Chapter Ⅰ 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
Chapter Ⅱ Conforming Finite Elements 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
Ellipticity in H1 44
Boundary-value problems with natural bound-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
4．The RitzGalerkin 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-mials 64
Remarks on C1 elements 67
Bilinear elements 68
Quadratic rectangular elements 69
Affine families 70
Choice of an element 74
Problems 74
6．Approximation Properties 76
The BrambleHilbert lemma 77
Triangular elements with com-plete polynomials 78
Bilinear quadrilateral elements 81
In-verse estimates 83
Clément's interpolation 84
Appendix:On the optimality of the estimates 85
Problems 87
7．Error Bounds for Elliptic Problems of Second Order 89
Remarks on regularity 89
Error bounds in the energy norm 90
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
Chapter Ⅲ Nonconforming and Other Methods 105
1．Abstract Lenmas and a Simple Boundary Approximation 106
Generalizations of Céa's lemma 106
Duality methods 108
The Crouzeix-Raviart element 109
A simple approximation to curved boundaries 112
Modifications of the duality argument 114
Problems 116
2．Isoparametric Elements 117
Isoparametric triangular elements 117
Isoparametric quadrilateral elements 119
Problems 121
3．Further Tools from Functional Analysis 122
Negative norms 122
Adjoint operators 124
An abstract exis-tence theorem 124
An abstract convergence theorem 126
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
Saddle point problems with penalty term 138
Typical applications 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 P1 ele-ment 170
Problems 171
8．A Posteriori Error Estimates 172
Residual estimators 174
Lower estimates 176
Remark on other estimators 179
Local mesh refinement and convergence 179
9．A Posteriori Error Estimates via the Hypercircle Method 181
Chapter Ⅳ The Conjugate Gradient Method 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
The general gradient method 196
Gradient methods and quadratic functions 197
Convergence behavior in the case of large condition 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
Remarks on parallelization 216
Nonlinear problems 217
Prob-lems 218
5．Saddle Point Problems 221
The Uzawa algorithm and its variants 221
An alternative 223
Problems 224
Chapter Ⅴ Multigrid Methods 225
1．Multigrid Methods for Variational Problems 226
Smoothing properties of classical iterative methods 226
The multi-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 the V-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．Multigrid Analysis via Space Decomposition 261
Schwarz alternating method 262
Assumptions 265
Direct con-sequences 266
Convergence of multiplicative methods 267
Verification of A1 269
Local mesh refinements 270
Problems 271
6．Nonlinear Problems 272
The multigrid-Newton method 273
The nonlinear multigrid method 274
Starting values 276
Problems 277
Chapter Ⅵ Finite Elements in Solid Mechanics 278
1．Introduction to Elasticity Theory 279
Kinematics 279
The equilibrium equations 281
The Piola trans-form 283
Constitutive Equations 284
Linear material laws 288
2．Hyperelastic Materials 290
3．Linear Elasticity Theory 293
The variational problem 293
The displacement formulation 297
The mixed method of Hellinger and Reissner 300
The mixed method of Hu and Washizu 302
Nearly incompressible material 304
Locking 308
Locking of the Timoshenko beam and typical remedies 310
Problems 314
4．Membranes 315
Plane stress states 315
Plane strain states 316
Membrane ele-ments 316
The PEERS element 317
Problems 320
5．Beams and Plates:The Kirchhoff Plate 323
The hypotheses 323
Note on beam models 326
Mixed methods for the Kirchoff plate 326
DKT elements 328
Problems 334
6．The Mindlin-Reissner Plate 335
The Helmholtz decomposition 336
The mixed formulation with the Helmholtz decomposition 338
MITC elements 339
The model without a Helmholtz decomposition 343
Problems 346
References 348
Index 361