Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the ε -uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods. The first part of the book explores boundary value problems for elliptic and parabolic reaction-diffusion and convection-diffusion equations in n -dimensional domains with smooth and piecewise-smooth boundaries. The authors develop a technique for constructing and justifying ε uniformly convergent difference schemes for boundary value problems with fewer restrictions on the problem data.
Containing information published mainly in the last four years, the second section focuses on problems with boundary layers and additional singularities generated by nonsmooth data, unboundedness of the domain, and the perturbation vector parameter. This part also studies both the solution and its derivatives with errors that are independent of the perturbation parameters.
Co-authored by the creator of the Shishkin mesh, this book presents a systematic, detailed development of approaches to construct ε uniformly convergent finite difference schemes for broad classes of singularly perturbed boundary value problems.
Author(s): Grigory I. Shishkin, Lidia P. Shishkina
Series: Monographs and Surveys in Pure and Applied Mathematics
Publisher: Chapman & Hall
Year: 2008
Language: English
Pages: 392
Cover Page......Page 1
Title Page......Page 2
DIFFERENCE METHODS FOR SINGULAR PERTURBATION PROBLEMS......Page 4
Dedication......Page 6
Contents......Page 7
Preface......Page 12
References......Page 0
Part I: Grid approximations of singular perturbation partial differential equations......Page 15
1.1 The development of numerical methods for singularly perturbed problems......Page 16
1.2 Theoretical problems in the construction of difference schemes......Page 19
1.3 The main principles in the construction of special schemes......Page 21
1.4 Modern trends in the development of special difference schemes......Page 23
1.5 The contents of the present book......Page 24
1.6 The present book......Page 25
1.7 The audience for this book......Page 29
2.1 Problem formulation. The aim of the research......Page 30
2.2 Estimates of solutions and derivatives......Page 32
2.3.1 Suffcient conditions for epsilon-uniform convergence of difference schemes......Page 39
2.3.2 Suffcient conditions for epsilon-uniform approximation of the boundary value problem......Page 42
2.3.3 Necessary conditions for distribution of mesh points for epsilon-uniform convergence of difference schemes. Construction of condensing meshes......Page 46
2.4.1 Problems on uniform meshes......Page 51
2.4.2 Problems on piecewise-uniform meshes......Page 57
2.4.3 Consistent grids on subdomains......Page 64
2.4.4 epsilon-uniformly convergent difference schemes......Page 70
2.5.1 A domain-decomposition-based difference scheme for the boundary value problem on a slab......Page 71
2.5.2 A difference scheme for the boundary value problem in a domain with curvilinear boundary......Page 80
3.1 Problem formulation. The aim of the research......Page 87
3.2 Estimates of solutions and derivatives......Page 88
3.3 Suffcient conditions for epsilon-uniform convergence of a difference scheme for the problem on a parallelepiped......Page 97
3.4 A difference scheme for the boundary value problem on a parallelepiped......Page 101
3.5 Consistent grids on subdomains......Page 109
3.6 A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary......Page 114
4.1 Monotonicity of continual and discrete Schwartz methods......Page 120
4.2 Approximation of the solution in a bounded subdomain for the problem on a strip......Page 123
4.3 Difference schemes of improved accuracy for the problem on a slab......Page 131
4.4 Domain-decomposition method for improved iterative schemes......Page 136
5.1 Problem formulation......Page 144
5.2 Estimates of solutions and derivatives......Page 145
5.3 epsilon-uniformly convergent difference schemes......Page 156
5.3.1 Grid approximations of the boundary value problem......Page 157
5.3.2 Consistent grids on a slab......Page 158
5.3.3 Consistent grids on a parallelepiped......Page 165
5.4.1 The problem on a slab......Page 169
5.4.2 The problem on a parallelepiped......Page 172
6.1 Problem formulation......Page 175
6.2.1 The problem solution on a slab......Page 176
6.2.2 The problem on a parallelepiped......Page 179
6.3 On construction of epsilon-uniformly convergent difference schemes under their monotonicity condition......Page 186
6.3.1 Analysis of necessary conditions for epsilon-uniform convergence of difference schemes......Page 187
6.3.2 The problem on a slab......Page 190
6.3.3 The problem on a parallelepiped......Page 193
6.4 Monotone epsilon-uniformly convergent difference schemes......Page 195
7.1 Problem formulation......Page 200
7.2 Estimates of the problem solution on a slab......Page 201
7.3 Estimates of the problem solution on a parallelepiped......Page 208
7.4 Necessary conditions for epsilon-uniform convergence of difference schemes......Page 215
7.5 Sufficient conditions for epsilon-uniform convergence of monotone difference schemes......Page 219
7.6 Monotone epsilon-uniformly convergent difference schemes......Page 222
Part II: Advanced trends in epsilon-uniformly convergent difference methods......Page 227
8.1 Introduction......Page 228
8.2 Problem formulation. The aim of the research......Page 229
8.3 A priori estimates......Page 231
8.4 Grid approximations of the initial-boundary value problem (8.2), (8.1)......Page 237
9.1 Introduction......Page 241
9.2.1 Problem (9.2), (9.1)......Page 243
9.2.2 Some definitions......Page 244
9.2.3 The aim of the research......Page 246
9.3 A priori estimates......Page 247
9.4 Classical finite difference schemes......Page 249
9.5 Construction of epsilon-uniform and almost epsilon-uniform approximations to solutions of Problem (9.2), (9.1)......Page 252
9.6 Difference scheme on a grid adapted in the moving boundary layer......Page 257
9.7 Remarks and generalizations......Page 260
10.1 Introduction......Page 264
10.2.1 Problem with sufficiently smooth data......Page 266
10.2.2 A finite difference scheme on an arbitrary grid......Page 267
10.2.4 Special epsilon-uniform convergent finite difference scheme......Page 268
10.2.5 The aim of the research......Page 269
10.3 A priori estimates for problem with sufficiently smooth data......Page 270
10.4 The defect correction method......Page 271
10.5 The Richardson extrapolation scheme......Page 275
10.6 Asymptotic constructs......Page 278
10.7 A scheme with improved convergence for finite values of epsilon......Page 280
10.8 Schemes based on asymptotic constructs......Page 282
10.9.1 Problem (10.56) with piecewise-smooth initial data......Page 285
10.9.2 The aim of the research......Page 286
10.10 A priori estimates for the boundary value problem (10.56) with piecewise-smooth initial data......Page 287
10.11 Classical finite difference approximations......Page 290
10.12 Improved finite difference scheme......Page 292
11.1 Introduction......Page 294
11.2 Problem formulation. The aim of the research......Page 295
11.3 Grid approximations on locally refined grids that are uniform in subdomains......Page 298
11.4 Difference scheme on a priori adapted grid......Page 302
11.5 Convergence of the difference scheme on a priori adapted grid......Page 308
11.6 Appendix......Page 312
12.1 Introduction......Page 314
12.2 Conditioning of matrices to difference schemes on piecewise-uniform and uniform meshes. Model problem for ODE......Page 316
12.3 Conditioning of difference schemes on uniform and piecewise-uniform grids for the model problem......Page 321
12.4 On conditioning of difference schemes and their matrices for a parabolic problem......Page 328
13.1 Introduction......Page 331
13.2 Problem formulation. The aim of the research......Page 332
13.3 Compatibility conditions. Some a priori estimates......Page 334
13.4 Derivation of a priori estimates for the problem (13:2) under the condition (13:5)......Page 337
13.5 A priori estimates for the problem (13.2) under the conditions (13.4), (13.6)......Page 345
13.6 The classical finite difference scheme......Page 347
13.7 The special finite difference scheme......Page 349
13.8 Generalizations......Page 352
14.1 Application of special numerical methods to mathematical modeling problems......Page 353
14.2 Numerical methods for problems with piecewise-smooth and nonsmooth boundary functions......Page 355
14.3 On the approximation of solutions and derivatives......Page 356
14.4 On difference schemes on adaptive meshes......Page 358
14.5.1 Problem formulation in an unbounded domain. The task of computing the solution in a bounded domain......Page 361
14.5.2 Domain of essential dependence for solutions of the boundary value problem......Page 363
14.5.3 Generalizations......Page 367
14.6 Compatibility conditions for a boundary value problem on a rectangle for an elliptic convection-diffusion equation with a perturbation vector parameter......Page 368
14.6.1 Problem formulation......Page 369
14.6.2 Compatibility conditions......Page 370
References......Page 375
