Finite elements: theory, fast solvers, and applications in elasticity theory

385 Pages·2007·1.358 MB·English

by  Dietrich Braess

This page intentionally left blank
FINITE ELEMENTS
This definitive introduction to finite element methods has been thoroughly
updatedforthisthirdedition,whichfeaturesimportantnewmaterialforboth
researchandapplicationofthefiniteelementmethod.
The discussionof saddlepointproblemsis ahighlightof thebookand has
beenelaboratedtoincludemanymorenonstandardapplications.Thechapter
on applications in elasticity now contains a complete discussion of locking
phenomena.
Thenumericalsolutionofellipticpartialdifferentialequationsisanimport-
ant application of finite elements and the author discusses this subject com-
prehensively. These equations are treated as variational problems for which
the Sobolev spaces are the right framework. Graduate students who do not
necessarily have any particular background in differential equations but
requireanintroductiontofiniteelementmethodswillfindthistextinvaluable.
Specifically,thechapteronfiniteelementsinsolidmechanicsprovidesabridge
betweenmathematicsandengineering.
DIETRICH BRAESS is Professor of Mathematics at Ruhr University
Bochum,Germany.

FINITE ELEMENTS
Theory, Fast Solvers, and Applications in Elasticity Theory
DIETRICH BRAESS
TranslatedfromtheGermanbyLarryL.Schumaker
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521705189
© D. Braess 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-27910-2 eBook (NetLibrary)
ISBN-10 0-511-27910-8 eBook (NetLibrary)
ISBN-13 978-0-521-70518-9 paperback
ISBN-10 0-521-70518-5 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
v
Contents
Preface to the Third English Edition page x
Preface to the First English Edition xi
Preface to the German Edition xii
Notation xiv
ChapterI
Introduction 1
§1. Examples and Classification of PDE’s 2
Examples 2 — Classification of PDE’s 8 —Well-posed problems 9
— Problems 10
§2. The Maximum Principle 12
Examples 13 — Corollaries 14 — Problem 15
§3. Finite Difference Methods 16
Discretization 16 — Discrete maximum principle 19 — Problem 21
§4. A Convergence Theory for Difference Methods 22
Consistency 22 — Local and global error 22 — Limits of the con-
vergence theory 24 — Problems 26
ChapterII
ConformingFiniteElements 27
§1. Sobolev Spaces 28
Introduction to Sobolev spaces 29 — Friedrichs’ inequality 30 —
Possible singularities of H1 functions 31 — Compact imbeddings
32 — Problems 33
§2. Variational Formulation of Elliptic Boundary-Value Problems of
Second Order 34
Variational formulation 35 — Reduction to homogeneous bound-
ary conditions 36 — Existence of solutions 38 — Inhomogeneous
boundary conditions 42 — Problems 42
§3. The Neumann Boundary-Value Problem.A Trace Theorem 44
EllipticityinH1 44—Boundary-valueproblemswithnaturalbound-
ary conditions 45 — Neumann boundary conditions 46 — Mixed
boundary conditions 47 — Proof of the trace theorem 48 — Practi-
cal consequences of the trace theorem 50 — Problems 52
vi Contents
§4. The Ritz–Galerkin Method and Some Finite Elements 53
Model problem 56 — Problems 58
§5. Some Standard Finite Elements 60
Requirements on the meshes 61 — Significance of the differentia-
bility properties 62 — Triangular elements with complete polyno-
mials64—RemarksonC1 elements67—Bilinearelements68—
Quadraticrectangularelements69—Affinefamilies70—Choice
of an element 74 — Problems 74
§6. Approximation Properties 76
The Bramble–Hilbert lemma 77 — Triangular elements with com-
plete polynomials 78 — Bilinear quadrilateral elements 81 — In-
verse estimates 83 — Cle´ment’s interpolation 84 — Appendix: On
the optimality of the estimates 85 — Problems 87
§7. Error Bounds for Elliptic Problems of Second Order 89
Remarksonregularity89—Errorboundsintheenergynorm90—
L2 estimates 91 — A simple L∞ estimate 93 — The L2-projector
94 — Problems 95
§8. Computational Considerations 97
Assembling the stiffness matrix 97 — Static condensation 99 —
Complexity of setting up the matrix 100 — Effect on the choice of
a grid 100 — Local mesh refinement 100 — Implementation of the
Neumann boundary-value problem 102 — Problems 103
ChapterIII
NonconformingandOtherMethods 105
§1. Abstract Lemmas and a Simple BoundaryApproximation 106
GeneralizationsofCe´a’slemma106—Dualitymethods108—The
Crouzeix–Raviartelement109—Asimpleapproximationtocurved
boundaries 112 — Modifications of the duality argument 114 —
Problems 116
§2. Isoparametric Elements 117
Isoparametrictriangularelements117—Isoparametricquadrilateral
elements 119 — Problems 121
§3. Further Tools from FunctionalAnalysis 122
Negative norms 122 — Adjoint operators 124 — An abstract exis-
tencetheorem124—Anabstractconvergencetheorem126—Proof
of Theorem 3.4 127 — Problems 128
§4. Saddle Point Problems 129
Saddle points and minima 129 — The inf-sup condition 130 —
Mixed finite element methods 134 — Fortin interpolation 136 —
Contents vii
Saddlepointproblemswithpenaltyterm138—Typicalapplications
141 — Problems 142
§5. Mixed Methods for the Poisson Equation 145
The Poisson equation as a mixed problem 145 — The Raviart–
Thomas element 148 — Interpolation by Raviart–Thomas elements
149 — Implementation and postprocessing 152 — Mesh-dependent
norms for the Raviart–Thomas element 153 — The softening be-
haviour of mixed methods 154 — Problems 156
§6. The Stokes Equation 157
Variational formulation 158 —The inf-sup condition 159 — Nearly
incompressible flows 161 — Problems 161
§7. Finite Elements for the Stokes Problem 162
An instable element 162 — The Taylor–Hood element 167 — The
MINI element 168 — The divergence-free nonconforming P ele-
1
ment 170 — Problems 171
§8. A Posteriori Error Estimates 172
Residualestimators174—Lowerestimates176—Remarkonother
estimators 179 — Local mesh refinement and convergence 179
§9. A Posteriori Error Estimates via the Hypercircle Method 181
ChapterIV
TheConjugateGradientMethod 186
§1. Classical Iterative Methods for Solving Linear Systems 187
Stationary linear processes 187 — The Jacobi and Gauss–Seidel
methods 189 — The model problem 192 — Overrelaxation 193 —
Problems 195
§2. Gradient Methods 196
Thegeneralgradientmethod196—Gradientmethodsandquadratic
functions197—Convergencebehaviorinthecaseoflargecondition
numbers 199 — Problems 200
§3. Conjugate Gradient and the Minimal Residual Method 201
The CG algorithm 203 —Analysis of the CG method as an optimal
method 196 — The minimal residual method 207 — Indefinite and
unsymmetric matrices 208 — Problems 209
§4. Preconditioning 210
Preconditioning by SSOR 213 — Preconditioning by ILU 214 —
Remarksonparallelization216—Nonlinearproblems217—Prob-
lems 218
viii Contents
§5. Saddle Point Problems 221
The Uzawa algorithm and its variants 221 —An alternative 223 —
Problems 224
ChapterV
MultigridMethods 225
§1. Multigrid Methods forVariational Problems 226
Smoothingpropertiesofclassicaliterativemethods226—Themulti-
grid idea 227 — The algorithm 228 — Transfer between grids 232
— Problems 235
§2. Convergence of Multigrid Methods 237
Discrete norms 238 — Connection with the Sobolev norm 240 —
Approximation property 242 — Convergence proof for the two-grid
method 244 —An alternative short proof 245 — Some variants 245
— Problems 246
§3. Convergence for Several Levels 248
A recurrence formula for the W-cycle 248 — An improvement for
the energy norm 249 — The convergence proof for theV-cycle 251
— Problems 254
§4. Nested Iteration 255
Computation of starting values 255 — Complexity 257 — Multi-
grid methods with a small number of levels 258 —The CASCADE
algorithm 259 — Problems 260
§5. MultigridAnalysis via Space Decomposition 261
Schwarzalternatingmethod262—Assumptions265—Directcon-
sequences 266 — Convergence of multiplicative methods 267 —
Verification ofA1 269 — Local mesh refinements 270 — Problems
271
§6. Nonlinear Problems 272
Themultigrid-Newtonmethod273—Thenonlinearmultigrid
method 274 — Starting values 276 — Problems 277
ChapterVI
FiniteElementsinSolidMechanics 278
§1. Introduction to Elasticity Theory 279
Kinematics279—Theequilibriumequations281—ThePiolatrans-
form283—ConstitutiveEquations284—Linearmateriallaws288
§2. Hyperelastic Materials 290