Numerical Analysis of Partial Differential Equations

Numerical Analysis of Partial Differential Equations......Page 5
Contents......Page 7
Preface......Page 11
Acknowledgments......Page 15
1.1 Second-Order Approximation for Δ......Page 17
1.2 Fourth-Order Approximation for Δ......Page 31
1.3 Neumann Boundary Condition......Page 35
1.4 Polar Coordinates......Page 40
1.5 Curved Boundary......Page 42
1.6 Difference Approximation for Δ2......Page 46
1.7 A Convection-Diffusion Equation......Page 48
1.8 Appendix: Analysis of Discrete Operators......Page 51
1.9 Summary and Exercises......Page 53
2.1 Function Spaces......Page 61
2.2 Derivatives......Page 64
2.3 Sobolev Spaces......Page 68
2.4 Sobolev Embedding Theory......Page 72
2.5 Traces......Page 75
2.6 Negative Sobolev Spaces......Page 78
2.7 Some Inequalities and Identities......Page 80
2.8 Weak Solutions......Page 83
2.9 Linear Elliptic PDEs......Page 90
2.10 Appendix: Some Definitions and Theorems......Page 98
2.11 Summary and Exercises......Page 104
3.1 Approximate Methods of Solution......Page 111
3.2 Finite Elements in 1D......Page 117
3.3 Finite Elements in 2D......Page 125
3.4 Inverse Estimate......Page 135
3.5 L2 and Negative-Norm Estimates......Page 138
3.6 Higher-Order Elements......Page 141
3.7 A Posteriori Estimate......Page 149
3.8 Quadrilateral Elements......Page 152
3.9 Numerical Integration......Page 154
3.10 Stokes Problem......Page 160
3.11 Linear Elasticity......Page 172
3.12 Summary and Exercises......Page 176
4.1 Condition Number......Page 185
4.2 Classical Iterative Methods......Page 189
4.3 Krylov Subspace Methods......Page 194
4.4 Direct Methods......Page 206
4.5 Preconditioning......Page 212
4.6 Appendix: Chebyshev Polynomials......Page 224
4.7 Summary and Exercises......Page 226
5.1 Trigonometric Polynomials......Page 237
5.2 Fourier Spectral Method......Page 248
5.3 Orthogonal Polynomials......Page 258
5.4 Spectral Galerkin and Spectral Tau Methods......Page 278
5.5 Spectral Collocation......Page 280
5.6 Polar Coordinates......Page 294
5.7 Neumann Problems......Page 296
5.8 Fourth-Order PDEs......Page 297
5.9 Summary and Exercises......Page 298
6 Evolutionary PDEs......Page 307
6.1 Finite Difference Schemes for Heat Equation......Page 308
6.2 Other Time Discretization Schemes......Page 327
6.3 Convection-Dominated equations......Page 331
6.4 Finite Element Scheme for Heat Equation......Page 333
6.5 Spectral Collocation for Heat Equation......Page 337
6.6 Finite Difference Scheme for Wave Equation......Page 338
6.7 Dispersion......Page 343
6.8 Summary and Exercises......Page 348
7 Multigrid......Page 361
7.1 Introduction......Page 362
7.2 Two-Grid Method......Page 365
7.3 Practical Multigrid Algorithms......Page 367
7.4 Finite Element Multigrid......Page 370
7.5 Summary and Exercises......Page 379
8 Domain Decomposition......Page 385
8.1 Overlapping Schwarz Methods......Page 386
8.2 Orthogonal Projections......Page 390
8.3 Non-overlapping Schwarz Method......Page 398
8.4 Substructuring Methods......Page 403
8.5 Optimal Substructuring Methods......Page 411
8.6 Summary and Exercises......Page 426
9 Infinite Domains......Page 435
9.1 Absorbing Boundary Conditions......Page 436
9.2 Dirichlet–Neumann Map......Page 440
9.3 Perfectly Matched Layer......Page 443
9.4 Boundary Integral Methods......Page 446
9.5 Fast Multipole Method......Page 449
9.6 Summary and Exercises......Page 452
10.1 Newton\'s Method......Page 457
10.2 Other Methods......Page 462
10.3 Some Nonlinear Problems......Page 465
10.4 Software......Page 483
10.5 Program Verification......Page 484
10.6 Summary and Exercises......Page 485
Answers to Selected Exercises......Page 487
References......Page 493
Index......Page 499