R)4W y k Z ) ,

(4.3.36)

where K is the reference n-simplex o f type (2) o f the family. Then provided the diameters hK are small enough, there exists a constant C such that, f o r all finite elements in the family, and all functions u E W3*p(K),

Ib - nKv1Im.q.K d c(meas(R))”q-”ph~-m(IUI~.p.K + IuI~.~.K). (4.3.37)

Proof. The inclusion (4.3.35) guarantees the existeye of the interpolation operators fi and &, which satisfy the relation (4.3.12). Combining the inequalities of Theorems 4.3.2 and 4.3.3, we obtain, if m = 0, 1 or 2, respectively,

1 +-I6 hK

- fi~lr,q,k)).

By virtue of the inclusions (4.3.35) and (4.3.36), we may argue as in Theorem 3.1.4 and infer that there exists a constant C depending only on

242

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

the set

[Ch. 4,64.3.

k such that for all 8 E W3mP(k),

s ,m. (8- ~?ijl,.~,,ts C ( G ) ~ , ~ I, R Upon combining the above inequalities, we obtain 1 hKm and another application of Theorems 4.3.2 and 4.3.3 yields: Iu - &u

Irn.q.Ks

~(mea~(R))”~-Iij)~.~.,t,

s CIJF~~I~!MIFK I:.m.d~13.p,~+ + IFK I I FK I2 , m . d v I2.p.K + IFK I3

I~I3,p.l~

I

. d u I ,p.K)

s ~ ( m e a s ~ R ~ ) - ” ~ h ~ I u IIVI~.~.K). 2.~,K+

Thus inequality (4.3.37) is proved for the values rn = 0, 1 and 2. The 0 case m = 3 is left as a problem (Exercise 4.3.7). It is interesting to compare the estimates of the above theorem with the analogous estimates obtained for a regular afine family of n-simplices of type (2) (cf. Theorem 3.1.6): (Iu - n

K ~ l l ~ . ~ ~. K( sm

eas~R~)~’~-”~~~~~(u(~,~.~.

We conclude that the two estimates coincide except for the additional semi-norm Iu (2.p.K (which appears when one differentiates a function composed with other than an affine function; cf. the end of the proof of Theorem 4.3.2). Also, the present estimates have been established under the additional assumption that the diameters hK are sufficiently small, , k (cf. basically to insure the invertibility of the derivative? D F K ( ~f) E the proof of Theorem 4.3.3).

Remark 4.3.3. (i) Just as in the case of affine families (cf. Remark 3.1.3), the parameter meas@) can be replaced by h; in inequality (4.3.37), since it satisfies (cf. (4.3.26)) the inequalities u,,u-”h~ s meas& s u,,hk,

where a. denotes the dx-measure of the unit sphere in R”. (ii) If necessary, the expression ( I U ( ~ , ~ . K + I V ( 3 , p . K ) appearing in the right-hand side of inequality (4.3.37) can be of course replaced by the 0 expression 1(. I$,p.K+ Iul$.p.K)””* Similar analyses can be carried out for other types of simplicia1 finite

Ch. 4, 9: 4.3.1

ISOPARAMETRIC FINITE ELEMENTS

243

elements, such as the isoparametric n-simplex of type (3) (cf. Exercise 4.3.8). For the general theory, which also applies to isoparametric Hermite finite elements, see CIARLET& RAVIART(1972b). If we turn to quadrilateral finite elements, the situation is different. Of course, we could again consider this case as a perturbation of the affine case. But, as exemplified by Fig. 4.3.4, this would reduce the possible shapes to "nearly parallelograms". Hopefully, a new approach can be developed whereby the admissible shapes correspond to mappings FK which are perturbations of mappings in the space (QI(R))", instead of the space (PI(&)". Accordingly, a new theory has to be developed, in particular for the quadrilateral of type (l), as indicated in Exercise 4.3.9.

Exercises 4.3.1. Let (k,fi,$) be a Hermite finite element where the order of directional derivatives occuring in the definition is one, i.e., the set $ is of the form

with degrees of freedom of the following form:

4;: @ +@(Ci!), 4:: @ + D@(Cif)&. Let F: k +R" be a differentiable one-to-one

(i) mapping. Let a! = F(Ciio), 1 d i d No; at = F(&') and 6; = DF(df)&, 1 d k d di, 1 d i d NI. Then show that the triple (K, P,Z)is a Hermite finite element, where

1

K

P = (p: K 4 R ; p = @ F-', @ E fi}, 1 d i d No; pi, 1 d k d di, 1 d i zs NJ,

= F(@,

Z = {Q!, Q!: P

+P(a!),

Qik P

+DP(ai?6:,

and show that (nu)* = I%. (ii) If the mapping F belongs to the space (fi)", one obtains in this fashion an isoparametric Hennite finite element. In this case, write the mapping F in terms of the basis functions of the finite element (k,fi, 3). Deduce that, in practice, the isoparametric finite element is completely determined by the data of the points a! and of the vectors 6:. Why are the points af not arbitrary? Show that this is again a generalization of affine equivalence.

244

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.64.3.

(iii) Using (ii). construct the isoparametric Hermite triangle of type (3) which is thus defined by the data of three “vertices” a,, 1 d i d 3, two directions at each point a,, 1 S i d 3, and a point ~ 1 2 3 . (iv) Examine whether the construction of (i) and (ii) could be extended to Hermite finite elements in the definition of which higher order directional derivatives are used. For a reference about questions (i), (ii) and (iii), see CIARLET& RAVIART (1972b). 4.3.2. Let ( K , P, Z)be an isoparametric finite element derived from a finite element ( k ,@, 2) by the construction of Theorem 4.3.1. Show that if the space @ contains constant functions, then the space P always contains polynomials of degree 1 in the variables xI,x2,. . . ,xn. Isn’t there a paradox? 4.3.3. Give a description of the space PK corresponding to the isoparametric triangle of type (2). In particular show that one has P K # P 2 ( K ) in general (although the inclusion P I ( K )c PK holds; cf. Exercise 4.3.2). 4.3.4. (i) Let a, and a,, i # j , be two “vertices” of an isoparametric triangle of type (2). Show that the curved “side” joining these two points is an arc of parabola uniquely determined by the following conditions: It passes through the points a,, a,, a,, and its asymptotic direction is parallel to the vector a,, - ( a , + a,)/2. (ii) Use the result of (i) to show that the mapping F corresponding to the following data is not invertible: a l = ( 0 , 0 ) , a 2 = ( 2 , 0 ) , a3=(072), a23 = (0, 1).

al2=(190), a l 3 = ( 1 ,

11,

4.3.5. Given a regular isoparametric family of n-simplices of type (2), d o we have diam K = diam K for hK = diam K sufficiently small? 4.3.6. In Theorem 4.3.3, it was shown that the isoparametric mappings FK are one-to-one (for hK small enough) by an argument special to isoparametric n-simplices of type (2). Give a more general proof, which would apply to other isoparametric finite elements. 4.3.7. Complete the proof of inequalities (4.3.37) by considering the case m = 3. 4.3.8. (i) Carry out an analysis similar to the one given in the text for the isoparametric n-simplex of type (3) (cf. Fig. 4.3.2 for n = 2). Introducing the unique affine mapping FK which satisfies PK(~,) = a,, 1 S i S n + 1, $how that one obtains interpolation error estimates of the

Ch. 4, 5 4.3.1

245

ISOPARAMETRIC FINITE ELEMENTS

form IIv - H

~ m,q,K I s C(meas(R))"q-l'pht m l I4 . p . K 1

i.e., as in the affine case, provided we consider a regular isoparametric family for which condition (iii) of (4.3.27) is replaced by the following: (*)

1laiij.K

- fiiij.AI=

hi),

1 =S i, j

sn

+ I,

i+ j,

(**) I ~ u ~ ~ ~ , ~ - ~ ~ ~ ~ 1 c, iJ--. 4

(4.4.68)

Then, if hypothesis (Hl) holds, there exists a constant C independent o f h such that

(4.4.69)

where h = maxKETh hK. Proof. By Theorem 4.4.2, the approximate bilinear forms are uniformly Vh-elliptic and therefore, we can use the abstract error estimate (4.4.21) of Theorem 4.4.1. (i) Since n C 5 , the inclusion H3(d)4 % O ( f i ) holds, and by Theorem 4.4.3, the function nhii belongs to the space x o h since on the boundary r, we have u = ii = 0. Thus we may let V h = nhii in the term inf,,,,,(. . .}

Ch.4, 5 4.4.1

2nd-ORDER PROBLEMS OVER CURVED DOMAINS

261

(4.4.70)

By Theorem 4.4.3, we know that llfi

- nhclll,fihd ChzIIfil13,nh 6 Ch21(fi((,,n.

(4.4.71)

(ii) To evaluate the two consistency errors, a specific choice must be made for the functions a'ij which appear in the bilinear form hh(. ,.): We shall choose precisely the fun_ctions given in (4.4.67). Notice that, since the inclusion W z * m ( f4 i ) Ul(fi) holds, the functions a'ij are in particular defined everywhere on the set 6.Then we have, for all wh E vh,

and, since all the quadrature nodes bl,K belong to the set 6,we have aij(bl,K) = Zij(b1.K). Consequently, we can rewrite the above expression as

Using the estimates of Theorem 4.4.4 and the Cauchy-Schwarz inequality, we obtain

268

F I N I T E E L E M E N T M E T H O D S F O R 2nd O R D E R P R O B L E M S

[Ch. 4.8 4.4.

By another application of Theorem 4.4.3,

IIfilI2.nh+Chllfil13.nh

CIlfil13,fi,

and thus, we have shown that SUP whEVh

Ia‘h(flhfi7

wh)-

11 wh 11 I

ah(nhfi, wh)l

n

2

s ch2

IIfiijII~,m,~IfiII~,fi~

i.j=l

,ah

(4.4.72) (iii) Let us next examine the expression which appears in the numerator of the second consistency error. First it is easily verified that assumptions (4.4.67) imply in particular that the functions (aijaifi)belong to the space H ’ ( 0 ) . Therefore Green’s formula yields

=

Since we have

I,,

fWh

dx.

the function fgiven in (4.4.68) is an extension of the function f. Besides, using once again the fact that all integration nodes bl,Kbelong to the set 6,we obtain f ( b / . K ) = f ( b l . Kand ) conse-quently, we can write

Using the estimates of Theorem 4.4.5, we get

Ch. 4, 0 4.4.1

2nd-ORDER PROBLEMS OVER CURVED DOMAINS

269

By construction, the interiors of the n-simplices do not overlap and therefore the quantity ZKEThmeas(g) = meas( K~~~ k ) is clearly bounded independently of h. Thus, we have shown that

u

(4.4.73)

and inequality (4.4.69) follows from inequalities (4.4.70), (4.4.7l), (4.4.72) and (4.4.73). 0 We have therefore reached a remarkable conclusion: In order t o retain the same order of convergence a s in the case of polygoral domains (when only straight finite elements are used), the same quadrature scheme should be used, whether it be f o r straight or f o r isoparametric finite elements. Thus, if n = 2 for instance, we can use the quadrature scheme of (4.1.17), which is exact for polynomials of degree s 2. Remark 4.4.4. (i) As one would expect, it is of course true that, in the absence of numerical integration, the order of convergence is the same, i.e., one has Ilk- fihlll,fZh= O ( h 2 ) , where 6 is now the solution of the discrete problem (4.4.6). To show this is the object of Exercise 4.4.3. (ii) To make the analysis even more complete, it would remain to show that for a given domain with a curved boundary (irrespectively of whether or not numerical integration is used), isoparametric n-simplices of type (2) yield better estimates than their straight counterparts! Indeed, STRANG & BERGER(1971) and T H O M ~(1973b) E have shown that one gets in the latter case an O(h3”) convergence. In this direction, see Exercise 4.4.4. 0 Remark 4.4.5. By contrast with the case of straight finite elements (cf. Remark 4.1.8), the integrals J K aijaiuhajvh dx are no longer computed exactly when the coeficients aij are constant functions. If K is an isoparametric n-simplex of type (2), we have

270

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4 , 8 4.4.

and (cf. (4.4.46)), JF

(-f )(aiP I ) ( 2) A

-

n

- I?

a $ ' ( i ) det(a,F(i), . . . ,

X a k - ~ F ( i )ei, , a,+iF(f), . . . a n F ( i ) ) 9

= {polynomial of degree

s n in

a},

(ajp)^(x)= (JF(?))-' x {polynomial of degree d n in 2). Since

. . . ,& F ( i ) )

J F ( i ) = det(alF(i),

= {polynomial of degree S n in

we eventually find that dip'dip dx =

I

a},

{polynomial of degree d 2n in i } di.

g {polynomial of degree s n in 2)

Therefore the exact computation of such integrals would require a quadrature scheme which is exact for rational functions of the form NID with N E P 2 , , ( k >D , E P.(k). 0 Exercises With the ;otations of Fig. 4.4.3, show that the image bK = F K ( ~ ) of any point 6 E K belongs to the set d f l K provided h~ is small enough. 4.4.2. With the same assumptions as in Theorem 4.4.4, show that the estimates lEK(~aip'ajp)l ChKllaIl2.m.dlP'II1,KIP) 1.K 4.4.1.

hold. Deduce from these another proof of the uniform Vh-ellipticity of the approximate bilinear forms (this type of argument is used by ZLAMAL(1974)). 4.4.3. Analyze the case where isoparametric n-simplices of type (2) are used without numerical integration, i.e., the discrete problem is defined as in (4.4.6). [Hint: After defining appropriate extensions of the functions aii so that the discrete bilinear forms are uniformly Vh-elliptic, use the abstract error estimates of Theorem 4.4.1. This type of analysis is carried out in SCOTT (1973a).]

Ch. 4, 94.4.1

27 1

2nd-ORDER PROBLEMS OVER CURVED DOMAINS

Fig. 4.4.4

4.4.4. Assume that the set R is a bounded convex domain in R2.Given a triangulation made up of triangles with straight sides only, let Xh denote the finite element space whose generic finite element is the triangle of type (2), and let v h = { o h E xh; U h = 0 on r h } , where r h is the K~~~ K (cf. Fig. 4.4.4(a)). boundary of the set Show that (cf. STRANG & BERGER(1971), THOMBE(1973b); see also STRANG & FIX(1973, Chapter 4))

u

IIu - uhlll,Rh = O(h3'*)9

where

uh

E

v h

is the solution of the equations

(one should notice that in this case, the Xh-interpolant of the solution u does not belong to the space v h ) . In other words, triangulations of type (b) are asymptotically better than triangulations of type (a) (cf. Fig. 4.4.4). 4.4.5. Analyze the case where n-simplices of type (1) are used, with or without numerical integration, over a curved domain R. It is assumed that all the vertices which are on the boundary of the set 6,= K ~ KY are also on the boundary r.

u

~

272

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

Bibliography and comments

The content of this section is essentially based on CIARLET& RAVIART (1972c, 1975) and RAVIART (1972). The abstract error estimate of Theorem 4.1.1 is based on STRANG (1972b). The proof of the uniform Vh-ellipticitygiven in Theorem 4.1.2 is based on, and generalizes, an idea of G. Strang (STRANG& FIX(1973, Section 4.3)). Theorem 4.1.3 is due to BRAMBLE& HILBERT(1970). It is recognized as an important tool in getting error estimates in numerical integration and interpolation theory (although we did not use it in Section 3.1). In CIARLET& RAVIART(1975), the content of this section is given a general treatment so as to comprise as special cases the inclusions P k ( k )C I; c P k f ( k )(cf. Exercise 4.1.6), the case of quadrilateral elements (cf. Exercise 4.1.7), etc.. . As regards in particular the error estimate in the norm J . J O , ~ (cf. the abstract error estimate of Exercise 4.1.3), the following is proved: Assuming that the adjoint problem is regular and that P = P k ( k ) ,one has Iu - Uh(0.R= O(hk+')if the quadrature scheme is exact for the space P 2 k - 2 ( k if) k 2 2, or if the quadrature scheme is exact for the space P I ( & if k = 1. For other references concerning the effect of numerical integration, (1968) see BABUSKA& AZIZ (1972, Ch. 9), FIX(l972a, 1972b), HERBOLD where this problem was studied for the first time, HERBOLD, SCHULTZ & VARGA (1969), HERBOLD& VARGA (1972), ODEN & REDDY(1976a, Section 8.8), SCHULTZ (1972), STRANG & FIX(1973, Section 4.3). Comparisons between finite element methods (with or without numerical integration) and finite-difference methods are found in BIRKHOFF & GULATI (l974), TOMLIN (1972), WALSH (1971). Examples of numerical quadrature schemes used in actual computations are found in the book of ZIENKIEWICZ (1971, Section 8.10). For general introductions to the subject of numerical integration (also known as : n um erica I quadrat ure, approximate integration, approximate quadrature), see the survey of HABER(1970), and the books of DAVIS& RABINOWITZ(1974), STROUD (1971). For studies of numerical integration along the lines developed here, see also MANSFIELD(1971, 1972a). In ARCANGELI& GOUT(1976) and MEINGUET (1979, the constants appearing in the quadrature error estimates are evaluated. 4.2. The abstract error estimate of Theorem 4.2.2 is due to STRANG 4.1.

Ch. 4.1

BIBLIOGRAPHY A N D C O M M E N T S

273

(1972b). The description of Wilson’s brick is given in WILSON& TAYLOR (197 1). In analyzing the consistency error, we have followed the method set up in CIARLET(1974a) for studying nonconforming methods, the main idea being to obtain two polynomial invariances in the functions DK(. , .) so as to apply the bilinear lemma. For the specific application of this method to Wilson’s brick, we have extended to the three-dimensional case the analysis which LESAINT (1976) has made for Wilson’s rectangle. P. Lesaint has considered the use of this element for approximating the system of plane elasticity, for which he was able to show the uniform ellipticity of the corresponding approximate bilinear forms. In this fashion, P. Lesaint obtains an O ( h ) convergence in the norm ll.llh and an O(hz) convergence in the norm (the corresponding technique is indicated in Exercise 4.2.3). Also, the idea of introducing the degrees of freedom J K ajjp dx is due to P. Lesaint. In his pioneering work on the mathematical analysis of nonconforming methods, G. Strang (cf. STRANG(1972b), and also STRANC& FIX (1973, Section 4.2) where the study of Wilson’s brick is sketched) has shown in particular the importance of the patch test of B. Irons (cf. IRONS & RAZZAQUE(1972a)). For more recent developments on the connection with the patch test, see OLIVEIRA (1976). There are other ways of generating nonconforming finite element methods. See for example RACHFORD& WHEELER(1974). In NITSCHE (1974), several types of such methods are analyzed in a systematic way. See also CEA (1976). References more specifically concerned with nonconforming methods for fourth-order problems are postponed till Section 6.2. 4.3. This section is based on CIARLET& RAVIART(1972b), where an attempt was made to establish an interpolation theory for general isoparametric finite elements (in this direction see Exercises 4.3.1, 4.3.8 and 4.3.9). A survey is given in CIARLET(1973). To see that our description indeed coincides with the one used by the Engineers, let us consider for example the isoparametric triangle of type (2) as described by FELIPPA & CLOUGH(1970, p. 224): Given six points ai = (ali, a2i) 1 C i G 6, in the plane (the points ad, u5 and a6 play momentarily the role of the points which we usually call a I 2 , 023 and a133 respectively), a “natural” coordinate system is defined, whereby the following relation (written in matrix form) should hold between the Cartesian coordinates xI and x2 describing the finite element and the

I.OR

274

HNITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

“new” coordinates A l , A2 and A S :

Then we observe that the first two lines of the above relation precisely represent relation (4.3.7), with F(f)= ( F , ( i ) ,Fz(f))now denoted (xI,x2). The last line of the above matrix equation implies either h l + h2+ h3 = 1 or A l + h 2 + h s = -f, so that the solution A 1 + A 2 + h 3= 1 is the only one which is acceptable if we impose the restriction that hi 3 0, 1 d i d 3. Therefore, the “natural” coordinates h l , h2 and A 3 are nothing but the barycentric coordinates with respect to a fixed triangle K, and the isoparametric finite element associated with the points ai, 1 d i d 6, is in this formulation the set of those points (xl, x2) given by the first two lines of the above matrix equation when the “natural” coordinates hi (also known as “curvilinear” coordinates) satisfy the inequalities 0 d hi d 1, 1 C i G 3, and the equality B t l hi = 1. A general description of isoparametric finite elements along these lines (1971, chapter 8). The first references is also found in ZIENKIEWICZ where such finite elements are found are ARGYRIS& FRIED(1968) and ERGATOUDIS, IRONS& ZIENKIEWICZ(1968). In case of isoparametric quadrilateral elements, JAMET (1976b) has significantly contributed to the interpolation error analysis, by relaxing some assumptions of CIARLET& RAVIART(1972b). Curved finite elements of other than isoparametric type have also been considered, notably by ZLAMAL (1970, 1973a, 1973b, 1974) and SCOTT (1973a). Both authors begin by constructing a curved face K ’by approximating a smooth surface through an ( n - 1)-dimensional interpolation process. This interpolation serves to define a mapping FK which in turn allows to define a finite element with K’ as a curved face. Then the corresponding interpolation theory follows basically the same pattern as here. In particular, R. Scott constructs in this fashion a curved finite element which resembles the isoparametric triangle of type (3) and

Ch. 4.1

BIBLIOGRAPHY AND COMMENTS

275

for which an interpolation theory can be developed which requires weaker assumptions than those indicated in Exercise 4.3.8. In ARCANGBLI& GOUT (1976), a polynomial interpolation process over a curved domain is analyzed. For curved finite elements based on the so-called blending function interpolation process, see notably CAVENDISH,GORDON & HALL (1976), GORDON & HALL (1973), the paper of BARNHILL(1976a) and the references therein. WACHSPRESS (1971, 1973, 1975) uses rational functions for constructing general polygonal finite elements in the plane with straight or curved sides. For additional references, see LEAF, KAPER & LINDEMAN(1976), LuKAS (1974), MCLEOD & MITCHELL(1972, 1976), MITCHELL(1976), MITCHELL & MARSHALL(1975). 4.4. The error analysis developed in this section follows the general approach set up in CIARLET& RAVIART(1972~)(however it was thought at that time that more accurate quadrature schemes were needed for isoparametric elements), where an estimate of the error in the norm I.lo,n was also obtained. An analogous study is made in ZLAMAL(1974), where it is shown that, for two-dimensional curved elements for which fi = P&), k even, it is sufficient to use quadrature schemes exact for polynomials of degree S 2 k - 2, in order to retain the O ( h k ) convergence in the norm Il.l l,nh. ZLAMAL(1973b) has also evaluated the error in the absence of numerical integration. For complementary results, see VEIDINGER(1975). Likewise, SCOTT(1973a) has shown that quadrature schemes of higher order of accuracy are not needed when curved finite elements are used. However, the finite elements considered by M. Zltimal and R. Scott are not of the isoparametric type as understood here. For such elements, a general theory is yet to be developed, in particular for quadrilateral finite elements. In spite of the absence of a uniform V,,-ellipticity condition, GIRAULT (1976a) has successfully studied the use of quadrilaterals of type (1) in conjunction with a one-point quadrature scheme. Alternate ways of handling Dirichlet problems posed over domains with curved boundaries have been proposed, which rely on various alterations of the bilinear form of the given problem. In this direction, we notably mention (i) penalty methods, as advocated by AUBIN (1969) and BABUSKA (1973b), and later improved by KING(1974), (ii) methods where the boundary condition is considered as a con-

276

FINITE E L E M E N T METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

straint and as such is treated via techniques from duality theory, as in BABUSKA (1973a), (iii) least square methods as proposed and studied in BRAMBLE& SCHATZ (1970, 19711, BRAMBLE & NITSCHE(1973), BAKER(1973), (iv) methods where the domain is approximated by a polygonal domain, as in BRAMBLE, DUPONT& THOMBE(1972), (v) various methods proposed by NITSCHE(1971, 1972b). For additional references for the finite element approximation of boundary value problems over curved boundaries, see BABUSKA (1971b), BERGER(1973), BERGER, SCOTT& STRANG(1972), BLAIR (l976), BRAMBLE (1979, NITSCHE(1972b), SCOTT(1975), SHAH(1970), STRANG & BERGER(1971), STRANG & FIX(1973, Chapter 4), THOMBE (1973a, 1973b). See also Chapter 6 for fourth-order problems. We finally mention that, following the terminology of STRANG (1972b), we have perpetrated in this chapter three variational crimes: numerical integration, nonconforming methods, approximation of curved boundaries.

Additional bibliography and comments

Problems on unbounded domains Let us consider one physical example: Given an electric conductor which occupies a bounded volume d in R3,and assuming that the electric potential uo is known along the boundary r of the set R, the electric conductor problem consists in finding the space distribution of the electric potential u. This potential u is the solution of

r

Au = O AU = O

in in u = uo on

R,

nr=C6,

r.

Thus, in addition to a standard problem on the set 6, we have to solve a boundary value problem on the unbounded set 6’. Classically, this problem is solved in the following fashion: Denoting by d,u the normal derivative of U I A across r and by (a,u)’ the normal derivative of ulfi. across r (both normals being oriented in the same direction), let q = a,u - (8,u)’.

Ch. 4.1

277

ADDITIONAL BIBLIOGRAPHY AND COMMENTS

Then if the function q is known on I’, the solution u is obtained in R3as a single layer potential through the formula Vx E R3, u(x) =

4,r

I

rllx-

dy(y).

YII

By specializing the points x to belong to the boundary r in the above formula, we are therefore led to solve the integral equation:

in the unknown q. For details about this classical approach, see for instance PETROVSKY (1954). Interestingly, this integral equation can be given a variational formulation which, among other things, make it amenable to finite element approximations, as shown by NBDBLEC & PLANCHARD (1973). First we need a new Soboleu space, the space

~ ” ’ (=r{ r)E L’(T);3u E ~ ‘ ( 0tr )u =;r

on r ) ,

which is dense in the space L’(r). It is a Hilbert space when it is equipped with the quotient norm

r E ~ ” ’ ( r(JrJJHqr) ) + = inf{llu((l.a;u E ~

‘ ( atr )u ,= r

on

r}.

We shall denote by H-’/’(T)its dual space, and by l ~ ~ l ~ H - i / z ( r )the dual norm. Denoting by (., *>rthe duality pairing between the spaces H-”’(I‘) and H”’(r), we note that V r E L’(r)c H-”’(T), Vs E H”’(I‘),

(r, s)r =

r

rs dy.

For details about these spaces (and more generally about the spaces H ‘ ( r ) , t E R ) , see LIONS& MAGENES(1968). The bilinear form

and it is continuous when is well-defined over the space 9 ( T )x 9(r), the space 9(r)is equipped with the norm ~ ~ ~ ~ I H - i ~ 2 ( ,Consequently, -,. it has a unique extension over the space H - ’ / ’ ( r )x H - ” ’ ( r ) , which shall be denoted by a ( . ,*). and one can show that this bilinear form is H-”’(r)-

278

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

elliptic. Therefore, the natural variational formulation of the problem posed above as an integral equation consists in finding the unique function q which satisfies q E H - ” 2 ( r ) and

V r E H - ’ ” ( r ) , a(q, r ) = (uo, r)r,

assuming the data uo belongs to the space H”’(T). Once the function q is found in this fashion, the solution u of the original problem is obtained as follows. Define the space W,’(R’) as being the completion of the space 9(R3) with respect to the norm I * l l r 3 . This space (which does not coincide with the space H1(R3),i.e., the completion of the space 9(R3) with respect to the norm 11.111,~3) can be equally characterized by (cf. BARROS,NETO (1963, DENY & LIONS (1953-1954))

W,’(R3)= { u E L6(R’); aiu E L2(R3); 1 d i S 3)

Then for each function q E 9(r), the function u : x E R3+u(x) =

1‘

-dy(y)

4~ rllx-~ll

belongs to the space Wd(R3) and, besides, the mapping q E 9(T)+u E Wd(R3) defined in this fashion is continuous when the space 9(r)is equipped with the norm l l - l l H - i n ( r ) . Therefore, it has a unique extension over the space H-”’(r). In other words, we have solved the original problem via the mappings uoE H ’ l 2 ( r += ) q E H-’I2(r)+ u E Wd(R3) (as indicated in NBDBLEC& PLANCHARD( 1973),one can also solve directly the problem uoE H ’ ” ( r )+= u E Wi(R3)). We mention that the related perfect dielectric problem can be also handled in an analogous manner (the boundary condition u = uo on r is then replaced by d,u - c(&u)’ = ul on

r, c > 0).

J.C. NCdClec and J. Planchard then construct a general finite element approximation of the above problem. Given a subspace of the space H - ’ ” ( r ) , they first derive an abstract error estimate: Let U0h E v h be an approximation of the function uo and let the discrete solution qh be such that vrh E

vh,

a(qh, r h ) = ( U O h , rh)r.

Ch. 4.1

279

ADDITIONAL BIBLIOGRAPHY A N D COMMENTS

Then there exists a constant C independent of the subspace

v h

such that

To apply this error estimate the authors assume that the boundary r is polygonal so that it may be triangulated in an obvious fashion, i.e., the set r is written as a union KETh K of triangles K. Then they look for a discrete solution in either space

u

VOh={Uh:r’R;

V l h

= {uh E

V K E Y h ,

V o ( r ) ;V K E y h ,

~~~K€Po(K)}CL~(~)CH-”~(~) VhlK

(note that the functions in the space adjacent triangles) and they show that

E P , ( K ) } C H ’ ( r )c H - ’ ” ( r )

V0h

are discontinuous across

assuming the function 4 is smooth enough. To conclude their analysis, they compute the function

and they obtain in both cases

Ilu - Uhll Wb(R3)

C(4)h3l2 + ((u- UOh11H”2(r)*

Of course, the major computational difficulty in this approach is the evaluation of the coefficients of the resulting linear system. For a review of the numerical aspects of such integral equation techniques for solving problems on unbounded domains arising in the study of 2- or 3-dimensional incompressible potential flows around obstacles, see HESS(1975a, 1975b). NBDBLEC(1976) next considers the case of a curved surface which needs therefore to be approximated by another surface r h made up of finite elements of isoparametric type (such a construction is related to - and is of interest for - the surface approximation found in the shell problem; cf. Section 8.2). Again error estimates for the differences (4- 4 h ) and ( u - u h ) are obtained in appropriate Hilbert spaces. LE Roux (1974, 1977) considers the finite element approximation of the analogous problem in dimension two. In this case, the kernel in the

r

280

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch.4.

integral transform is In Ilx - yll instead of l/llx - y(1. A similar analysis is found in HSIAO& WENDLAND(1976). There are other ways of handling problems on unbounded domains. In particular, there are methods where the unbounded domain is triangulated and then the triangulation is “truncated” in some fashion. In this spirit, BABUSKA (1972~)considers the model problem: Find u E H’(R”) such that - d u + u = f in R”,f E L2(R”).Using an “abstract” variational approximation (cf. BABUSKA(1970, 1971a)), he obtains orders of convergence on compact subsets of R”which are arbitrarily close to the orders of convergence obtained in the case of bounded domains. By contrast with the method of FIX& STRANG (1969), the discrete solution is obtained via the solution of a linear system with a finite number of unknowns. In SILVESTER & HSIEH (1971), a bounded subdomain is triangulated in the usual way while the remaining unbounded part is represented by a single “finite element” of a special type. As we shall mention in the section “Bibliography and Comments” of Section 5.1, problems on unbounded domains which typically arise in the study of 2-dimensional compressible flows may be reduced to variational inequalities, as in CIAVALDINI & TOURNEMINE (1977) and Roux (1976).

The Stokes problem Classically, the Stokes problem for an incompressible viscous fluid in a domain d c R”,n = 2 or 3, consists in finding functions u = (ui)?=,and p defined over the set 6,which satisfy (Au = (Aui)Z1) - vAu

+V p =f

div u = 0 in u = o on r.

in 0, 0,

The vector function u represents the velocity distribution, the scalar function p is the pressure, and the given vector function f = (fi)Y=l E ( L ‘ ( 0 ) ) ”represents the volumic forces per unit mass. The constant Y > 0 is the dynamic viscosity, a constant which is inversely proportional to the Reynolds number. In order to derive the variational formulation of this problem, we introduce the space V = { u E ( H d ( 0 ) ) ” ;div u = 0 )

Ch. 4.1

ADDITIONAL BIBLIOGRAPHY AND COMMENTS

28 1

provided with the norm

and we introduce the bilinear form u, u E ( H ' ( 0 ) ) "+ a ( u , u ) =

i.j= I

I 0

a,uiaivi dx,

which is clearly V-elliptic. We shall use the notation (u, u ) =

In

uu dx,

(u , u ) =

u

*

u dx.

Since one has (Vp, u ) = - ( p , div u ) for all u E 9(R)for smooth functions p , the natural variational formulation of this problem consists in finding a pair (u, p ) such that (u,p ) €

v x (L2(n)/PO(n))rand

V u E (Hd(O))", va(u, u ) - ( p , div u ) =

v. U )

(notice that the definition of the space L 2 ( n ) / P o ( n reflects ) the fact that the unknown p can be determined up to additive constants only). Then we observe that the relations V u E V,

ua(u, u ) =

v, u )

determine uniquely the function u (a word of caution: Since the space (9(n))"is not contained in the space V, the above variational problem cannot be interpreted in the usual way as a boundary value problem). Once the function u is known, it remains to find a function p E L 2 ( n ) / P o ( Rsuch ) that V u E (Hd(fW", ( p , div u ) = g ( u ) ,

where the linear form g: u E

)" -,g(u)

= va(u, u ) - ~ f u, )

is continuous over the space (Hd(0))"and vanishes over its subspace V, by definition of the function u. It then follows that there exists a function p E L2(f2), unique up to an additive constant factor, such that the linear form can also be written as V u E (HA(R))", g ( u ) =

I, div p

u dx.

282

FINITE ELEMENT METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

This is a nontrivial fact whose proof may be found in de RHAM(1955) (the converse is clear). Notice that if n = 2, the Stokes problem can be reduced to a familiar problem: Since div u = 0, there exists a stream function such that uI = a,+ and u2 = - &+. Then a simple computation shows that uAz+ = f with f = &fz- af,. When the set 0 is simply connected, we may impose the boundary condition $ = 0 on r,so that we also have &+ = 0 on r as a consequence of the boundary condition u = 0 on r. Therefore the solution o f the Stokes problem is reduced in this case to the solution of a biharmonic problem (cf. Section 1.2). Observe that the vorticity - A + is then nothing but the value of the rotational of the velocity u. Finite element methods for this problem will be described in Section 6.1 and Chapter 7. As regards the finite element approximation of the general Stokes problem, it is realized that a major difficulty consists in taking properly into account the incompressibility condition div u = 0. A first approach is to use standard finite element spaces v h in which the condition div tch = 0 is exactly imposed. However, this process often results in sophisticated elements. Methods of this type have been extensively (1972a, 1972b). studied by FORTIN In a second approach, whose applicability seems wider, the incompressibility condition is approximated. This is the method advocated by & RAVIART (1973), who seek the discrete solution in a space CROUZEIX of the form

+

where X0h is a product of standard finite element spaces and @h appears as an appropriate space of “Lagrange multipliers”, following the terminology of duality theory (cf. the section “Additional Bibliography and Comments” in Chapter 7). For instance if the generic finite element in the space XOh is the triangle - or tetrahedron - of type (k),the space @h is the product IIKETh Pk-&). In their remarkable paper, M. Crouzeix and P.-A. Raviart construct both conforming and nonconforming finite elements of special type (cf. Exercise 2.3.9) and they obtain estimates Iu - uhl:,K)1/2, and for the error (ui- Uihli.fl)”2 for the error (zKETh through an extension of the Aubin-Nitsche lemma. They also compute an approximation P h of the pressure p and they evaluate the norm (Ip - PhllL2(n),po(n).Finally, they briefly consider the case of the in-

Ch. 4.1

ADDITIONAL BIBLIOGRAPHY A N D COMMENTS

283

homogeneous boundary condition u = uo on I‘. As usual the error estimates depend upon the smoothness of the solution, a question studied in KELLOGG& OSBORN(1976), OSBORN(1976b), and TBMAM (1973). It seems however that the most promising finite element approximations of the Stokes problem are of the so-called mixed type. For such methods, the reader is referred to the section “Additional Bibliography and Comments” at the end of Chapter 7. Further references concerning the finite element approximation of the Stokes problem are FALK(1976a, 1976c), FALK& KING(1976), and the thorough treatment given by T I ~ M A (1977). M We also mention that CROUZEIX& LE Roux (1976) have proposed and analyzed a finite element method for two-dimensional irrotational fluid flows, in which the unknown u = (u,,u2) satisfies rotu = O in 0, div u = f in R, ( u . v = g on r.

Eigenvalue problems

Given an elliptic operator 9 defined on a bounded open subset R of R” and given the boundary condition u = 0 on r, the associated eigenvalue problem classically consists in finding real numbers A and functions u f 0 such that

i

9 u = A u in 0, u = 0 in r.

Indeed, eigenvalue problems may be associated with any other homogeneous boundary conditions but, for simplicity, we shall consider only the Dirichlet condition. Such problems typically arise when one looks for periodic (in time) solutions of evolution problems of the form &,u + 2 u = 0, u = 0 on I’, where am denotes the second partial derivative with respect to the time variable t. Such particular solutions being of the form u(x)eir”,p E R, the pair (A, u ) , A = p2, is therefore obtained through the solution of an eigenvalue problem. This is why such problems are of fundamental importance, in the analysis of vibrations of structures for instance.

284

FINITE E L E M E N T METHODS FOR 2nd ORDER PROBLEMS

[Ch. 4.

We shall in fact consider the variational formulation of this eigenualue problem, which consists in finding pairs (A, u ) , A E R, u E V - {0}, such that Vu E V

a ( u , u ) = A(u, u ) ,

where V = Hd(J.2) or H&O), depending upon whether 3 is a secondorder or fourth-order operator, a ( - ,.) is the associated bilinear form (i.e., which satisfies a ( u , u ) = ( P u , u ) for all u E 9(O)), and (., .) is the innerproduct in the space L ’ ( 0 ) . If (A, u ) is a solution, then u is called an eigenfunction associated with the eigenualue A. Let us make the usual assumptions that the bilinear form is continuous and V-elliptic, so that for each f € L 2 ( n ) there , exists a unique function u E V which satisfies a ( u , u ) = ( f , u ) for all u E V (if we identify the function f with an element of V’, we have u = A-If with the notations of Theorem 1.1.3). In this fashion, we define a mapping

G : f E L 2 ( a ) + u = G f E V, which is continuous (cf. Remark 1.1.3), and consequently, the mapping

G: V + V is compact, by Rellich theorem. Since ( u , u ) = a ( G u , u ) for all u, u E V by definition of the mapping G, the eigenualue problem amounts to finding the inverses of the eigenualues of the mapping G : V + V (clearly, zero cannot be an eigenvalue of the mapping G nor of the original problem). If we finally add the assumption of symmetry of the bilinear form, then the problem is reduced to that of finding the eigenvalues and eigenfunctions of a compact symmetric operator in the Hilbert space V, considered as equipped with the inner-product a ( . , .) (the symmetry is a consequence of the equalities a ( G u , u ) = ( u , u ) = ( u , u ) = a(Gu, u ) = a ( u , Gu)). Consequently, an application of the spectral theory of such operators (cf. e.g. RIESZ & NAGY (1952)) yields the following result concerning the existence and characterizations of the solutions of the eigenvalue problem: There exists an increasing sequence of strictly positive eigenualues :

0 < hi

4 A2 4

* * *

< hk < hk+l
)0 with limh4 ~ ( h=)0 such that the functions ( 1 + € ( h ) ) - ’ n h u belong to the set uh. Using this result, show that IIu - !.dhI(l,O= O(h”2-”p).

Ch. 5 , 5 5.2.1

5.2.

T H E MINIMAL SURFACE PROBLEM

30 I

The minimal surface problem

A formulation of the minimal surface problem

Let R be a bounded open subset of the plane R2, and let uo be a function given on the boundary of the set R. The minimal surface problem consists in finding a function u which minimizes the functional

r

J: v - + J ( v ) = / o m d x

(5.2.1)

over an appropriate space of functions which equal U o on r. In other words, among all surfaces given by an equation x3 = u ( x I ,XZ), x = (x,, x2) E R (for which the area can be defined) and which pass through a given curve of the form x 3 = u o ( x Ix2), , ( x , , x2) E r,one looks for a surface whose area is minimal. The mathematical analysis of this problem is not easy. In particular, it is not straightforward to decide which function space is more appropriate to insure existence and uniqueness of a solution. However, we shall not go here into such matters, refering instead the reader to the section “Bibliography and Comments” for additional information. See also Exercise 5.2.1. In this section, we shall make the following hypotheses: The set R is conuex and has a Lipschitz-continuous boundary, and the function uo is the trace over of a function (still denoted U o ) of the space H 2 ( R ) . Then, for our subsequent analysis, it will be convenient to consider that the minimal surface problem consists in finding a function u such that

r

uEU

and

J ( u ) = fif,J(u),

(5.2.2)

(5.2.3) Remark 5.2.1. The functional J of (5.2.1) is defined over any Sobolev space W’.’(R), 1 d p d 00. One reason for the present choice p = 2 is that it is easily seen that the functional J is differentiable over the 0 space H 1 ( R ) ,as we next show. For notational simplicity in the subsequent computations, it will be

302

[Ch. 5, 0 5.2.

SOME NONLINEAR PROBLEMS

convenient to introduce the function

f : x = ( x l , ~ 2 ) E R Z + f ( x ) = ~~ l=+ x : + x ~ . (5.2.4) Then, for all points x E R2 and all vectors 5 =

(el,

52)

E R2,

(5.2.5) so that, for all u, w E H'(R),

where

Therefore, the functional J is differentiable over the space HI r 3 Jh(fih) <Jh(uh).

Therefore the solutions of the minimization problem (5.2.10) coincide with the solutions of an analogous minimization problem, with the set u h replaced by the set o h

=

uh n {uh E xh;lluhIlh

r}.

Since the set o h is now compact, we have shown that problem (5.2.10) has at least one solution. Let us next turn to the question of uniqueness. It follows from the equality of (5.2.5) that the function f defined in (5.2.4) is strictly convex. This will in turn imply that the function Jh: U + R is also strictly convex: To prove this, let uh and Wh be two distinct elements of the set Uhr let

and let 8 be a given number in the interval 10, l[. Then

Ch. 5 , 8 5.2.1

THE MINIMAL SURFACE PROBLEM

305

Since the set Uh is convex, the minimization problem (5.2.10) has a unique solution. 0 Remark 5.2.2. The same argument shows that the minimization problem (5.2.2) has at most one solution. 0

We next obtain an error estimate in the norm

Il. l ll, n h.

Theorem 5.2.2. Assume that the solution u of the minimization problem (5.2.2) exists and is in the space H 2 ( 0 )fl W'*"(n).Then, given a regular family o f triangulations, there exists a constant C ( u )independent of h such that

(lu- uhlll.nh C(u)h.

(5.2.13)

Proof. In what follows, the notation C ( u ) stands for various constants solely dependent upon the solution u. For clarity, the proof has been subdivided in five steps. The first four steps consist in establishing that (U - UhII.Rh = O(h). (i) Let us first record some relations which are consequences o f the minimizing properties o f the functions u h and u. Using (5.2.7), we know that vvh E

uh,

JL(uh)(vh

- u h ) 3 0.

But in view of the particular form of the set inequalities are equivalent to the equations vwh EXOhy

JA(Uh)Wh

u h

(cf. (5.2.9)), these

= 0,

where, as usual, (5.2.14)

SOME NONLINEAR PROBLEMS

306

[Ch. 5, 5 5.2.

B y a computation similar to that which led to (5.2.6), we deduce that (5.2.15) Using again (5.2.7) and the particular form of the set U (cf. (5.2.3)), we see that v w E Hd(O), J ' ( u ) w = 0, and therefore, by (5.2.6), (5.2.16) Clearly, this application of (5.2.6) supposes that each function W h E X0h be identified with its extension to the space H,'(O) obtained by prolongating it by zero on the set O - Oh. (ii) Let us next show that, with the assumption that the solution u belongs to the space H 2 ( 0 )f l W'*"(O), there exists a constant C ( u ) such that the quantity (5.2.17) satisfies an inequality of the form Ah

C(u)h.

(5.2.18)

Let u h be an arbitrary function in the set uh, so that the function wh = u h - uh belongs to the space Xoh. Then, using relations (5.2.15) and (5.2.16) established in step (i), we can write (f(x) = cf. (5.2.4)):

m;

(5.2.19) The first integral can be bounded as follows:

Ch. 5 , 8 5.2.1

THE MINIMAL SURFACE PROBLEM

307

Therefore,

(5.2.21)

(5.2.22)

Combining relations (5.2.19), (5.2.20) and (5.2.21), we obtain v v h

Eu

h ,

A h

‘)‘(U)&

-k

(1 -I-y(u))lu - v h l t . n h .

Since the constant y ( u ) of (5.2.22) is strictly less than one, it follows that

with C ( u )= (1 + y(u))/(l- ~ ( u ) ) . Since the function u belongs to the space H’(l2) by assumption, its Xh-interpolant is well defined, and it belongs to the set u h . Thus Uh

uh

lu

- v h ) l . n h s lul*.fh,

(5.2.24)

and inequality (5.2.18) is a consequence of inequalities (5.2.23) and (5.2.24).

308

[Ch. 5, 8 5.2.

SOME NONLINEAR PROBLEMS

(iii) Let us show that Iuh(l,m,Rh

(5.2.25)

c(u).

Let K be an arbitrary triangle in the triangulation. Using step (ii) (cf. inequality (5.2.18)), one obtains

Because the restriction write

VUhlK

is constant over the triangle K , we may

(5.2.27) for some constant C independent of h. Then the conjunction of inequalities (5.2.26) and (5.2.27) implies that

Therefore, the norms llVuhlKll are bounded independently of K E y h(lim,, ( x 2 / d l+ x2) = Q))and thus property (5.2.25) is proved. (iv) Combining steps (ii) and (iii). we obtain

h

and

(5.2.28) (v) Let us add triangles K E T,” to each triangulation in Fig. 5.2.2, i.e., in such a way that, for all h,

y h

as indicated

so that the triangulations Yh U T,” again constitute a regular family (such a construction is certainly possible).

Ch. 5,

B 5.2.1

THE M I N I M A L SURFACE PROBLEM

309

Fig. 5.2.2

Because the boundary r is Lipschitz-continuous, there exists (cf. LIONS (1962, Chapter 2) or N E ~ A S(1967, Chapter 2)) an extension i.e., such that for all u E H 2 ( 0 ) ,the funcoperator E: H2(0)+H2(Rn), tion Eu E H2(R")satisfies EuJn= u and, besides, this operator is continuous: There exists a constant C ( 0 ) such that

vu E H 2 ( 0 ) ? l l E ~ I I 2 . R " Let then Eu

= u*.

C(0)llullz.n.

(5.2.29)

We define an extension uX: f?X+R of the function

uh by letting

(5.2.30) where nK denotes the P1(K)-interpolant associated with triangles of type (1). Observe that, since the function uh belongs to the set UI, as defined in (5.2.9), the function uX is continuous over the set f?$ by virtue of the second condition (5.2.30) and thus, it is in the space HWX). Finally, we shall use the following inequality, due to Friedrichs (cf. NECAS (1967), Theorem 1.9): There exists a constant C ( 0 ) such that

vu E H W ) , Let then u

= u*

an)(l~ll.n+Il~llLZU-d.

Ilu1ll.n~

- uX in this inequality. We obtain, upon combining with

310

SOME NONLINEAR PROBLEMS

[Ch. 5 , 5 5.2.

inequality (5.2.28),

Using inequality (5.2.29), we get

s

ChlIUlI2.R9

(5.2.33)

through two applications of Theorem 3.1.5. Inequality (5.2.33) implies that

and inequality (5.2.13) follows from inequalities (5.2.31), (5.2.32) and (5.2.34). 0

Exercises 5.2.1. Let 0 = { x E R2; 1 < llxll< 2}, uo= y for llxll= 1 and uo = 0 for llxll= 2, where y is a constant. Show that the associated minimal surface problem has a solution if y is smaller than a quantity y* while there is no solution if y > y* (cf. Fig. 5.2.3). [Hint: Reduce this problem to a minimization problem for functions in one variable.] This is a very simple example of a general phenomenon that R. TCmam has analyzed through the introduction of “generalized solutions” (cf. the section “Bibliography and Comments”). 5.2.2. (i) Show that the minimal surface problem amounts to formally solving a boundary value problem of the form (*)

[I

2

i z l

ai(aij(Vu)aiu)= 0 in 0,

u = uo sur

r,

Ch. 5 , 5 5.2.1

31 1

T H E MINIMAL S U R F A C E PROBLEM

Fig. 5.2.3

i.e., a nonhomogeneous Dirichlet problem for the nonlinear operator 2

u += i.i= 1

aj(a;j(vu)a;u)

and that this operator satisfies an ellipticity condition in the sense that, for any smooth enough function u, 2

3 P ( u ) > 0, V&,

i = 1,2,

2

aij(vu)& i,j= I

3p(u)

2 6:. i= I

However, the constant p ( u ) cannot be bounded below away from zero independently of u. (ii) Show that, for smooth functions, the boundary value problem (*) can also be written

312

SOME NONLINEAR PROBLEMS

Kh.

5 , § 5.3.

5.3. Nonlinear problems of monotone type A minimization problem over the space Wd.”(l2), 2 d p , and its finite

element approximation with n-simplices of type (1) Let there be given a convex open subset R of R” and let p be a number such that 2ap

(5.3.1)

(for the case where 1 < p < 2, see Exercise 5.3.2). We consider the minimization problem: Find a function u such that N

E Wd.”(R) and J(u) =

inf UE

w&n)

J(v)

(5.3.2)

where the functional J is given by

‘I

J(v)=IIVUI(”dx-f(v), P o

(5.3.3)

for some given element f of the dual space of the space Wd@(f2).We use the standard notation

For computational convenience, we shall consider throughout this section that the space Wd.”(R) is equipped with the norm (5.3.4) which is clearly equivalent to the standard semi-norm I-ll,p,o, itself a norm equivalent to the norm I . ( I . p . o over the space Wd.”(R). Finally, we shall use the notation ~~~~~* for the norm in the dual space (W$”(R))‘ of the space Wd*”(l2). Remark 5.3.1. For p = 2, this minimization problem reduces to the familiar homogeneous Dirichlet problem - A u = f in 0,u = 0 on r. El

Our proof of the existence of a solution of the minimization problem (5.3.2) (cf. Theorem 5.3.1) uses the simplest finite element approximation of this problem, which we now proceed to describe: We consider triangulations .Th made up of n-simplices K E .Th,in such a way that all

Ch. 5, Q 5.3.1

NONLINEAR PROBLEMS OF MONOTONE TYPE

313

the vertices situated on the boundary r h of the set 6, = U K E y h K also belong to the boundary r of the set R (a similar situation was considered in the previous section; see in particular Fig. 5.2.1 for n = 2). Then with each such triangulation, we associate the finite element space x h whose generic finite element is the n-simplex of type (1) (notice that the functions in the space xh are defined only on the set &), and we k t as usual

X0h

= {vh E xh; uhlr,, = 0).

Then we denote by v h the space formed by the extensions of the functions of the space x o h which vanish over the set d - a h . In fact we shall not distinguish between the functions in XOh and their corresponding extensions in the space vh. Notice that, because the set d was assumed to be convex, the inclusion v h

c wd’”(n)

(5.3.5)

and the relations u E g0(d)

and

u=O

on

r j nhv E vh

(5.3.6)

hold. Then the discrete problem consists in finding a function uh such that

uh E vh

and

J(&)

=

inf J(vh),

(5.3.7)

vhE “h

where the functional J is defined as in (5.3.3).

Theorem 5.3.1. The minimization problems (5.3.2) and (5.3.7) both have one and only one solution. Their respective solutions u E Wd*”(R) and u h E v h are also the unique solutions of the variational equations V v E Wd.”(R),

I~VU~P-*VU . V Udx = f ( v ) ,

(5.3.8) (5.3.9)

respectively. Proof. We begin by proving several properties of the functional J of (5.3.3).

[Ch. 5 , 8 5.3.

SOME NONLINEAR PROBLEMS

314

(i) Since

we deduce that (5.3.10)

lim J ( u ) = CQ.

Il4l+

(ii) Let us next establish the strict convexity of the functional J. The functional f being convex, it suffices to establish the strict convexity of the mapping uE

WdmP(fl) +

1

F ( V u ( x ) )dx, with F: S E R"+ p 11511". (5.3.1 1)

Let u and u be two different elements in the space Wd*p(fl)such that meas d > 0, where

d = {x E 0 ;V u # Vu},

and let 8 E 30, 1[ be given. Then write

so that the conclusion follows by making use of the strict convexity of the mapping F (which is itself a straightforward consequence of the strict convexity of the mapping t E R- ItlP).For a similar argument, see the proof of Theorem 5.2.1. Notice at this stage that the property of strict convexity implies the uniqueness of the solution of both minimization problems (5.3.2) and (5.3.7).

(iii) We then show that the functional J i s dijferentiable, and in so doing, we compute its derivative. Clearly, it suffices to examine the differentiability properties of the mapping considered in (5.3.11). We first observe that the mapping F is twice differentiable, with

aiF(()= lltjlp-2ei, 1 c i c n,

a , ~ ( t =) ( p - 2)11e()p-4eiej+ )1e1r-26ijr

1

i, j

n.

Ch. 5, Q 5.3.1

NONLINEAR PROBLEMS OF MONOTONE TYPE

InF(V(u

315

+ u)(x)) dx - In F(Vu(x)) dx=

with

On the one hand, we have

and thus the linear mapping

I,

u E Wd*’( 2.

0

Suficient condition f o r limh4 IJu- uh(l I.p.R = 0 Using the last part of the proof of the above theorem, we are in addition able to prove the convergence of the discrete solutions towards the solution u, as we now show. Theorem 5.3.2. Let there be given a family of finite element spaces as previously described, i.e., made up of n-simplices of type ( l ) , associated with a regular family of triangulations. Then with the sole assumption we have that the solution u is in the space W,,’*”(O)

Proof. We continue the argument used in part (v) of the proof of the previous theorem. Since the weak limit u is unique, we deduce that the whole family (uh) weakly converges to the solution u. Thus,

f(u)=$Zf(Uh).

318

SOME NONLINEAR PROBLEMS

[Ch. 5, 0 5.3.

On the other hand, we have

$3sup J(nh4) = $2J(nh4) = J(4).

v4 E g(f2), $3sup J ( u h )

Since the functions 4 can be chosen arbitrarily close to the solution u (in the norm of the space Wd*”(f2)),we deduce from the above relations that

J(u) = $5J(uh), i.e., in view of the expression for the functional J, that (5.3.17) since limh4 f(uh) = f(u). The space Wd*”(f2)being uniformly convex, the weak convergence and the convergence (5.3.17) imply the convergence in the norm. El

The equivalent problem Au

= f.

Two properties of the operator A

In order to have an approach similar to that of the linear case, let us introduce, for any function u E WdSP(f2),the element Au E ( Wd*”(O))‘ defined by (cf. the proof of Theorem 5.3.1)

V VE Wd*”(O), A u ( v )=

I,

IIVUJI”-*VU* V V dx.

(5.3.18 )

Notice that the element Au is nothing but the derivative of the mapping of (5.3.11), so that relation (5.3.12) may be equivalently written as

J ‘ ( u ) = Au - f.

(5.3.1 9)

In other words, the original minimization problem (5.3.2) is equivalent to the solution of the (nonlinear if p > 2) equation Au = f. Our next task is to establish (cf. Theorem 5.3.3) two properties of the operator A : Wd@(f2)-,(Wd-”(f2))‘

defined in (5.3.18) and whose bijectivity has been proved in Theorem 5.3.1. The first property (cf. (5.3.20)) is a generalization of the usual ellipticity condition in the linear case, while the second property (cf. (5.3.21)) is a generalization of the continuity of the operator A in the linear case (cf. the inequality IIA114v;v,s M established in (1.1.21)). In

Ch. 5 , 5 5.3.1

319

NONLINEAR PROBLEMS OF MONOTONE TYPE

order to simplify the exposition, we shall henceforth assume that n = 2 (the extension to higher dimensions is indeed possible, but at the expense of additional technicalities). Theorem 5.3.3. For a given number p in the interval [ 2 , 4 , let A : Wd*p(f2)+(Wdap(n))‘ be the operator as defined in (5.3.18). Then,

3a > 0, V u , u E WA.p(f2), allu - uIIp C (Au - Au)(u - u ) , (5.3.20)

3 M > 0, V U ,u E Wp(n), llAu - Avll*

d

M(llull + I)v(~)”-*(~u

-

uI(.

(5.3.21)

Proof. Let us introduce the auxiliary function

4: (E, 7) E 0 = {(E, 7 7 ) E R2 x R2; E # q b -+

4(E9 7 7 ) =

(11E11p-2E- l11111p-211) ( E - 7 7 ) , IIE - 77lr *

(5.3.22)

where . denotes as usual the Euclidean inner-product in the space R2. We shall show that

3a > 0, V ( E , q )E 0, a

4(E,7 7 ) .

(5.3.23)

a property which is easily seen to imply inequality (5.3.20). First, we notice that since V11#0,

4(0,77)= 1,

(5.3.24)

it suffices to consider the case where .f# 0. Next, we prove that

W E , 7 7 ) E 0, 4(E,77) > 0.

(5.3.25)

This follows from the relations

(llElr-*E - 11771r-211)( E - 7 7 ) = *

= l E1lp - (11E11p-2+ 117711p-2)(E * 11) + llllllp

llE1lp - 11Ellp-1117711- l17711p-’11E11 + Il77IlP = (llsll”-’ - l17711p-’~~l -ls117711) l

> 0 unless 11E11 = 111111. Since the penultimate inequality is an equality if and only if 77

= pt

for

320

[Ch. 5 , P 5.3.

SOME NONLINEAR PROBLEMS

some p E R,the only remaining case is that where 77

= - 2.

But then

(Ilzl1”-’2 - 117711”-277) * (2 - 7 7 ) = 411211” > 0. Finally, we observe that we may restrict ourselves to the case where 2 = = (1,O) since 4 ( A & AT) = 4(2,7 7 ) for all A > 0 on the one hand and since the Euclidean inner product is invariant through rotations around the origin on the other. Because it remains to study the behavior of the function 7) = (v,,v2)E (R2-f ) + 4(c 7 ) in the neighborhood of the point f. For this purpose, let

v1= 1 + p cos 8, v2= p

sin 8.

Then a simple computation shows that

4<s;7 7 ) =

I + ( p - 2) C O S ~e + ~ ( pe), Pp-2

1

with limp+,, ~ ( p0), = 0 uniformly with respect to 8 E [ 0 , 2 ~ [Therefore, .

(5.3.27) and relation (5.3.23) follows from the conjunction of relations (5.3.24) through (5.2.27). To prove the second relation (5.3.21), we introduce the auxiliary function $: ( 2 , ~ E ) 0 = ( ( 2 , ~ )E R2x R2,x # y}+

(5.3.28)

(5.3.29)

Vrl# 0,

* ( O , 7 7 ) = 1,

(5.3.30)

we may assume that 2# 0. In fact, it suffices to consider the case where 2 = i= ( 1 , O ) since $(A& A q ) = t,b(e, 7 7 ) for all A > 0 on the one hand, and since the Euclidean norm is invariant through rotations around the origin

Ch. 5 , 5 5.3.1

NONLINEAR PROBLEMS OF MONOTONE TYPE

32 1

on the other. We also have Iim $(f, 7 ) = 1 .

(5.3.31)

Iltlll-

To study the behavior of the function 7 = (q,,q 2 )E (Rz- i)+$(g 7 7 ) in the neighborhood of the point & we let v1= 1 + p cos 8, q2= p sin 8 as before. In this fashion we obtain

$(l,q ) = 22-p(i + p ( p - 2) C O S ~e)1/2+ ~ ( pe), , with Iimp4 ~ ( p8, ) = 0 uniformly with respect to 8 E [ O , 27r[ and there(5.3.32) 9-4

Then relation (5.3.29) follows from relations (5.3.30) to (5.3.32). As a consequence, we have V 6 17 E R2,

II l1771P-277 - 1151)”-z511

M1 I77 - fll(llall+ 11511)p-2~ (5.3.33)

To prove inequality (5.3.21), we shall use the characterization (5.3.34)

d

Mllu - ~ll(llUII+ II~II)p-211wII.

and inequality (5.3.21) follows from the above inequality coupled with characterization (5.3.34). 0

Strongly monotone operators. Abstract error estimate We are now in a position to describe an abstract setting particularly appropriate for this type of problem and its approximation: We are given

322

SOME NONLINEAR PROBLEMS

a (generally nonlinear) mapping A: V+V’

acting from a space V, with norm 1(-11, into its dual space V’, with norm ~ ~which ~ ~possesses ~ * the , two following properties: (i) The mapping A is strongly monotone, i.e., there exists a strictly increasing function x:[0, +a[+ R such that

~ ( 0=) 0 and lim ~ ( t=)01,

(5.3.35)

f-

V U ,u E V, (Au - A u ) ( u- U ) 3 ~

( 1 -1 ~Il)llu ~ - uIJ.

(5.3.36)

In particular, the operator A as defined in (5.3.18) is strongly monotone, with (cf. Theorem 5.3.3) (5.3.37)

X(t) = atp-’.

(ii) The mapping A is Lipschitz-continuous for bounded arguments in the sense that, for any ball B ( 0 ; r ) = { u E V; IIu(Jc r } , there exists a constant T ( r ) such that V u , u E B ( 0 ; r ) , IIAu - Au(l* C T(r)llu - 01 .

(5.3.38)

Thus, the operator A as defined in (5.3.18) is Lipschitz-continuous for bounded arguments, with (cf. Theorem 5.3.3)

T ( r )= M(2r)P-*.

(5.3.39)

Let there be given an element f E V’. For operators which satisfy assumptions (i) and (ii), we are able to obtain in the next theorem an abstract estimate for the error IIu - uhll, where u and uh are respectively the solutions of the equations V v E V,

A u ( u ) = f ( u ) (equivalently,

VVh E vh, AUh(uh) = f(Uh),

Au

= f),

(5.3.40) (5.3.41)

where Vh is a (finite-dimensional in practice) subspace of the space V (we showed in Theorem 5.3.1 that, with the operator A of (5.3.18), problems (5.3.40) and (5.3.41) have solutions; for general existence results, see “Bibliography and Comments”). Theorem 5.3.4. Let there be given a mapping A : V + V’ which is strongly monotone and Lipschitz-continuous f o r bounded arguments.

Ch. 5, 5 5.3.1

323

NONLINEAR PROBLEMS OF MONOTONE TYPE

Then there exists a constant C independent of the subspace Vhsuch that

Proof. To begin with, we show that the assumption of strong monotonicity for the operator A implies that the same a priori bound holds for both solutions u and uh: The conjunction of inequality (5.3.36) and relations (5.3.40) implies that

x~llull>llull ( A u - AO)u = f(u)- (AO)u

== (Ilfll* + IIA0ll*>llull9

and a similar inequality holds with u replaced by uh. Therefore, the function x being strictly increasing with x ( 0 ) = 0 and limt- X ( t ) = by assumption, we have IIuII, lluhII x-'(IlfII*+ IIAoll*). (5.3.43) Q)

Next, let t+, be an arbitrary element in the space Vh. Using the inclusion Vhc V and relations (5.3.40) and (5.3.41), we obtain (Au -Auh)wh = 0 for all w hE Vh so that, in particular, (AU - AUh)(Uh - O h ) = 0.

Combining the above equations with inequalities (5.3.36), (5.3.38) and the a prion bound (5.3.43), we obtain

x(llu - uhII)IIu - uhll s (Au - AUh)(u - uh) = (Au - A u ~ ) ( u- ~ h

)

d IlAu - AuhII*IIu - uhII

s r(x-l(llfll*+ IIAOll*>>llu- Uhll Ib - 41, and thus inequality (5.3.42) is proved, with C = r(x-'(llfII* + IIAOll*)). 0 Remark 5.3.3. The abstract error estimate of the previous theorem is another generalization of CCa's lemma, since in the linear case one has X ( t )= a t . 0 Remark 5.3.4. In the particular case of the operator A of (5.3.18), we have A 0 = 0, so that with the function x of (5.3.37), we obtain

324

SOME NONLINEAR PROBLEMS

[Ch. 5, 8 5.3.

If we argue as in part (iv) of the proof of Theorem 5.3.1, however, we obtain the improved a prion bound

Ilull, lluhll

((lf(I*)”(p-”.

0

Estimate o f the error Ilu - uhlll.p.n Let us now return to the minimization problem (5.3.2) and its finite element approximation as described at the beginning of this section. For simplicity, we shall assume that the set fi is polygonal. Then we get as an application of Theorem 5.3.4:

Theorem 5.3.5. Let there be given a family of finite element spaces made up o f triangles of type ( l ) , associated with a regular family o f triangulations. Then, if the solution u E Wd*p(f2)of the minimization problem (5.3.2) is in the space W2.p(f2), there exists a constant C(llfll*,J U ) ~ , ~ . ~ ) such that

Proof. Since A 0 = 0, the constant which appears in inequality (5.3.42) is a function of IlfII* only. Next, for some constants C independent of the subspace vh, we have inf

“hEVh

IlU - u h l l c

ClU -nhuli.p.nS Chlul2.p.a-

It then remains to apply inequality (5.3.42) with the function ~ ( t= )atP-’.

0 One should be aware that the above error estimate may be somehow illusive in that the solution u need not be in the space W2.p(f2)even with very smooth data (cf. Exercise 5.3.1). This is why it was worth proving convergence with the minimal assumption that u E Wd,P(f2) (Theorem 5.3.2). This is also why we did not consider the (otherwise straightforward) case where the generic finite element in the spaces vh would be for example the triangle of type (k).

Exercises 5.3.2. Following GLOWINSKI& MARROCCO(1973, consider the one-

Ch. 5 , B 5.3.1

NONLINEAR PROBLEMS OF MONOTONE TYPE

325

dimensional analog of the minimization problem (5.3.2), where

p2

= 1- 1 ,

[J(u) =

+ 1[,

I

P a

Ju’Ipdx - y l n u

dx, y E R.

Show that the unique solution u E Wd*p(0)of this problem is given by y‘/(P-yl - lxp/(P-l)), and that u E W * * ~ ( Oif)

3+v3 1 < p 0, v u , u E v,

(YIIU

- U1l2 C

Cllull+ Ilvl()Z-p(A~ - Av)(u - u ) ,

3 M > 0, V U ,u E V, llAu - A ~ l l *d Mllu - ~1Jp-l. (iv) Deduce from (iii) that (GLOWINSKI & MARROCCO (1975))

Ib - uhIll.p.0 cllr - (6 )IlmW4.P(R):

Wm.q(lZ)b

(6.1.27)

A(;)

where, for each 6 E 8, denotes the Pg-interpolation operator associated with the corresponding reference finite element (K, B(ci), S(ci)). With self-explanatory notations, we have, for all functions v^ E W4.P(K),

+

c {Dv^(cii)(dj 3

- c i i ) } G i j ( c i , .)

i=l b-il=l 3

+x i=l

{D6(6i)(ci- 6j)}ti(a^, .).

(6.1.28)

and

where the constants C ( k ) are independent of 6. Let us then consider the norm I l . I J m, q . K of any one of the basis functions f i i ( c i , .), Gii(ci, .) and ti(& .). On each of the triangles ki(ci), 1 6 i s 3, which subdivide the triangle K, the restriction of any one of these basis functions is a polynomial of degree c 3 , whose coefficients are obtained through the solution of a linear system with an invertible matrix (the set &(ci) is @(ci)-unisolvent as long as the point ci belongs to the interior of

346

[Ch. 6, 0 6.1.

THE PLATE PROBLEM

the set k).This matrix depends continuously on the point B since its coefficients are polynomial functions of the coordinates of the point 4. Consequently, each coefficient is in turn a continuous function of the point Ci and there exists a constant such that

e

SUP {IISi(i, .)JJm,q,lQ, II8ij(B,

*)IIm.q.R,

IIfi(&

*>IIm,q,R}

a d

d

6,

(6.1.30)

since the set B is compact. Then it follows from relations (6.1.27) to (6.1.30) that sup C(6, R) = C(B, R) < a. a d

Combining this result with inequality (6.1.26), we obtain v u E W‘*P(K), V K , Iu

(6.1.31) - A ~ t ) l ~ ,C(E, ~ , ~R)(mea~(K))”~-”~hf;”~u1~.~,~. d

(iii) By an argument similar to that used in the proof of Theorem 6.1.1 (cf. (6.1.16)), we find that

2 a , ( ~- AKu)(bi){(a3

IIKU- AKU=

(6.1.32)

b i ) * Vi}Ti,

1-1

where the functions ri, 1 d i d 3, are the basis functions associated with PK,2 ~ ) . the degrees of freedom {Dp(bi)(a- b,)} in the finite element (K, Applying Theorem 3.1.5 with m = 1, 4 = w and k = 3, we find that h i Iul4.p.K. la.(u - A K O ) ( ~ ~ )sI f i ( u - A K U I ~ , C(meas(K))-”P~,K~ PK

(6.1.33)

Next we have

.

I{(a - bi) vi)l

s hK,

(6.1.34) (6.1.35)

and we deduce from relations (6.1.30) and (6.1.32) to (6.1.35) that

C

C(meas(K)) l’q-l’phf;mlu l4 *POK .

(6.1.36)

Then the proof is completed by combining the above inequality with inequality (6.1.3 1). 0

Ch. 6, 0 6.1.1

341

CONFORMING METHODS

Remark 6.1.4. When q = 2, it is easily seen that the highest admissible value for the integer m compatible with the inclusion PK C H " ( K ) is m = 2 (that the integer m is at least 2 follows from an application of Theorem 2.1.2 to the partitionned triangle K = ;=, K i ; to prove that the integer m cannot exceed 2 requires an argument which shall be used later, cf. Theorem 6.2.1). Notice that this is the first instance of a restriction on the possible inclusions PK C W m S q ( KThe ) . next finite element under study will be another instance. Fortunately, the inclusion PK C H 2 ( K )is precisely that which is needed to insure convergence, as we shall show at the end of this section. 0

u

A singular finite element of class

%I:

The singular Zienkiewicz triangle

Let us next turn to an example of a triangular finite element, which is of class '&I as a result of the addition of appropriate rational functions to a familiar space of polynomials. The singular Zienkiewicz triangle is defined as follows (Fig. 6.1.5): The set K is a triangle with vertices ai, 1 d i d 3, the space PK is the space P ' ; ( K )of the Zienkiewicz triangle (cf. (2.2.39)) to which are added three

a3

THE PLATE PROBLEM

348

[Ch. 6 . 5 6.1.

functions q i : K +R, 1 S i s 3, defined by

1

4i =

4Aih f+Ih f+2

(hi + hi+l)(Ai+ hi+d qi(ai+l)= qi(ai+z) = 0,

for O S h i C 1, O S A i + l , A i + z < 1, (6.1.37)

where the functions hi, 1 C i < 3 , denote the barycentric coordinates in the triangle K (notice that the function given in the first line of definition (6.1.37) is not defined for hi + hi+l= 0 or hi + hi+2= 0 , i.e., for hi+2= 1 or hi+l= 1; this is why we have to assign values to the function qi at the vertices ai+l and u ~ + ~ Finally ). the set SK is the same as for the Hsieh-Clough-Tocher triangle. As usual, we begin by examining the question of unisolvence. Observe that this finite element is an instance where the validity of the inclusions PK C V'(K) (which is part of the definition of elements of class % I ) and PK c H 2 ( K )requires a proof. Theorem 6.1.4. With the definitions of Fig. 6.1.5, the set ZK is PKunisolvent. The resulting singular Zienkiewicz triangle is a finite element of class %', and the inclusion PK c H 2 ( K ) holds. Proof. (i) To begin with, let us verify the inclusions PK C % ' ( K ) and PK C H 2 ( K ) . Since such properties are invariant through affine transformations, we may consider the case where the set K is the unit triangle I? with vertices h , = (1, 0), h2 = (0,l), and h3 = (0,O). Then it suffices to study the behavior of the function GI: K + R in a neighborhood of the origin in K.We have

(6.1.38) where the function f ( x ) = 4( 1 - x1- x2)2/(1 - x 2 ) and its derivatives have no singularity at the origin. Since liix,,x24+x I x i / ( x l+ x2) = 0 , we deduce that limx,,x2-.o+ G 1 ( x )= 0. Therefore the function GI is continuous at the origin. For x I . x2 3 0 and x # 0, we have:

Ch. 6 , 5 6.1.1

349

CONFORMING METHODS

and thus we conclude that

(6.1.40) ajdl(x)= 0 = ajal(O), j = 1,2, which proves that the function GI is continuously differentiable at the lim

x,.x24+

origin. Arguing analogously with the vertex &, and next with the functions and 43, we conclude that the inclusion PK

holds. obtain partial origin.

G2

c %l(K)

This inclusion implies the inclusion PK C H ' ( K ) and thus, to the inclusion P K C H Z ( K ) it , remains to show that the second derivatives of the function GI are square integrable around the For x # 0, we find

(6.1.41)

where the functions g , , , g12an4 g22are continuous around the origin. Since the three functions factoring the function f(x) are bounded on the set k,the inclusion PK c HZ(K) follows. (ii) The inclusion PK c V'(K) proved in (i) guarantees that the degrees of freedom of the set XK are well-defined for the functions in the space PK. The PK-unisolvence of the set ZK will be an easy consequence of the PK-unisolvence of the set E K

= { p ( a i ) , D p ( a i ) ( a j - ai), D p ( b i ) ( a i - bi), l a i , j C 3 , l j - i( = I},

(6.1.42) which we proceed to show. Let us denote by pi, 1 c i a 3, and pij, 1 Q i, j C 3, - il = 1, the basis functions of the space P ' ; ( K ) as given in (2.2.39). By definition, they satisfy Pi(ak) = &k, Dpi(ak)(al - ak) = 0, (6.1.43) for 1 Q i, k, 1 C 3, (k - I ( = 1, and Pij(ak) = 0, Dpij(ak)(al - ak) = &&,

(6.1.44)

350

THE PLATE PROBLEM

(Ch. 6. 8 6.1.

for 1 d i, j, k, 16 3, lj - i( = Ik - 11 = 1. We next show that they satisfy

Dpi(bk)(ak-bk)=-a+kk, Dpii(bk)(ak- bk) =

-a+:&

1 si,k s 3 ,

+f&, 1

i,j, k 6 3,

(6.1.45) l j - il = 1.

(6.1.46) For the purpose of proving these relations, it is convenient to compute the directional derivatives Dp(bi)(ai- bi) for a function p: K + R expressed in terms of barycentric coordinates (the computation below is not restricted to n = 2). Let then p ( x l ,x2) = q(AI,A2, A3) be such a function. Denoting as usual by B = (bij)the inverse matrix of the matrix A of (2.2.4), we find that

Let us compute for example the quantity

where aii, j = 1,2, denote the coordinates of the vertex ai. By definition of the matrices B and A,

so that

Dp(bl)(al- 61)=

i, 4) - i(azq(0,f, i) + & d o , 4, 5)).

(6.1.47)

Then relations (6.1.45) and (6.1.46) follow from the above result (and analogous computations for Dp(bi)(ai- bi), i = 2,3) and the following expressions of the basis functions p i and pii (which are easily derived from relations (2.2.37) and (2.2.38)): pi = -2A!+ 3A!+ 2AlA2A3,

(6.1.48)

On the other hand, the functions qi as defined in (6.1.37) satisfy

qi(ak) = 0, 1 d i, k d 3, Dqi(ak)(al- ak) = 0 , 1 s i, k, 1 G 3, ( k - I( = 1 , Dqi(bk)(ak- bk) = &, 1 d i, k d 3.

(6.1 S O )

Ch. 6, 8 6.1.1

35 1

CONFORMING METHODS

The second equalities have been obtained in (6.1.40). The last ones are obtained through another application of relations of the form (6.1.47). Then it follows from relations (6.1.43) to (6.1.50) that the functions (which all belong to the space PK):

{ P i j - i (1q i

+ 3qj-2q,)],

1 d i, j, 1 Q 3, { i , j , I } = {1,2,3},

(6.1.51) qi, 1 aia3,

form a basis of the space PK.corresponding to the degrees of freedom of the set SKof (6.1.42). Thus this set is a PK-unisolvent set. It remains to prove that the set ZK is also PK-unisolvent. To prove this, we make the following observation: Along each side K' of the triangle K, the restrictions plK., p E PK,are polynomials of degree 3 in one variable, while the restrictions Dp(.)hK,,p E PK,of any directional derivative are polynomials of degree a 2 in one variable. This is clearly true for the functions in the space P ; ( K ) , and it is a straightforward consequence of the definition for the functions qi. Notice in particular that this property implies that the finite element is of class % I . Let then p E PK be a function which satisfies

p ( a i )= a , p ( a i )= & p ( a i ) = &p(bi) = 0,

1 d i s 3.

The conjunction of these relations and of the above property implies that the normal derivative and the tangential derivative vanish along any side of the triangle K. Consequently, the directional derivatives D p ( b i ) ( a i - b i ) , 1 d i d 3, vanish, and therefore the function p is identically zero 0 since the set SKis PK-unisolvent. Remark 6.1.5. Just as for the Hsieh-Clough-Tocher triangle, the normal derivatives at the mid-point of the sides can be eliminated by requiring that the normal derivatives vary linearly along the sides. Then we obtain in this fashion another finite element of class %' for which dim(PK)= 9 (cf. Exercise 6.1.6). 0 Theorem 6.1.5. A regular family of singular Zienkiewicz triangles is almost afine: For all p E ] l , m ] and all pairs ( m , q ) with m a 0 and

352

THE PLATE PROBLEM

[Ch. 6 , 0 6.1.

4 € [ 1 , w ] such that (6.1.52)

there exists a constant C independent of K such that

Proof. We shall simply give some indications. The proof of inequality (6.1.53) rests on the inclusion

P2(K) C P K (notice that the inequality p > 1 is required so as to guarantce the inclusion W 3 * p ( K 4) % ‘ ( K )= dom DK). One first argues with the finite element (K, PK,S K ) , with Z K as in (6.1.42), which can be imbedded in an affine family. Then one uses the same device as in the proofs of Theorems 6.1.1 and 6.1.3. 0 Remark 6.1.6. The second partial derivatives of the basis function 41 (as given in (6.1.41)) are not defined at the origin. In fact, for each slope t > 0, an easy computation shows that

This phenomenon is observed in ZIENKIEWICZ(1971, p. 199), where it is stated that “second-order derivatives have non-unique values at nodes”. Hopefully, this observation carries no consequence since it does not prevent the function q l from being in the space %“(K)n H 2 ( K ) . 0

Estimate of the error

111.4

- uh112.R

Let us now return to the finite element approximation of the clamped plate problem (6.1.1).We shall consider families of finite element spaces Xh, with the same generic finite element (K, P K ,ZK),for which we shall need the following assumptions: (Hl*) The family ( K , P K , ZK), K E y h , f o r all h, is an alrnost-a$ne f a m i fy. (H2*) The generic finite element is of class % I .

Ch. 6, 0 6.1.1

CONFORMING METHODS

353

If we assume (as in the subsequent theorems) that the inclusion PK C H 2 ( K )holds, the inclusion x h C H 2 ( 0 ) is then a consequence of hypothesis (H2*). This being the case, we let v h

= XOOh = { V h E

xh;

Uh

= avUh = 0

on r).

(6.1.54)

Notice that the Xh-interpolation operator associated with any one of the finite elements of class %' considered in this section satisfy the implication VEdOmnh

and

U = a , U = o

On r j f l h U E X m h , (6.1.55)

which will accordingly be an implicit assumption in the remainder of this section. To begin with, we derive an error estimate in the norm ((.I(2.~. As usual, the letter C represents any constant independent of h and of all the functions appearing in a given inequality.

Theorem 6.1.6. In addition to (Hl*) and (H2*), assume that there exists an integer k 3 2 such that the following inclusions are satisfied: Pk(K)c PK

c H2(K),

Hk+'(K)4 V ( K ) ,

(6.1 S6) (6.1 S7)

where s is the maximal order of partial derivatives occuring in the definition of the set Z K . Then if the solution u E H i ( O ) o f the clamped plate problem is also in the space H k + ' ( 0 ) ,there exists a constant C independent of h such that (111 - U h l ( Z . o c Chk-'(ulk+i,a9

(6.1.58)

where &, E v h is the discrete solution. Proof. Using CCa's lemma, inequality (6.1.7) and relation (6.1.59, we obtain

IIu - Uh112.Rs C V inf hEVh

IIu - Vh(l2.R

354

T H E PLATE PROBLEM

[Ch. 6. 5 6.1.

Remark 6.1.7. By the previous theorem, the least assumptions which insure an O(h) convergence in the norm I(.112,~ are the inclusions P 2 ( K )C

PK on the one hand, and the fact that the solution u of the plate problem is in the space H 3 ( R ) on the other. It is remarkable that this last regularity result is precisely obtained if the right-hand side f is in the space L 2 ( n ) and , if d is a convex polygon, an assumption often satisfied for plates. Therefore, since one cannot expect better regularity in general, the choice PK = P 2 ( K )appears optimal from the point of view of convergence. However, by ieni9ek-s result, this choice is not compatible with the inclusion xh C %'(d). 0 Suficient conditions f o r

\izIJu

- uh(l2.R= 0

We next obtain convergence in the norm tions (cf. (6.1.59) below).

Theorem 6.1.7. clusions

((. l (2, n

under minimal assump-

In addition t o (Hl*) and (H2*), assume that the in-

PZ(K)c PK

c HZ(K)

(6.1.59)

are satisfied, and that the maximal order s of partial derivatives found in the set 2 K satisfies s s 2. Then we have

9% 11u -

Uh112.R

= 0-

(6.1.60)

Proof. The argument is the same as in the proof of Theorem 3.2.3 and, for this reason, will be only sketched. Using inequality (6.1.5) with k = 2, p = m, M = 2 and q = 2, one first shows that the space "I' =

w3qn)n ~ : ( n )

is dense in the space H i ( 0 ) . Then it suffices to use the inequality

valid for any function v E V .

0

Conclusions In the following tableau (Fig. 6.1.6), we have summarized the application of Theorem 6.1.6 to various finite elements of class %'.

Ch. 6, 5 6.1.1

355

CONFORMING METHODS

Finite element

dim PK

P , ( K ) C P,

Argyris triangle

21

P,(K)= P,

o(h4)

E HYR)

Bell’s triangle

18

P , ( K )c P ,

ow)

EH W )

Bogner-Fox-Schmit rectangle

16

P , ( K ) c P,

O(hZ)

u E H4(R)

Hsieh-Clough-Tocher triangle

12

P , ( K )c P,

O(h2)

u E H4(R)

9

P,(K)cP,

O(h)

UEH’(R)

12

P,(K)cP,

O(h)

u E H’(R)

9

P , ( K )C P ,

O(h)

u E H’(R)

Reduced Hsieh-CloughTocher triangle (cf. Exercise 6.1.3)

~

Singular Zienkiewicz triangle Reduced singular Zienkiewicz triangle (cf. Exercise 6.1.6)

IIu

-

~,,ll,,~

Assumed regularity

~~~~

Fig. 6.1.6

One should notice that, if the reduced Hsieh-Clough-Tocher triangle and the reduced singular Zienkiewicz triangle are optimal in that the dimension of the corresponding spaces PK is the smallest, this reduction in the dimension of the spaces PK is obtained at the expense of an increased complexity in the structure of the functions p E PKRemark 6.1.8. In order to get an O ( h k + ’convergence ) in the norm l.lO.~, it would be necessary to assume that, for any g E L 2 ( R ) , the corresponding solution cps of the plate problem belongs to the space H 4 ( 0 )n H i ( R ) and that there exists a constant C such that llq+l14,~=S Clglo.n for all g E L2(0). However, this regularity property is no longer true for conuex polygons in general. It is true only if the boundary r is sufficiently smooth: For example, this is the case if the boundary r is of class %‘. But then this regularity of the boundary becomes incompatible with our assumption that d be a polygonal set. 17

356

THE PLATE PROBLEM

[Ch. 6, 5 6.1.

Exercises Show that a regular family of Bell’s triangles (cf. Fig. 2.2.18) is almost affine, with the value k = 4 in the corresponding inequalities of the form (6.1.5). 6.1.2. The purpose of this problem is to give another proof of unisolvence for the Hsieh-Clough-Tocher triangle (as originally proposed in CIARLET(1974~)).Without loss of generality, it can be assumed that a = (0,O). Denoting by (xi, y i ) the coordinates of the vertex ai, let

6.1.1.

a i = det

?+I Xi+2

Yi+l) Yi+2

For definiteness, it shall be assumed that a = 1. Given a function p E PK whose degrees of freedom are all zero, let 60 = p ( a ) , 6; = ai(DP(a)(ai- a ) } , 1 c i s 3.

(i) Show that PIK, = p:{(--pi

+ 3)60+ ( p i + *- I*i)si+l+( ~ i + 2 - ~ i ) s i + 2 ) ,

ISis3, where for each i we denote by pi the unique function which satisfies pi E P , ( K i ) , p i ( a ) = 1,

pi(ai+i)= pi(ai+z)= 0.

(ii) Show that 3

2

si

= 0,

i=l

S o - z l Sk

L +a,& = 0,

1 s j d 3,

and conclude that Si = 0, 0 =s i s 3 (the first equality expresses that the function p is differentiable at the point a, while the other relations express the equalities

6.1.3. The reduced Hsieh-Clough-Tocher triangle is a triangular finite element whose corresponding data PK and X K are indicated in Fig. 6.1.7. Show that the set X K is PK-unisolvent and that a regular family of reduced Hsieh-Clough-Tocher triangles is almost affine, with the value k = 2 in the corresponding inequalities of the form (6.1.5).

Ch. 6 , 8 6.1.1

CONFORMING METHODS

357

W a2

Reduced Hsieh-Clough-Tocher triangle

1

= { p E %'(K); plK, E P&K;), 1 c i zs3, d v p E P , ( K ' ) for each side K ' of K}; dim PK = 9; S K = M a i ) , a , p ( a , ) .a 2 p ( a , ) 1, i ss 3).

PK

Fig. 6.1.7

6.1.4. Following PERCELL(19761, one may define a triangular finite element of class %' analogous to the Hsieh-Clough-Tocher triangle, as follows: With an identical subdivision K = U:=, Ki, let PK = {P E

% ' ( K ) ;plKi E ~ 4 ( K i ) , 1 =Z i s 3},

ZK = {p(ai),

aiP(ai),

p(aij),

1 S i <j

p(a),

a2p(ai), 1 b i b3;

3;

avp(aiii), 1 =S i, j d 3,

i# j ;

ads), ads>),

where a.. = ai + a . 11

2''

a,..= 2ai + aj IIJ

3

.

Then show that the set SK is PK-unisolvent. 6.2.5. The Fraeijs de Veubeke-Sander quadrilateral is a finite element (K, P K ,ZK)for which the set K is a convex nondegenerate quadrilateral with vertices ai, 1 C i S 4, and mid-points of the sides bi, 1 C i d 4. As indicated in Fig. 6.1.8, let K , denote the triangle with vertices a l , a2 and a4, and let K 2 denote the triangle with vertices a l , a2 and u3. The space PK and the set ZK are indicated in Fig. 6.1.8.

358

[Ch. 6, 8 6.1.

THE PLATE PROBLEM

1

Fraeijs de Veubeke-Sander quadrilateral

Fig. 6.1.8

(i) Show that the set ZK is PK-unisolvent (CIAVALDINI & NEDELEC (1974)). (ii) We shall say that a family of Fraeijs de Veubeke-Sander quadrilaterals is regular if it is a regular family of finite elements in the usual sense and if, in addition, the following condition is satisfied: For each quadrilateral K in the family, let FK denote the unique affine =a l . ~ and F~(d2) = a2.k mapping which satisfies FK(O)= aK, FK(dI) where aK is the intersection of the two diagonals of the quadrilateral K, and where d l = ( l , O ) , d 2 = ( 0 , l ) (cf. Fig. 6.1.9). Then there exist compact intervals f3 and f4 contained in the half-axes {(XI,XZ) E R2;X I< 0, xz = O},

{(XI,

XJ

E Rz;XI = 0,~2 < 0},

respectively, such that the points d i , = ~ F i ' ( a b K belong ) to the intervals 4, f o r j = 3 and 4. In other words, the quadrilateral F i ' ( K ) is in between the two extremal quadrilaterals k, and k, indicated in Fig. 6.1.9. Then, following CIAVALDINI& NBDBLEC(1974), show that a regular family of Fraeijs de Veubeke-Sander quadrilaterals is almost affine, with the value k = 3 in the corresponding inequalities of the form (6.1.5).

Ch. 6, 0 6.1.1

CONFORMING METHODS

359

Fig. 6. I .9

(iii) Carry out a similar analysis (unisolvence, interpolation error) for the reduced Fraeijs de Veubeke-Sander quadrilateral, whose characteristics are indicated in Fig. 6.1.10 (for the definition of the spaces R , ( K )and R 2 ( K ) ,see Fig. (6.1.8)). 6.1.6. The reduced singular Zienkiewicz triangle is a triangular finite element whose corresponding data PK and ZK are indicated in Fig. 6.1.11. Show that the set ZK is PK-unisolvent and that a regular family of

360

THE PLATE PROBLEM

Reduced Fraeijs de Veubeke-Sander quadrilateral

Fig. 6.1.10

I

Reduced singular Zienkiewicz triangle

v

PK = { p E P ; ( K ) @ {qi}; a.p E P , ( K ' )for each side K' of K ) (cf. (2.2.39) and (6.1.37));dim PK = 9; Z K = { p ( a i ) .a , p ( a , ) , a2p(ai).I s i s 3). Fig. 6.1.11

[Ch. 6, 5 6.1.

Ch. 6, 5 6.1.1

36 1

CONFORMING METHODS

reduced singular Zienkiewicz triangles is almost affine, with the value k = 2 in the corresponding inequalities of the form (6.1.5). 6.1.7. The purpose of this problem is to describe another instance where rational functions are added to a polynomial space so as to obtain a singular finite element of class %'. An analogous process yielded the singular Zienkiewicz triangle. (i) Following BIRKHOFF (1971), let T 3 ( K )denote, for any triangle K, the space of all polynomials whose restrictions along each parallel to any side of K are polynomials of degree < 3 in one variable. Show that the space T 3 ( K ) ,of so-called tricubic polynomials, is the space P3(K) to which are added linear combinations of the three functions h : h d 3 , A,hiA3 and h , h 2 h ~(which are not linearly independent). Show that dim P 3 ( K )= 12. (ii) Following BIRKHOFF& MANSFIELD(1974), we define the Birkhof-Mansfield triangle as indicated in Fig. 6.1.12 (as usual, L p ( b i ) = D2p(bi)(v, 7 ) where 7 is the unit tangential vector at the point bi). Show that, along each side of the triangle K, the functions in the space PK are polynomials of degree d 3 in one variable and that any directional derivative Dp(.)[, where [ is any fixed vector in R2, is also a polynomial of degree S 3 in one variable along each side of the triangle K. Show that the set Z K is PK-unisolvent. a

l n

a3

. 362

THE PLATE PROBLEM

[Ch. 6,

P 6.2.

Show that the resulting finite element is of class (e' and that the inclusion PK C H 2 ( K )holds. (iii) Show that a regular family of Birkhoff-Mansfield triangles is almost affine, with the value k = 3 in the corresponding inequalities of the form (6.1.5). (iv) Carry out a similar analysis (unisolvence and interpolation error) for the reduced Birkhof-Mansfield triangle, whose characteristics are indicated in Fig. 6.1.13. 6.2. Nonconforming methods

Nonconforming methods for the plate problem

To begin with, we shall give the general definition of a nonconforming method for solving the clamped plate problem (corresponding to the data (6.1,l)).Assuming the set d polygonal, so that it may be exactly covered with triangulations, we construct a finite element space Xh whose generic finite element is not of class %'. Then the space X h will not be a subspace of the space H*(L?),as a consequence of the next theorem (which is the converse of Theorem 2.1.2),whose proof is left to the reader (Exercise 6.2.1).

I

Reduced Birkhoff-Mansfield triangle

I

Ch. 6, 5 6.2.1

NONCONFORMING METHODS

363

Theorem 6.2.1. Assume that the inclusions PK C Ce'(K) for all K E Y and X h C H z ( 0 ) hold. Then the inclusion x h

h

c %I(&) D

holds. Let us henceforth assume that we have

V K E Y h , PK C H z ( K ) ,

(6.2.1)

so that, in particular, we have

Xh c LZ(0).

(6.2.2)

After defining an appropriate subspace XWh of xh, so as to take into account the boundary conditions u = d,u = 0 along I' as well as possible (this will be illustrated on one example), we define the approximate bilinear form:

+2aizuhanuh)} dx.

(6.2.3)

Observe that this definition is justified by the inclusions (6.2.1). Then the discrete problem consists in finding a function uh E Vh = XOOhsuch that

Vvh E

vhr

ah(Uhr O h ) = f ( V h )

(6.2.4)

(the linear form need not be approximated in view of the inclusion (6.2.2)). In analogy with the norm J.lz.R of the space V = H & 0 ) , we introduce the semi-norm (6.2.5)

over the space vh. Next we extend the domains of definition of the mappings ah(.,.) and ll.llh to the space V h + V . Thus there Zxists a constant M independent of the space Vh such that

v u , E (vh+

v),

lah(u, u)l

Rllullhllullh.

(6.2.6)

364

[Ch.6, 0 6.2.

THE PLATE PROBLEM

A n example of a nonconforming finite element: Adini's rectangle In the remainder of this section, we shall essentially concentrate on one example of a nonconforming finite element, in the sense that it yields a nonconforming method when it is used in the approximation of the plate problem. This element, known as Adini's rectangle, corresponds to the following data K, PK and &: The set K is a rectangle whose vertices ai, 1 =zi =s4, are counted as in Fig. 6.2.1. The space PK is composed of all polynomials of the form

I

Adini's rectangle

Fig. 6.2.1

Ch. 6, 0 6.2.1

365

NONCONFORMING METHODS

i.e., we have

PK = P3(K) 0

v {xIx;,

(6.2.7)

x:xz}.

Notice that the inclusion

P3(K) c PK

(6.2.8)

holds, and that (6.2.9)

dim(PK)= 12.

To see that the set

ZK = { p ( a i ) ,aiP(ai), azp(ai),

1c i

c 41

is a PK-unisofvent set, let us argue on the square we can write

(6.2.10)

k = [-1,

+112. Then

with

(6.2.12)

etc.. . Let us assume that the set d is rectangular, so that it may be covered by triangulations made up of rectangles. With such a triangulation we associate a finite element space x h whose functions v h are defined as follows: (i) For each rectangle K E y h , the restrictions v h ( K span the space PK of (6.2.7). (ii) Each function v h E x h is defined by its values and the values of its first derivatives at all the vertices of the triangulation. Along each side K' of an Adini's rectangle K, the restrictions plK,, p E P K , are polynomials of degree s 3 in one variable. Since such polynomials are uniquely determined by their values and the values of their first derivative at the end points of K', we conclude that Adini's rectangle is a finite element of class Ce0. It is not of class %'I, however:

366

[Ch.6,5 6.2.

THE PLATE PROBLEM

Along the side K ; = [a4,a , ] for instance (cf. Fig. 6.2.1), the normal derivative is a polynomial of degree s 3 in the variable x2 on the one hand, and on the other the only degrees of freedom that are available for the normal derivative along the side K ; are its two values at the end points. We let vh = X W h r where X W h denotes the space of all functions Uh E x h such that Uh(b) = alUh(b) = azUh(b) = 0 at all the boundary nodes b. Then the functions z)h E V,, vanish along the boundary but their derivatives &2ih do not vanish along the boundary r in general, although they vanish at the boundary nodes. To sum up, we have constructed a finite element space V,, whose functions vh satisfy

r,

1

oh

E HA(n) n%'(d), vhll<E H z ( K ) for all K E Yh,

(6.2.13)

a,vh(b) = 0 at the boundary nodes.

Observe that the associated Xh-interpolation operator 2)

E Hi(&?)n dom n h

nhU E

x ~=,vh.

Ilh

is such that

(6.2.14)

We shall use this implication in particular for functions in the space H 3 ( n )n H i ( n )c %'(d)= dom nh. Prior to the error analysis, we must examine whether the mapping 11.llh of (6.2.5) is indeed a norm. Theorem 6.2.2.

The mapping

i s a norm over the space

vh.

Proof. Let t+, be a function in the space vh such that IIuhllh = 0. Then the functions aj(u,,lK),j = 1,2, are constant over each rectangle K E Yh. Since they are continuous at the vertices, the functions djvh, j = 1,2, are therefore constant over the set 6,and since they vanish at the boundary nodes, they are identically zero. Thus the function uh E vh is identically zero, as a consequence of the inclusion V,, C Hd(0) n %'(n). 0 Notice that the approximate bilinear forms ah(. ,.) are uniformly Vh-elliptic, since one has (cf. (6.2.3)) vvh

E

Vhr

(1

-a)llUhllZhs

ah(Uh9 Uh)r

(6.2.15)

Ch. 6, Q 6.2.1

NONCONFORMING METHODS

367

and the Poisson coefficient a lies in the interval reasons).

lo,{[ (for physical

Remark 6.2.1. Had we tried to use nonconforming finite element methods for the biharmonic problem (in which case the approximate bilinear form reduces to Z K E y h J duhduh K dx). the uniform Vh-ellipticity is no longer automatic, and this is essentially why we restrict ourselves to plate problems. In contrast, conforming methods as described in the previous section apply equally well to any fourth-order elliptic boundary value problem. 0

Consistency error estimate. Estimate of the error

(XKcrh

( u - Uhl:,K)I12

We are now in a position to apply the abstract error estimate of Theorem 4.2.2, which we recall here for convenience:

IIu - U h ( l h d c( inf IIu - VhIlh + sup UhEVh

lah(u, wh) - f ( W h ) [

whEVh

11 Whllh

)*

(6.2.16)

In what follows, the solution u will be assumed to be in the space H 3 ( 0 )f l H t ( 0 ) (this is true for any f E L 2 ( 0 )if d is a convex polygon, i.e., a rectangle in the present case). Observing that any family of Adini’s rectangles is affine, we obtain for a regular family of triangulations, (6.2.17)

and this estimate takes care of the first term in the right-hand side of inequality (6.2.16). The estimate of the second term, i.e., the consistency error estimate, rests on a careful decomposition of the difference

Dh(u, wh) = ah(u, wh) - f ( W h ) r Let us first show that the term the form

wh E vh*

f ( W h ) = J n f w hdx

(6.2.18)

can be rewritten in

(this equality is obvious if u E H4(0)f l H t ( n ) , in which case

f(Wh)

=

In A2u wh dx, but we only assume here that u E H 3 ( 0 )n H i ( 0 ) ) .To see this, let wh E vh be given, and let ( w 3 be a sequence of functions W ~ g(0) E such that ~ i r n k - ~ ~ wWh(II,R ~ - = o (recall that wh E vh c

368

[Ch. 6, 9 6.2.

THE PLATE PROBLEM

H d ( 0 ) ) .By making use of Green’s formulas (1.2.5) and (1.2.9), we obtain for all integers k,

lo

dud w; dx

=-

I,

V(Au) V w; dx,

~o{2alzua12w;-alIuaz2whk- az2uallwadx= 0,

r, and

since a,wk = a,wi = 0 along problem (cf. (6.1.1)), fw; dx

=

-

I,

thus, by definition of the abstract

-

V(Au) V w; dx.

Therefore, fwh

dx

(1

= lim k-

- a)

JK

fwhk dx = ;&

{-

amKuavKwh

lo

dud W; dx

avrKuaTKwh) d r .

When the above expressions are added up so as to form the approximate bilinear form of (6.2.3), we first find that

using the inclusion find that

v h

C H ’ ( 0 ) and equality (6.2.19), and next-we shall

Ch. 6, 5 6.2.1

NONCONFORMING METHODS

369

To prove this last relation, consider separately the case where K’ C aK is a side common to two adjacent rectangles K I and Kz , and the case where K‘C aK is a portion of the boundary r. In the first case the two corresponding integrals cancel because u E H 3 ( n ) and Wh E %‘(d), and in the second case the integral vanishes because wh = 0 along r. To sum up, we have found that vwh E

vh,

Dh(u, w h ) = ah(u, W h ) - f ( W h )

i.e., we have obtained one decomposition of the expression &(u, wh) as a sum Dh(u, wh) =

ZTh

D K ( U I K 9 WhIIK),

where each mapping DK(., .) appears as a bilinear form over the space H 3 ( K )x P K . Just as in the proof of Theorem 4.2.6, the key argument will consist in obtaining another decomposition of the form (6.2.20) (cf. (6.2.23)), which in this case takes into account the “conforming” part of the first order partial derivatives of the functions in the space vh (for related ideas, cf. Remark 4.2.5). This will in turn allow us to obtain appropriate estimates of the difference Dh(u, wh). as we shall show in the proof of the next theorem. Theorem 6.2.3. Assume that the solution u of the plate problem is in the space Hi(R) f l H3(R).Then, for any regular family of triangulations, there exists a constant C independent of h such that

(6.2.21) Proof. In view of the decomposition (6.2.20), we are naturally led to study the bilinear form Dh(. .): ( u , 9

wh)

E H 3 ( n )x

vh

370

THE PLATE PROBLEM

[Ch. 6, 0 6.2.

with

-

I,:

(Au - (1 - a ) a r r ~ ~ ) ~ j ( w h dy], lK)

j = 1729

where, for each K E T h , the sides K ; and Ky, j = 1 , 2 , are defined as in Fig. 6 . 2 . 1 . For each triangulation T h , we let Yh denote the finite element space whose generic finite element is the rectangle of type ( 1 ) and we let z h = YOh denote the space of all functions wh E Yh which vanish at the boundary nodes. Clearly, the inclusion z h

c~ ~ ( n ~6d ) (n)

implies that

vu E H 3 ( 0 ) ,

vzh E z h ,

Di(v, z h ) = 0, j = 1, 2,

with

Consequently, if, for each K E Yh,AK denotes the Q,(K)-interpolation operator, we can also write

Ch. 6, 5 6.2.1

371

NONCONFORMING METHODS

Using the definition of the operator invariance:

AK,

we find a first polynomial

V v E H 3 ( K ) , V q E Q , ( K ) , Aj,K(v, q ) = 0 , j = 1,2, (6.2.26)

with Aj,K(v, 9 ) =

I -IK7 K;

(Av - (1 - u)amKv)(q- AKq) d'Y

(Av - ( 1 - a)a,v)(q - A&) d y , j = 1,2.

We next proceed to obtain the second polynomial invariance: VV E P2(K), V q E ajPK, Aj,K(v, q ) = 0, j = 1,2,

(6.2.27)

where the spaces ajPK

= {ajp ;p E PK}, j = 1,2,

(6.2.28)

both contain the space Q1(K).To see this, it suffices to show that V q E a p K , IKi(q-AKq)dy=IKj(q-AKq)dyi

j = 192.

(6.2.29) Let us prove this equality for j = 1, for instance. Each function q E is of the form 4 = YO(X1) + Y I ( X l ) X 2 + yzx:

+ ysx:,

where yo and yI are polynomials of degree a 2 in the variable xl. Given r the linear function any function r defined on a side K', let h ~ *denote along K' which assumes the same values as the function r at the end points of K'. Then we have ( 9 - AKq)(K;(x2)= y2x: + y3x: - AK;(yZx:+ y3x:)

and therefore ( 4 - A K ~ ) I K ; (= X (Z4)- k ? ) ( ~ ~ ( x z ) ,

which proves (6.2.29).Consequently, the polynomial invariance of (6.2.27) holds. To estimate the quantities Aj.K( v , sip) of (6.2.25), it suffices to estimate the similar expressions

372

[Ch. 6, 8 6.2.

THE PLATE PROBLEM

i

for cp E H ' ( K ) , q E ajPK, = 1,2. Using the standard correspondences between the functions 5: K + R and u : K + R, we obtain aI.K((P9

4 ) = hZal,K(4,.4),

82,K((P7

4) = h16z.g(4,4),

(6.2.31)

and we shall also take into account the fact that a function 4 belong to the space a p t when the function q belongs to the space ail',. Paralleling the polynomial invariances (6.2.26) and (6.2.27), we now have: v4 E H ' ( R ) , v4 E Po(&, Sj,R(@.4) = 0, (6.2.32) ~4 E P ~ ( R ) VG , E ajpg, aj,g(4,4) = 0, j = I , 2.

i

Then if we equip the spaces ajPK with the norm VG E H I ( & ) , VG E a p t ,

ll.lll.K,

we obtain

411 ~ I I ~ I I = ~ ~ ~ d ~ ~ II II ~~I.dI4II I II ILI .~t , ~ ~ ~ Z ) and thus each bilinear from ai,t(.,.) is continuous over the space I ~ j . ~ ( ~d*

HI(&) X ajPg. Using the bilinear lemma (Theorem 4.2.5), there exists another constant such that

c

(6.2.33) By Theorem 3.1.2 and the regularity assumption, there exists a constant C such that

I4II.IzC CIcpIl.K?

(6.2.34)

IciIl,d~ Clql1.K.

Combining relations (6.2.3l), (6.2.33) and (6.2.34), we conclude that (6.2.35) Let then v E H 3 ( K )and p E PK be two given functions, so that the functions cp = Av - ( 1 - a ) d z 2 v and q = a l p belong to the spaces H ' ( K ) and alpK, respectively. Then we have

IAI.K(v,P)I = I S I . K ( A-~ ( 1 - a)azzu7 ~ I P ) I

C

C~KI~I~.KIPIZ.K-

Arguing analogously with the term lA2,K(u, p ) l , we obtain vu E H ' ( K ) , j= I

vp E pK,

Ch. 6, 8 6.2.1

373

NONCONFORMING METHODS

Then we are able to estimate the second term in the abstract error estimate (6.2.16): We find that, for all wh E V,,,

Ia,,(u, w h ) - f ( w , , N s

2 I&(w w)I

K€Yh

~hIuIdl~hIIh

0

and the proof is complete.

Further results The error estimate (6.2.21) can be improved when all the rectangles K E Yhare equal. In this case, LASCAUX & LESAINT (1975) have shown that IIu - uh(lhC Ch21u14.aif the solution u is in the space H4(0). For an error estimate in the norm IJ.I(I,R, see Exercise 6.2.2. Another nonconforming finite element for solving the plate problem is the Zienkiewicz triangle (cf. BAZELEY, CHEUNG, IRONS & ZIENKIEWICZ (1965)) which was described in Section 2.2 (cf. Fig. 2.2.16). Through a refinement of the argument used in the proof of Theorem 6.2.3, LASCAUX & LESAINT(1975) have shown that the necessary polynomial invariances in the difference Dh(u,w,,) (which in turn imply convergence) are obtained if and only if all sides of all the triangles found in the triangulation are parallel to three directions o n l y . In this case, one gets I(u - u,,((hS Chlul3.0 and IIu - u h ( I I , a 6Ch21u13,nassuming the solution u is the space H 3 ( 0 ) . This is therefore an answer to the Union Jack problem: As pointed out in ZIENKIEWICZ(1971, p. 188-189), the engineers had empirically discovered that configuration (a) systematically yields poorer results than configuration (b) (Fig. 6.2.2). The reason why the degree of freedom ~ ( ~ 1 2 3(which ) is normally found in the Hermite triangle of type (3)) should be eliminated is that the presence of the associated basis function A lA2h3 (cf. (2.2.37)) would destroy the required polynomial invariances.

(b) Fig. 6.2.2

374

[Ch. 6 , 8 6.2.

T H E PLATE PROBLEM

Whereas Zienkiewicz triangles yield finite element spaces which satisfy the inclusion V,,C %“(6) flHA(R) (just as Adini’s rectangles), there exist nonconforming finite elements for the plate problem which are not even of class g o . Two such finite elements, the Morley triangle and the Fraeijs de Veubeke triangle, are analyzed in Exercise 6.2.3.

Exercises

6.2.2. Prove Theorem 6.2.1 (cf. Theorem 4.2.1 for a similar argument). 6.2.2. Using the abstract error estimate of Exercise 4.2.3, show that (LASCAUX & LESAINT (1975))

Ilu - 4 l l I I . R ~Ch2143.R9 for finite element spaces whose generic element is the Adini rectangle. 6.2.3. Following LASCAUX & LESAINT (1975), the object of this problem is the study of two nonconforming finite elements which are not of class (e0. The first element, known as Morley’s triangle (cf. MORLEY (1968)) corresponds to the data indicated in Fig. 6.2.3. The second element, known as Fraeijs de Veubeke triangle (cf. FRAEIJS DE VEUBEKE (1974)) is an example of a finite element where some degrees of freedom are averages (another related instance is Wilson’s

Morley ’s triangle

P,

= P , ( K ) ; dim

PK = 6;

ZK = { p ( ~ , ) , I s i s 3: a,p(a,,), Fig. 6.2.3.

I s i < j s 3)

Ch. 6, 9 6.2.1

NONCONFORMING METHODS

375

Fraeijs de Veubeke triangle

Fig. 6.2.4

brick; cf. Section 4.2). All the relevant data are indicated in Fig. 6.2.4 where, for each i = 1,2,3, IK:I denotes the length of the side K : . (i) In each case, prove the PK-unisolvence of the given sets SK and that, for regular families, one has E H 3 ( K ) C d O m n ~ ,~ OSmc3, VU

U - ~ K U I , , , K Ch3-"'IU13.~, ~

i.e., regular families of Morley's triangles or Fraeijs de Veubeke triangles are almost-afine. Prove in particular that the space PK corresponding to the Fraeijs de Veubeke triangle contains the space P 2 ( K ) . (ii) For each finite element, describe the associated finite element space X,, and then let vh =XOOhrwhere XWh is composed of the functions in xh whose degrees of freedom vanish along the boundary r. Show that neither element is of class Vo.However, show that in each case the averages of the first order partial derivatives are the same

376

T H E PLATE P R O B L E M

[Ch. 6.

across any side common to two adjacent finite elements, while the same averages vanish along a side included in r. (iii) Show that for both elements the semi-norm 11.llh of (6.2.5) is a norm over the space Vh. (iv) Show that if the solution u belongs to the space H 4 ( 0 ) ,the error estimates

Ilu - U h l l h =Z C(hluI3.0+ h21u14.n) holds. Therefore, contrary to the Zienkiewicz triangle, no restriction need to be imposed on the geometry of the triangulations so as to obtain convergence. [Hint: The decomposition (6.2.20) is here to be replaced by v w h

E

vh,

Dh(u, wh)

=ah(u,

wh)-f<Wh)

and the key idea is again to subtract off appropriate “conforming” parts in the above expression. Then it is possible to apply the bilinear lemma (one side at a time rather than one element at a time, as in the case of Wilson’s brick or Adini’s rectangle).]

Bibliography and comments 6.2 and 6.2. The first interpolation error estimates for the Argyris triangle are due to ZLAMAL(l968), who obtained estimates in the spaces %‘“(K).The results and methods of M. Zl6mal were extended by i E N f S E K (1970) to finite elements which yield inclusions of the form x h C %‘“(d). BRAMBLE& ZLAMAL(1970) have obtained estimates in Sobolev norms, which are contained in the estimates of Theorem 6.1.1. The Hsieh-Clough-Tocher triangle appeared in CLOUGH& TOCHER (1965). It is also named after Hsieh who was the first to conceive in 1962 the idea of matching three polynomials so as to get a finite element of class %’. The interpolation theory given in Theorem 6.1.3 is based on CIARLET(1974~)where a proof of unisolvence was also given along the lines indicated in Exercise 6. I .2. The proof of unisolvence given in

Ch. 6.1

BIBLIOGRAPHY A N D C O M M E N T S

377

Theorem 6.1.2 is due to PERCELL(1976). See also DOUGLAS,DUPONT, PERCELL& SCOTT (1976). The Fraeijs de Veubeke-Sander quadrilateral (cf. Exercise 6.1.5) is due to SANDER(1964) and FRAEIJS DE VEUBEKE (1965a, 1968), and it has been theoretically studied by CIAVALDINI & NEDELEC(1974). The singular Zienkiewicz triangle is found in Section 10.10 of ZIENKIEWICZ (1971), where alternate singular finite elements are also described. Since the second derivatives of the functions in the space PK have singularities at the vertices (Remark 6.1.6), very accurate quadrature schemes are used in practical computations. IRONS& RAZZAQUE (1972b) (see also RAZZAQUE (1973)) obviate this computational difficulty by “smoothing” the second derivatives. Other ways of adding rational functions are mentioned in BIRKHOFF& MANSFIELD(1974) (cf. Exercise 6.1.7), MANSFIELD (1974, 1976b), DUPUIS& GOEL(1970a). Boolean sum interpolation theory can also be used to derive blending polynomial interpolants, which interpolate a function u E Cem(K)and all its derivatives of order C m on the (possibly curved) boundary of a triangle K. In BIRKHOFF& this direction, see BARNHILL (1976a, 1976b), BARNHILL, GORDON(1973), BARNHILL& GREGORY (1975a, 1975b). For a discussion about the use of finite elements of class Ce’ from the engineering viewpoint, see ZIENKIEWICZ (1971, chapter 10). There, finite elements of class Ce’ are called “compatible” while finite elements which are not of class Ce’ are called “incompatible”, and rational functions such as those which are used in the singular Zienkiewicz triangle are called “singular shape functions”. The Bogner-Fox-Schmit rectangle is not the only rectangular finite element of class Ce’ that may be used in practice. See for example GOPALACHARYULU (1973, 1976). The general approach followed in Section 6.2 is that of CIARLET & LESAINT (1979, a thorough study is made (1974a, 1974b). In LASCAUX not only of Adini’s rectangle, but of other nonconforming finite elements for the plate problem, such as the Zienkiewicz triangle, Morley’s triangle (cf. Exercise 6.2.3) and various instances of Fraeijs de Veubeke triangles (an example of which is given in Exercise 6.2.3). A survey of the use of such nonconforming elements, from an Engineering viewpoint, is found in ZIENKIEWICZ (1971, chapter 10). Adini’s rectangle is due to ADINI& CLOUGH(1961) and MELOSH (1963) and, for this reason, it is sometimes called the ACM rectangle. The convergence of Adini’s rectangle has also been studied by KIKUCHI (1975d, 1976a) and MIYOSHI(1972). KIKUCHI (1975d) considers in addi-

378

T H E PLATE PROBLEM

[Ch. 6.

tion the use of this element for the approximation of the eigenvalue problem. This last problem is also considered from a numerical stand& OLSON(1970) for conforming and nonconforming point by LINDBERG finite elements. An extension to the case of curved nonconfonning elements is considered in BARNHILL& BROWN(1975). Although some of the references given in Section 4.2 were more specifically concerned with second-order problems, some of them are also relevant in the present situation, notably CBA (1976), NITSCHE (1974), OLIVEIRA (1976). There are alternate definitions of nonconforming methods. For example, let us assume that we are given a finite element space v h which satisfies the inclusion v h C U o ( d )fl H d ( 0 ) . Assuming as usual that the functions in the spaces PK are smooth, the conformity would require the = 0 along any side K' comadditional conditions that a , , ( v h , K , ) + av(vhlK2) mon to two adjacent finite elements K , and KZ,and that a,?h = 0 along r. If these conditions cannot be exactly fulfilled, they may be considered as constraints, and accordingly, they may be dealt with either by a penalty method or by duality techniques. In the first approach, one minimizes a functional of the form 1

1

J%(vh)

=2 ah(vh,

vh)

- f ( u h ) + e(h) @ ( v h )

where

and c(-) is a function of h which approaches zero with h. The function c(-) is usually of the form c ( h ) = Ch", C > 0 , where the exponent u > 0 is to be chosen so as to maximize the order of convergence. A method of this type has been studied by BABUSKA & ZLAMAL (1973) who have shown that the use of the Hermite triangle of type (3) results in the error estimates

Ilu - uhllh d c f i b h n 9 IIu - uh1li.n d ChIIuII3.n if u E H3(f2),with the optimal choice c ( h )= Chz, and

IIu - UhIlh d ChIlullan,

(Iu - uh1li.n d ChIluI14,n9

if u E H4(0), with the optimal choice c ( h )= Ch3 (let us add however that this penalty method is analyzed in the case of the biharmonic

Ch. 6.1

BIBLIOGRAPHY A N D COMMENTS

379

problem instead of the plate problem). Such techniques are used in practice: See ZIENKIEWICZ (1974). The second approach consists in introducing an appropriate Lagrangian. This is done for example by HARVEY& KELSEY(1971) who use the Hermite triangle of type (3) for solving the plate problem. Let us next review further aspects of the finite element approximation of the plate problem and of more general fourth-order problems. RANNACHER (1976a) has obtained error estimates in the norm I-lo,.+ The effect of numerical integration is analyzed in BERNADOU& DUCATEL (1976). As regards the approximation of fourth-order problems on domains with curved boundaries, we mention MANSFIELD (1976b), who considers in addition the effect of numerical integration. Her approach parallels that given in CIARLET& RAVIART (1972~)for second-order problems. Curved isoparametric finite elements of a new type are suggested by ROBINSON (1973). In the case of the simply supported plate problem (cf. Exercise 1.2.7), we mention the BabuSka paradox (cf. BABUSKA (1963); see also BIRKHOFF (1969)): Contrary to second-order problems, no convergent approximation may be found if the curved boundary is replaced by a polygonal domain: This is because the boundary condition Au - (1 - a)a,u = 0 on I' (which is included in the variational formulation) is then replaced by the boundary condition avvu= 0. Additional references concerning the handling of curved boundaries and/or boundary conditions for the plate problems are NITSCHE (1971, 1972b), CHERNUKA, COWPER,LINDBERG& OLSON (1972), and the survey of SCOTT (1976b). Finite element approximation of variational inequalities of order four are considered by GLOWINSKI(1975, 1976b). See also GLOWINSKI, LIONS& TREMOLIBRES (1976b, Chapter 4). When large vertical displacements are considered, the plate problem amounts (cf. LANDAU & LIFSCHITZ(1967, Chapter 2)) to finding a pair (ul, u2)E ( H $ ( 0 ) ) 2 solution , of two coupled nonlinear equations, known as von Kamzann's equations:

alA2ul - [ul,1.421 = f,

f E L2(0),

a2A2u2+[ul, ul] = 0, where [ v , w ] = allua22w

+ a22vallw - 2a12va12w,

380

T H E PLATE PROBLEM

[Ch. 6.

and a], a2 are two strictly positive constants. The existence of a (possibly non unique) solution is proved in LIONS (1969, p. 53). For an analysis of a finite element approximation by a mixed method, see MIYOSHI(1976a, 1976b, 1976c, 1977). Another finite element method is proposed in BERGAN& CLOUGH(1973) to handle large displacements. For yet other types of finite element approximation of the plate problem, see ALLMAN (1976), FRIED(1973c), FRIED& YANG (1973), IRONS (1974b), KIKUCHI(1975e), STRICKLIN, HAISLER, TISDALE & GUNDERSON (1 969). Plates with cracks have been considered by YAMAMOTO & TOKUDA(1973), and YAMAMOTO & SUMI (1976). Further references are found in the next chapter, specially for the so-called mixed and hybrid methods.

CHAPTER

7

A MIXED FINITE ELEMENT METHOD

Introduction In this chapter, we consider the problem of approximating the solution of the biharmonic problem: Find u E H:(R), R C R2,such that

Our objective is to study a method based on a different variational formulation of the biharmonic problem (it being implicitly understood that the above variational formulation is the standard one). Such methods fall themselves into several categories (cf. the discussion in the section “Additional Bibliography and Comments” at the end of this chapter), and it is the purpose of this chapter to study one of these, of the so-called mixed type. Basically, it corresponds to a variational formulation where the function u is the first argument of the minimum ( u , cp) of a new functional. In this fashion, we shall directly get approximations not only of the solution u, but also of the second argument cp. Since this function cp turns out to be - A u in the present case, this approach is particularly appropriate for the study of two-dimensional steady-state flows, where -Au represents the uorticity. Thus our first task in Section 7.1 is to construct a functional 9 and a space 7 f such that (Theorem 7.1.2) $(u,

- A u ) = (U.,)E’Y inf

9(u, 4).

The space 7 f consists of pairs ( u , 4) E Hd(R) x L2(R)which satisfy specific linear relations of the form p ( ( u , $), p ) = 0 for all functions p EH’(0). Next, this problem is discretized in a natural way: Given a finite element space Xh contained in the space H ’ ( R ) ,one looks for a pair 38 1

382

A MIXED FINITE ELEMENT METHOD

[Ch. 7.

where the space y h consists of those pairs ( v h , $k) E XOh x x h which satisfy linear relations of the form p((vh, $h), p h ) = 0 for all functions p h E Xh. The major portion of Section 7.1 is then devoted to the study of convergence (Theorems 7.1.5 and 7.1.6): Our main conclusion is that, if the inclusions Pk(K) C PK,K E y h , hold, the error estimate IIu - uhlli.a+

IAU

4-(PhIo.a=O(hk-’)

holds. The main difficulty in this error analysis is that, in general, the space “v;, is not a subspace of the space .Y. (if this were the case, it would

suffice to use the convergence analysis valid for conforming methods). The advantages and drawbacks inherent in this method are easily understood: The main advantage is that it sufices to use finite elements of class g o , whereas finite elements of class %’ would be required for conforming methods. Another advantage (from the point of view of fluid mechanics) is that the present method not only yields a continuous approximation of the function u , but also of the vorticity -Au, whereas a standard approximation using finite elements of class V’ would result in a discontinuous approximation - A u h of the vorticity (which, in addition, needs to be computed). The major drawback is that the computation of the discrete solution (Uh, ( P h ) requires the solution of a constrained minimization problem. since the functions E x o h and $h E x h do not vary independently from one another. It is the object of Section 7.2 to show how such a problem may be solved, using duality techniques. The basic idea consists in introducing an appropriate space ./Uh C x h of “multipliers” and then in applying Uzawa’s method for solving the saddle-point equations (cf. Theorem 7.2.2) of the Lagrangian associated with the present variational formulation. The convergence of Uzawa’s method is established in Theorem 7.2.5. In the process, we find an answer to a problem which has been often considered for the biharmonic problem and its various possible discretizations: We show (Theorem 7.2.4) that, in this particular case, Uzawa’s method amounts to solving a sequence of discrete Dirichlet problems f o r the operator -A.

Ch. 7, 97.1.1

A MIXED FINITE ELEMENT METHOD

383

Therefore we have at our disposal a method for approximating the solution of a fourth-order problem which uses the same finite element programs as those needed f o r second-order problems. 7.1. A mixed finite element method for the biharmonic problem

Another variational formulation of the biharmonic problem

Consider the variational problem which corresponds to the following data:

a(u, u ) =

AuAu dx,

(7.1.1)

where the set d is a convex polygonal subset of R2 and the function f belongs to the space L 2 ( 0 ) We . recognize here the biharmonic problem, whose solution u E H i ( O ) also satisfies

J(u) = inf J(u),

(7.1.2)

UEH%(O)

with (7.1.3)

Thus we may equivalently consider that we are minimizing the functional (7. I .4)

over those pairs ( u , #) E H;(O) x L 2 ( 0 )whose elements u and 4 are related through the equality - A u = 4. This observation is the basis for another variational formulation of the biharmonic problem (Theorem 7.1.2), which depends on the fact that the space

{(u, 4) E H&O) X L 2 ( 0 ) ;- AU = +) can be described in an alternate way, as we now show.

384

[Ch. 7. 8 7 . 1 .

A MIXED FINITE ELEMENT METHOD

Theorem 7.1.1.

Define the space

where (7.1.6) Then the mapping (u7

+) E "cr+

I+lo.n

is a norm ouer the space V,which is equivalent t o the product norm +)E V+(JuI:.n+lt,bl&di'2, and which makes V a Hilbert space. In addition, we have

(u,

V = {(u, +) E H&R) x L2( R ) ;- A v = +}. Proof. Equipped with the product norm, the space V is a Hilbert space since it is a closed subspace of the space Hd(R) x L 2 ( R ) . Let ( u , +) be any element of the space V.The particular choice p = v in the definition (7.1.5) of this space gives

dx 6 C(R)l+l~.nlul~.n.

Iul:.a=

where C(R) is the constant appearing in the PoincarC-Friedrichs inequality (cf. ( I .2.2)). Therefore, (IUl:.n+

l+l:,n)i/2 6

((c(m2 + 1)i'21+lo.n9

and the first assertion is proved. Since the set R has a Lipschitz-continuous boundary formula

r, the

Green

v u E H 2 ( R ) , v p E Hi(R),

JnVu

. V p dx =

-Io

Au p dx +

8.u p

(7.1.7)

dy

+

E L2(R) be related holds. Let then the functions u E H&R) and For any function p E H ' ( R ) , an application of through -Au = Green's formula (7.1.7) shows that p((u, +), p ) = 0, since 8,u = 0 on r. Conversely, let the functions u E Hd(l.2) and E L2(R) satisfy

+.

+

Ch. 7, 0 7.1.1

385

A MIXED FINITE ELEMENT METHOD

p((u, JI), p ) = 0 for all

p

E H ' ( R ) .In particular then, we have

so that v appears as the solution of a homogeneous Dirichlet problem for the operator - A on the set R. Since the set d is convex, such a second-order boundary value problem is regular, i.e., the function u is in the space H*(R).Using Green's formula (7.1.7) with functions p in the space Hd(R), we first deduce that - A u = JI, and using the same Green formula with functions p in the space H ' ( R ) , we next deduce that a,u = 0 along r. El Theorem 7.1.2. Let u E H i ( R ) denote the solution of the minimization problem (7.1.2). Then we also have (7.1.8)

where the functional 9 and the space 'If are defined as in (7.1.4) and ( 7 . 1 3 , respectively. In addition, the pair (u,- A u ) E 'If is the unique solution of the minimization problem (7.1 3). Proof.

The symmetric bilinear form

( ( u ,cp).(v, JI)) E V X V + I n cpJI dx is continuous and 'If-elliptic (by Theorem 7.1.1), and the linear form ( u , $1 E ' I f + I n f v dx

is continuous. Therefore the minimization problem: Find an element

(u*,cp) E 'If such that

has one and only one solution, also solution of the variational equations

W v , JI) E 'If,

In

cpJI

dx =

dx.

(7.1.10)

Let us establish the relationship between this solution (u*,cp) and the solution of problem (7.1.2). Since the pair (u*,cp) is an element of the

386

[Ch. 7, 07.1.

A MIXED FINITE ELEMENT METHOD

space V , we deduce from Theorem 7.1.1 that the function u* belongs to the space Hi(L!) and that -Au* = cp. Applying again the same theorem in conjunction with relations (7.1. lo), we find that V u E H&L!),InAu*Au dx =

In

f u dx,

and thus the function u* coincides with the solution u of problem (7.1.2). 0

The corresponding discrete problem. Abstract error estimate We are now in a position to describe a discrete problem associated with this new variational formulation of the biharmonic problem. Let there be given a finite element space Xh which satisfies the inclusion x h

c H'(L!).

(7.1.1 1)

We define as usual the finite element space XOh

= {Vh E x

h ; uh

=0

on

r},

(7.1.12)

and we let (compare with (7.1.5)) y h

= {(uh,

$h)

EXOh

x h ; V F h

Ex h ,

p ( ( u h , $h),

ph)

=

o}, (7.1.13)

where the mapping p((. ,.), .) is defined as in (7.1.6). Then, in analogy with (7.1.8), we define the discrete problem as follows: Find an element (uh,( P h ) E y h such that (7.1.14)

where 8; is the functional defined in (7.1.4).

Remark 7.1.1. It is thus realized that the same space X h is used for the approximation of both spaces H ' ( 0 ) and L 2 ( 0 ) It . is indeed possible to develop a seemingly more general theory where another space, say Y h r is used for approximating the space L2(L!),but eventually the advantage is nil: As shown in CIARLET& RAVIART(1974), one is naturally led, in the process of getting error estimates, to assume that the inclusion YhC x h holds, and this is precisely contrary to what one would have naturally

Ch. 7, P7.1.1

387

A MIXED FINITE ELEMENT METHOD

expected. Besides, the assumption x h = Y h simplifications in the developments to come. Theorem 7.1.3. solution.

yields

significant El

The discrete problem (7.1.14) has one and only one

Proof. Arguing as in the proof of Theorem 7.1.1, we deduce that the mapping (uh, $h)

E “vh

~h~~ll.fl

is a norm over the space “vh. Thus, the existence and uniqueness of the solution of the discrete problem follows by an argument similar to that of Theorem 7.1.2. 0 As a consequence of this result, the element solution of the variational equations v ( u h , $h>

E vh,

lo

(Ph$h

d x = I,fuh

dx-

( u h , (Ph)

E “y;, is also

(7.1.15)

We next begin our study of the convergence of this approximation process. As usual, we shall first establish an abstract error estimate (in two steps; cf. Theorems 7.1.4 and 7.1.5) and we shall then apply this to some typical finite element spaces (Theorem 7.1.6). The abstract error estimate consists in getting an upper bound for the expression Iu - U h I I . f l + IAu -k

(Phl0.n

(recall that (Ph is an approximation of -Au, whence the unusual sign in the second term). Notice that the above expression is a natural analogue in the present situation of the error in the norm II.(IZ.~that arises in conforming methods. As a first step towards getting the error estimate, we prove:

388

A MIXED FINITE ELEMENT METHOD

[Ch. 7, 07.1.

Proof. Since the set R is convex, the solution u of problem (7.1.2) belongs to the space H3(R).Thus we can write

-la

V u E S(R),

V u . V ( A u )dx = f a AuAu dx

=Iafu

dx,

and consequently

vu E Hd(R),

-laVu

V ( A u )dx =

fa

f v dx.

Using the definition (7.1.6) of the mapping p((. , .), .), we have therefore shown that, given any function u E Hd(f2) and any function $ E L 2 ( R ) we , have P ( ( v , $1, - A u ) = l a f u dx

+

la$Au

dx.

(7.1.17)

Let then (uh. $4,) be an arbitrary element of the space "v;, and let ph be an arbitrary element of the space Xh. Using the definition (7.1.5) of the space V, the variational equations (7.1.15) and relation (7.1.17), we obtain

From this equality, we deduce that

( I n ( d u + ( P h ) ( ( P h - $ h ) dx(

o(fl)((P~~ - $~Io.RIIAU P~III.o,

where the constant D ( R ) depends solely on the constant C(R) of the PoincarC-Friedrichs inequality (argue as in the beginning of the proof of Theorem 7.1.1). Using this inequality, we get ((PI,

- $hIi.a =

la

( ( ~ h-

$h)(Au + (

dx

~ h )

(7.1.18)

Ch. 7. 5 7.1.1

389

A MIXED FINITE ELEMENT METHOD

To apply Theorem 7.1.4, we have to estimate on the one hand the expression infPhExhJlAu+ p h ( J l . nwhich , is a standard problem. On the other hand, we also have to estimate the expression inf)E vh (lu - U ~ I ~ . R + IAu + + h l O . d ,

(Uh,+h

and this is no longer a standard problem, because the functions u h and +h d o not vary independently in their respective spaces X O h and x h . Nevertheless, it is possible to estimate the above expression by means of the “unconstrained” terms infUhEXoh Iu - v h l 1 . n and infrhExhIAu - phlo,n (cf. (7.1.22)). provided we make use of an appropriate inverse inequality, as our next result shows. Theorem 7.1.5. V p h

Let a ( h ) be a strictly positive constant such that

Exh,

Iph(l.Rs

Then there exists a constant C independent of the space lu

-uh110+

(7.1.20)

~(h)(phIO.R* x h

such that

IAu + (Ph(O.ns

hoof. Let ( v h , + h ) be an arbitrary element in the space vh, and let Ph be an arbitrary element in the space Xh. The function vh = ph + +h belongs to the space X h and thus, p((vh,

+h),

vh) = 0.

390

A MIXED FINITE ELEMENT METHOD

Next, using the fact that a,u

=0

on

[Ch. 7, 0 7.1.

r,we get

so that, combining the two above equalities, we obtain

and thus,

To finish the proof, it suffices to combine inequalities (7.1.16) and (7.1.22). 0 Estimate of the error (lu - u h l l . 0 + (Au+ c$hlo,n) To apply the abstract error estimate proved in the previous theorem, we shall need the following standard assumptions on the family of finite element spaces xh: (Hl) The associated family of triangulations T h is regular. (H2) All the finite elements ( K , pK.&), K E u h Yh, are affineequivalent to a single reference finite element (Iz,B, S).

Ch. 7, 87.1.1

A MIXED FINITE ELEMENT METHOD

39 1

(H3) All the finite elements ( K , PK.ZK), K E uh Y,,, are of class go. (H4) The family of triangulations satisfies an inverse assumption (cf. 3.2.28)).

Theorem 7.1.6. In addition to (HI), (H2), (H3) and (H4), assume that there exists an integer k a 2 such that the following inclusions are satisfied: P k ( k )C B C H ' ( k ) . (7.1.23) Then if the solution u E H i ( 0 ) of the minimization problem (7.1.2) belongs to the space H k " ( 0 ) , there exists a constant C independent of h such that (7.1.24) IU - uhIl&!+ (AU -k (Ph(o.ns Chk-'((ulk+l.n+(Aulk,n). Proof. In view of the inclusions (7.1.23), there exist constants C independent of h such that

(7.1.25)

inf lldu - p,,I(~.ns Chk-'[AuIk.n.

(7.1.26)

@hEXh

Next, the inverse assumption allows us to conclude (cf. (3.2.35)) that the constants a ( h ) in inequalities (7.1.20) may be taken of the form C a ( h ) = -, h

(7.1.27)

for another constant C independent of h. Then the conclusion follows by using relations (7.1.29, (7.1.26) and (7.1.27) in the error estimate (7.1.21). 0 Remark 7.1.2. In principle, an inclusion such as H'+'(k)G %'(k) (where s is the maximal order of partial derivatives occurring in the definition of the set %) should have been added, but it is always satisfied in practice: Since s = 0 or 1 for finite elements of class '&O and since n = 2, the inclusion H 3 ( k )C V(k)holds. 0 Concluding remarks Let us briefly discuss the application of this theorem: The major conclusion is that one can solve the biharmonic problem with the same

392

A MIXED FINITE ELEMENT METHOD

[Ch. 7, 8 7.1.

finite element spaces that are normally used f o r solving second-order problems, provided the inclusions P 2 ( K )C PK,K E Y h , hold. If we are using in particular triangles of type (k), k 3 2, we get IIU

- Uh1II.n = O ( h k - ' ) and

(AU + (Ph(o,n= O ( h k - ' ) .

We shall therefore retain two basic advantages of this method: First, we get a convergent approximation to the solution u (albeit in the norm ~ ~ instead . ~ ~ ,of, the ~ norm 1( . ( 1 2. R ) with much less sophisticated finite element spaces than would be required in conforming methods. The second advantage is that we obtain a convergent approximation (Ph of the vorticity - Au, a physical quantity of interest in steady-state flows. Nevertheless, one should keep in mind that, in spite of the simplicity of the spaces xh, there remains the practical problem of actually computing the pair ( U h , ( P h ) . This is the object of the next section.

Exercise 7.1.1. Following CIARLET& GLOWINSKI(1975), the object of this problem is to show that the solution of the biharmonic problem can be reduced to the solution of a sequence of Dirichlet problems f o r the operator - A (indeed, the analysis which shall be developed in the next section is nothing but the discrete analogue of what follows). We recall that (cf. Theorem 7.1.2)

where the functional 9 and the space 5' are defined as in (7.1.4) and ( 7 . 1 3 , respectively. Let there be given a subspace A of the space H ' ( R ) such that we may write the direct sum

H ' ( l 2 ) = Hd(l2) @ A. We next introduce the space

w = { ( v , (I)€ Hd(W x L2(R);V P E HdW), P ( ( v , (I),P ) =

0 1 9

where the mapping P is defined a s in (7.1.6), and we define the Lagrangian a ( V ,

Ijl),P ) =

f(v9

(I)+ P ( ( D I (I),P h

(i) Show that, given a function A E A, the problem: Find an element

Ch. 7, 97.1.1

A MIXED FINITE ELEMENT METHOD

393

has one and only one solution, which may also be obtained by solving the following Dirichlet problems for the operator - A : (*) Find a function QA E H ' ( 0 ) such that (VA - A ) E Hd(0)-

V UE H d ( 0 ) , lnVqA * V U dx

=

I,

fu dx.

(**) Find a function uAE Hd(0) such that

(notice that since d is a convex polygon, the function uAis in fact in the 0Hd(0)). ) space ~ ' (n (ii) Let u denote the solution of problem (7.1.2), and let A* be that function in the space A which is such that the function (Au + A * ) belongs to the space Hd(0). Show that ( ( u , - A u ) , A * ) is the unique saddle-point of the Lagrangian 9 over the space W XA, in the sense that V(U, *) E u', v p E A,

B ( ( ~ , - A ~ ) , ~ ) C ~ ' ( ( U , - A U ) , A * ) ~$),A*). ~((U,

(iii) As a consequence of (ii), show that

9e((~, - A u ) , A *) = ~gg(A), where the function g: A + R is defined by g : A E .A +g(A) = &'@,2'((u,

$), A )

(this is a standard device in duality theory; see for example EKELAND & TEMAM (1974, chapter VI). (iv) We next apply the gradient method to the maximization problem of question (iii) (a technique known as Uzawa's method for the original minimization problem, then called the primal problem): Given any

394

A MIXED FINITE ELEMENT METHOD

[Ch. 7, 5 7.1.

function AoE A and a parameter p > 0 (to be specified in (v)), we define a sequence of functions A " € A by the recurrence relation: vp

(An+l-A",p)rn =-p(Dg(A"),p),

where (. ,.)& is an inner product in the space A,whose associated norm is assumed to be equivalent to the norm ll. l [l, n , and (. , .) denotes the pairing between the spaces A' (= dual space of A )and A.Show that the function g is indeed everywhere differentiable over the space A and that one iteration of Uzawa's method consists of the following steps: (*) Given a function A " E A, find the function p" E H'(R) which satisfies: (p"- A " ) E

~ d < m

V u E Hd(R), fnVp" * Vu dx = f n f v dx. (**) Find the function

U"

E Hd(l2) which satisfies

V u E Hd(R), I n V u " V u dx =

In

p"u dx.

(***) Find the function A"+' E A which satisfies

V p E A, ( A " + ' - A", p)rn = p P ( ( U " , p"),CL).

(v) Show that the method described in question (iv) is convergent, in the sense that

lim ((u"- U I I , , ~ = 0, nlim ip" + Aulo.n= 0, n-Pprovided that 0