Difference methods for singular perturbation problems

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 Notes

Shishkin, Grigory I.; Shishkina, Lidia P.

Preface	p. xiii
I Grid approximations of singular perturbation partial differential equations	p. 1
1 Introduction	p. 3
1.1 The development of numerical methods for singularly perturbed problems	p. 3
1.2 Theoretical problems in the construction of difference schemes	p. 6
1.3 The main principles in the construction of special schemes	p. 8
1.4 Modern trends in the development of special difference schemes	p. 10
1.5 The contents of the present book	p. 11
1.6 The present book	p. 12
1.7 The audience for this book	p. 16
2 Boundary value problems for elliptic reaction-diffusion equations in domains with smooth boundaries	p. 17
2.1 Problem formulation. The aim of the research	p. 17
2.2 Estimates of solutions and derivatives	p. 19
2.3 Conditions ensuring [epsilon]-uniform convergence of difference schemes for the problem on a slab	p. 26
2.3.1 Sufficient conditions for [epsilon]-uniform convergence of difference schemes	p. 26
2.3.2 Sufficient conditions for [epsilon]-uniform approximation of the boundary value problem	p. 29
2.3.3 Necessary conditions for distribution of mesh points for [epsilon]-uniform convergence of difference schemes. Construction of condensing meshes	p. 33
2.4 Monotone finite difference approximations of the boundary value problem on a slab. [epsilon]-uniformly convergent difference schemes	p. 38
2.4.1 Problems on uniform meshes	p. 38
2.4.2 Problems on piecewise-uniform meshes	p. 44
2.4.3 Consistent grids on subdomains	p. 51
2.4.4 [epsilon]-uniformly convergent difference schemes	p. 57
2.5 Boundary value problems in domains with curvilinear boundaries	p. 58
2.5.1 A domain-decomposition-based difference scheme for the boundary value problem on a slab	p. 58
2.5.2 A difference scheme for the boundary value problem in a domain with curvilinear boundary	p. 67
3 Boundary value problems for elliptic reaction-diffusion equations in domains with piecewise-smooth boundaries	p. 75
3.1 Problem formulation. The aim of the research	p. 75
3.2 Estimates of solutions and derivatives	p. 76
3.3 Sufficient conditions for [epsilon]-uniform convergence of a difference scheme for the problem on a parallelpiped	p. 85
3.4 A difference scheme for the boundary value problem on a parallelepiped	p. 89
3.5 Consistent grids on subdomains	p. 97
3.6 A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary	p. 102
4 Generalizations for elliptic reaction-diffusion equations	p. 109
4.1 Monotonicity of continual and discrete Schwartz methods	p. 109
4.2 Approximation of the solution in a bounded subdomain for the problem on a strip	p. 112
4.3 Difference schemes of improved accuracy for the problem on a slab	p. 120
4.4 Domain-decomposition method for improved iterative schemes	p. 125
5 Parabolic reaction-diffusion equations	p. 133
5.1 Problem formulation	p. 133
5.2 Estimates of solutions and derivatives	p. 134
5.3 [epsilon]-uniformly convergent difference schemes	p. 145
5.3.1 Grid approximations of the boundary value problem	p. 146
5.3.2 Consistent grids on a slab	p. 147
5.3.3 Consistent grids on a parallelepiped	p. 154
5.4 Consistent grids on subdomains	p. 158
5.4.1 The problem on a slab	p. 158
5.4.2 The problem on a parallelepiped	p. 161
6 Elliptic convection-diffusion equations	p. 165
6.1 Problem formulation	p. 165
6.2 Estimates of solutions and derivatives	p. 166
6.2.1 The problem solution on a slab	p. 166
6.2.2 The problem on a parallelepiped	p. 169
6.3 On construction of [epsilon]-uniformly convergent difference schemes under their monotonicity condition	p. 176
6.3.1 Analysis of necessary conditions for [epsilon]-uniform convergence of difference schemes	p. 177
6.3.2 The problem on a slab	p. 180
6.3.3 The problem on a parallelepiped	p. 183
6.4 Monotone [epsilon]-uniformly convergent difference schemes	p. 185
7 Parabolic convection-diffusion equations	p. 191
7.1 Problem formulation	p. 191
7.2 Estimates of the problem solution on a slab	p. 192
7.3 Estimates of the problem solution on a parallelepiped	p. 199
7.4 Necessary conditions for [epsilon]-uniform convergence of difference schemes	p. 206
7.5 Sufficient conditions for [epsilon]-uniform convergence of monotone difference schemes	p. 210
7.6 Monotone [epsilon]-uniformly convergent difference schemes	p. 213
II Advanced trends in [epsilon]-uniformly convergent difference methods	p. 219
8 Grid approximations of parabolic reaction-diffusion equations with three perturbation parameters	p. 221
8.1 Introduction	p. 221
8.2 Problem formulation. The aim of the research	p. 222
8.3 A priori estimates	p. 224
8.4 Grid approximations of the initial-boundary value problem	p. 230
9 Application of widths for construction of difference schemes for problems with moving boundary layers	p. 235
9.1 Introduction	p. 235
9.2 A boundary value problem for a singularly perturbed parabolic reaction-diffusion equation	p. 237
9.2.1 Problem (9.2), (9.1)	p. 237
9.2.2 Some definitions	p. 238
9.2.3 The aim of the research	p. 240
9.3 A priori estimates	p. 241
9.4 Classical finite difference schemes	p. 243
9.5 Construction of [epsilon]-uniform and almost [epsilon]-uniform approximations to solutions of problem (9.2), (9.1)	p. 246
9.6 Difference scheme on a grid adapted in the moving boundary layer	p. 251
9.7 Remarks and generalizations	p. 254
10 High-order accurate numerical methods for singularly perturbed problems	p. 259
10.1 Introduction	p. 259
10.2 Boundary value problems for singularly perturbed parabolic convection-diffusion equations with sufficiently smooth data	p. 261
10.2.1 Problem with sufficiently smooth data	p. 261
10.2.2 A finite difference scheme on an arbitrary grid	p. 262
10.2.3 Estimates of solutions on uniform grids	p. 263
10.2.4 Special [epsilon]-uniform convergent finite difference scheme	p. 263
10.2.5 The aim of the research	p. 264
10.3 A priori estimates for problem with sufficiently smooth data	p. 265
10.4 The defect correction method	p. 266
10.5 The Richardson extrapolation scheme	p. 270
10.6 Asymptotic constructs	p. 273
10.7 A scheme with improved convergence for finite values of [epsilon]	p. 275
10.8 Schemes based on asymptotic constructs	p. 277
10.9 Boundary value problem for singularly perturbed parabolic convection-diffusion equation with piecewise-smooth initial data	p. 280
10.9.1 Problem (10.56) with piecewise-smooth initial data	p. 280
10.9.2 The aim of the research	p. 281
10.10 A priori estimates for the boundary value problem (10.56) with piecewise-smooth initial data	p. 282
10.11 Classical finite difference approximations	p. 285
10.12 Improved finite difference scheme	p. 287
11 A finite difference scheme on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation	p. 289
11.1 Introduction	p. 289
11.2 Problem formulation. The aim of the research	p. 290
11.3 Grid approximations on locally refined grids that are uniform in subdomains	p. 293
11.4 Difference scheme on a priori adapted grid	p. 297
11.5 Convergence of the difference scheme on a priori adapted grid	p. 303
11.6 Appendix	p. 307
12 On conditioning of difference schemes and their matrices for singularly perturbed problems	p. 309
12.1 Introduction	p. 309
12.2 Conditioning of matrices to difference schemes on piecewise-uniform and uniform meshes. Model problem for ODE	p. 311
12.3 Conditioning of difference schemes on uniform and piecewise-uniform grids for the model problem	p. 316
12.4 On conditioning of difference schemes and their matrices for a parabolic problem	p. 323
13 Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters	p. 327
13.1 Introduction	p. 327
13.2 Problem formulation. The aim of the research	p. 328
13.3 Compatibility conditions. Some a priori estimates	p. 330
13.4 Derivation of a priori estimates for the problem (13.2) under the condition (13.5)	p. 333
13.5 A priori estimates for the problem (13.2) under the conditions (13.4), (13.6)	p. 341
13.6 The classical finite difference scheme	p. 343
13.7 The special finite difference scheme	p. 345
13.8 Generalizations	p. 348
14 Survey	p. 349
14.1 Application of special numerical methods to mathematical modeling problems	p. 349
14.2 Numerical methods for problems with piecewise-smooth and nonsmooth boundary functions	p. 351
14.3 On the approximation of solutions and derivatives	p. 352
14.4 On difference schemes on adaptive meshes	p. 354
14.5 On the design of constructive difference schemes for an elliptic convection-diffusion equation in an unbounded domain	p. 357
14.5.1 Problem formulation in an unbounded domain. The task of computing the solution in a bounded domain	p. 357
14.5.2 Domain of essential dependence for solutions of the boundary value problem	p. 359
14.5.3 Generalizations	p. 363
14.6 Compatibility-conditions for a boundary value problem on a rectangle for an elliptic convection-diffusion equation with a perturbation vector parameter	p. 364
14.6.1 Problem formulation	p. 365
14.6.2 Compatibility conditions	p. 366
References	p. 371
Index	p. 389

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