Adaptive Numerical Solution of PDEs

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
Outline......Page 14
1.1 Laplace and Poisson Equation......Page 18
1.1.1 Boundary Value Problems......Page 19
1.1.2 Initial Value Problem......Page 23
1.1.3 Eigenvalue Problem......Page 25
1.2 Diffusion Equation......Page 28
1.3 Wave Equation......Page 31
1.4 Schrödinger Equation......Page 36
1.5.1 Boundary Value Problems......Page 39
1.5.2 Time-harmonic Differential Equations......Page 40
1.6 Classification......Page 42
1.7 Exercises......Page 44
2.1.1 Maxwell Equations......Page 47
2.1.2 Optical Model Hierarchy......Page 50
2.2 Fluid Dynamics......Page 53
2.2.1 Euler Equations......Page 54
2.2.2 Navier-Stokes Equations......Page 57
2.2.3 Prandtl’s Boundary Layer......Page 62
2.2.4 Porous Media Equation......Page 64
2.3.1 Basic Concepts of Nonlinear Elastomechanics......Page 65
2.3.2 Linear Elastomechanics......Page 69
2.4 Exercises......Page 72
3.1 Discretization of Standard Problem......Page 75
3.1.1 Discrete Boundary Value Problems......Page 76
3.1.2 Discrete Eigenvalue Problem......Page 81
3.2 Approximation Theory on Uniform Grids......Page 84
3.2.1 Discretization Error in L2......Page 86
3.2.2 Discretization Error in L∞......Page 89
3.3.1 One-dimensional Special Case......Page 91
3.3.2 Curved Boundaries......Page 93
3.4 Exercises......Page 96
4.1.1 Weak Solutions......Page 99
4.1.2 Ritz Minimization for Boundary Value Problems......Page 102
4.1.3 Rayleigh-Ritz Minimization for Eigenvalue Problems......Page 106
4.2 Spectral Methods......Page 108
4.2.1 Realization by Orthogonal Systems......Page 109
4.2.2 Approximation Theory......Page 113
4.2.3 Adaptive Spectral Methods......Page 116
4.3.1 Meshes and Finite Element Spaces......Page 121
4.3.2 Elementary Finite Elements......Page 124
4.3.3 Realization of Finite Elements......Page 134
4.4.1 Boundary Value Problems......Page 141
4.4.2 Eigenvalue Problems......Page 144
4.4.3 Angle Condition for Nonuniform Meshes......Page 149
4.5 Exercises......Page 152
5 Numerical Solution of Linear Elliptic Grid Equations......Page 156
5.1 Direct Elimination Methods......Page 157
5.1.1 Symbolic Factorization......Page 158
5.1.2 Frontal Solvers......Page 160
5.2 Matrix Decomposition Methods......Page 163
5.2.1 Jacobi Method......Page 165
5.2.2 Gauss-Seidel Method......Page 167
5.3.1 CG-Method as Galerkin Method......Page 169
5.3.2 Preconditioning......Page 172
5.3.3 Adaptive PCG-method......Page 176
5.3.4 A CG-variant for Eigenvalue Problems......Page 178
5.4.1 Illustration for the Poisson Model Problem......Page 183
5.4.2 Spectral Analysis for Jacobi Method......Page 187
5.4.3 Smoothing Theorems......Page 188
5.5 Iterative Hierarchical Solvers......Page 193
5.5.1 Classical Multigrid Methods......Page 195
5.5.2 Hierarchical-basis Method......Page 203
5.5.3 Comparison with Direct Hierarchical Solvers......Page 206
5.6 Power Optimization of a Darrieus Wind Generator......Page 207
5.7 Exercises......Page 213
6.1 A Posteriori Error Estimators......Page 216
6.1.1 Residual Based Error Estimator......Page 219
6.1.2 Triangle Oriented Error Estimators......Page 224
6.1.3 Gradient Recovery......Page 228
6.1.4 Hierarchical Error Estimators......Page 232
6.1.5 Goal-oriented Error Estimation......Page 235
6.2 Adaptive Mesh Refinement......Page 236
6.2.1 Equilibration of Local Discretization Errors......Page 237
6.2.2 Refinement Strategies......Page 242
6.3 Convergence on Adaptive Meshes......Page 246
6.3.1 A Convergence Proof......Page 247
6.3.2 An Example with a Reentrant Corner......Page 249
6.4 Design of a Plasmon-Polariton Waveguide......Page 253
6.5 Exercises......Page 257
7.1 Subspace Correction Methods......Page 259
7.1.1 Basic Principle......Page 260
7.1.2 Sequential Subspace Correction Methods......Page 263
7.1.3 Parallel Subspace Correction Methods......Page 268
7.1.4 Overlapping Domain Decomposition Methods......Page 272
7.1.5 Higher-order Finite Elements......Page 279
7.2 Hierarchical Space Decompositions......Page 284
7.2.1 Decomposition into Hierarchical Bases......Page 285
7.2.2 L2-orthogonal Decomposition: BPX......Page 291
7.3.1 Additive Multigrid Methods......Page 295
7.3.2 Multiplicative Multigrid Methods......Page 299
7.4.1 Theoretical Derivation......Page 302
7.4.2 Adaptive Realization......Page 308
7.5 Eigenvalue Problem Solvers......Page 313
7.5.1 Linear Multigrid Method......Page 314
7.5.2 Rayleigh Quotient Multigrid Method......Page 316
7.6 Exercises......Page 319
8 Adaptive Solution of Nonlinear Elliptic Problems......Page 323
8.1 Discrete Newton Methods for Nonlinear Grid Equations......Page 324
8.1.1 Exact Newton Methods......Page 325
8.1.2 Inexact Newton-PCG Methods......Page 329
8.2.1 Hierarchical Grid Equations......Page 332
8.2.2 Realization of Adaptive Algorithm......Page 334
8.2.3 An Elliptic Problem Without a Solution......Page 338
8.3 Operation Planning in Facial Surgery......Page 341
8.4 Exercises......Page 344
9.1 Time Discretization of Stiff Differential Equations......Page 346
9.1.1 Linear Stability Theory......Page 347
9.1.2 Linearly Implicit One-step Methods......Page 353
9.1.3 Order Reduction......Page 360
9.2 Space-time Discretization of Parabolic PDEs......Page 366
9.2.1 Adaptive Method of Lines......Page 367
9.2.2 Adaptive Method of Time Layers......Page 375
9.2.3 Goal-oriented Error Estimation......Page 384
9.3.1 Mathematical Models......Page 387
9.3.2 Numerical Simulation......Page 388
9.4 Exercises......Page 391
A.1 Fourier Analysis and Fourier Transform......Page 393
A.2 Differential Operators in R3......Page 394
A.3 Integral Theorems......Page 396
A.4 Delta-Distribution and Green\'s Functions......Page 400
A.5 Sobolev Spaces......Page 405
A.6 Optimality Conditions......Page 410
B.1 Adaptive Finite Element Codes......Page 411
B.3 Nonlinear Solvers......Page 412
Bibliography......Page 414
Index......Page 428