Difference Methods for Singular Perturbation Problems

by Shishkin; Grigory I.

• ISBN13: 9781584884590
• ISBN10: 1584884592
• Format: Hardcover
• Copyright: 2008-09-22
• Publisher: Chapman & Hall/

Summary

  Difference Methods for Singular Perturbation Problemsfocuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the -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 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 uniformly convergent finite difference schemes for broad classes of singularly perturbed boundary value problems.

Table of Contents

Preface	p. xiii
Grid approximations of singular perturbation partial differential equations	p. 1
Introduction	p. 3
The development of numerical methods for singularly perturbed problems	p. 3
Theoretical problems in the construction of difference schemes	p. 6
The main principles in the construction of special schemes	p. 8
Modern trends in the development of special difference schemes	p. 10
The contents of the present book	p. 11
The present book	p. 12
The audience for this book	p. 16
Boundary value problems for elliptic reaction-diffusion equations in domains with smooth boundaries	p. 17
Problem formulation. The aim of the research	p. 17
Estimates of solutions and derivatives	p. 19
Conditions ensuring [epsilon]-uniform convergence of difference schemes for the problem on a slab	p. 26
Sufficient conditions for [epsilon]-uniform convergence of difference schemes	p. 26
Sufficient conditions for [epsilon]-uniform approximation of the boundary value problem	p. 29
Necessary conditions for distribution of mesh points for [epsilon]-uniform convergence of difference schemes. Construction of condensing meshes	p. 33
Monotone finite difference approximations of the boundary value problem on a slab. [epsilon]-uniformly convergent difference schemes	p. 38
Problems on uniform meshes	p. 38
Problems on piecewise-uniform meshes	p. 44
Consistent grids on subdomains	p. 51
[epsilon]-uniformly convergent difference schemes	p. 57
Boundary value problems in domains with curvilinear boundaries	p. 58
A domain-decomposition-based difference scheme for the boundary value problem on a slab	p. 58
A difference scheme for the boundary value problem in a domain with curvilinear boundary	p. 67
Boundary value problems for elliptic reaction-diffusion equations in domains with piecewise-smooth boundaries	p. 75
Problem formulation. The aim of the research	p. 75
Estimates of solutions and derivatives	p. 76
Sufficient conditions for [epsilon]-uniform convergence of a difference scheme for the problem on a parallelpiped	p. 85
A difference scheme for the boundary value problem on a parallelepiped	p. 89
Consistent grids on subdomains	p. 97
A difference scheme for the boundary value problem in a domain with piecewise-uniform boundary	p. 102
Generalizations for elliptic reaction-diffusion equations	p. 109
Monotonicity of continual and discrete Schwartz methods	p. 109
Approximation of the solution in a bounded subdomain for the problem on a strip	p. 112
Difference schemes of improved accuracy for the problem on a slab	p. 120
Domain-decomposition method for improved iterative schemes	p. 125
Parabolic reaction-diffusion equations	p. 133
Problem formulation	p. 133
Estimates of solutions and derivatives	p. 134
[epsilon]-uniformly convergent difference schemes	p. 145
Grid approximations of the boundary value problem	p. 146
Consistent grids on a slab	p. 147
Consistent grids on a parallelepiped	p. 154
Consistent grids on subdomains	p. 158
The problem on a slab	p. 158
The problem on a parallelepiped	p. 161
Elliptic convection-diffusion equations	p. 165
Problem formulation	p. 165
Estimates of solutions and derivatives	p. 166
The problem solution on a slab	p. 166
The problem on a parallelepiped	p. 169
On construction of [epsilon]-uniformly convergent difference schemes under their monotonicity condition	p. 176
Analysis of necessary conditions for [epsilon]-uniform convergence of difference schemes	p. 177
The problem on a slab	p. 180
The problem on a parallelepiped	p. 183
Monotone [epsilon]-uniformly convergent difference schemes	p. 185
Parabolic convection-diffusion equations	p. 191
Problem formulation	p. 191
Estimates of the problem solution on a slab	p. 192
Estimates of the problem solution on a parallelepiped	p. 199
Necessary conditions for [epsilon]-uniform convergence of difference schemes	p. 206
Sufficient conditions for [epsilon]-uniform convergence of monotone difference schemes	p. 210
Monotone [epsilon]-uniformly convergent difference schemes	p. 213
Advanced trends in [epsilon]-uniformly convergent difference methods	p. 219
Grid approximations of parabolic reaction-diffusion equations with three perturbation parameters	p. 221
Introduction	p. 221
Problem formulation. The aim of the research	p. 222
A priori estimates	p. 224
Grid approximations of the initial-boundary value problem	p. 230
Application of widths for construction of difference schemes for problems with moving boundary layers	p. 235
Introduction	p. 235
A boundary value problem for a singularly perturbed parabolic reaction-diffusion equation	p. 237
Problem (9.2), (9.1)	p. 237
Some definitions	p. 238
The aim of the research	p. 240
A priori estimates	p. 241
Classical finite difference schemes	p. 243
Construction of [epsilon]-uniform and almost [epsilon]-uniform approximations to solutions of problem (9.2), (9.1)	p. 246
Difference scheme on a grid adapted in the moving boundary layer	p. 251
Remarks and generalizations	p. 254
High-order accurate numerical methods for singularly perturbed problems	p. 259
Introduction	p. 259
Boundary value problems for singularly perturbed parabolic convection-diffusion equations with sufficiently smooth data	p. 261
Problem with sufficiently smooth data	p. 261
A finite difference scheme on an arbitrary grid	p. 262
Estimates of solutions on uniform grids	p. 263
Special [epsilon]-uniform convergent finite difference scheme	p. 263
The aim of the research	p. 264
A priori estimates for problem with sufficiently smooth data	p. 265
The defect correction method	p. 266
The Richardson extrapolation scheme	p. 270
Asymptotic constructs	p. 273
A scheme with improved convergence for finite values of [epsilon]	p. 275
Schemes based on asymptotic constructs	p. 277
Boundary value problem for singularly perturbed parabolic convection-diffusion equation with piecewise-smooth initial data	p. 280
Problem (10.56) with piecewise-smooth initial data	p. 280
The aim of the research	p. 281
A priori estimates for the boundary value problem (10.56) with piecewise-smooth initial data	p. 282
Classical finite difference approximations	p. 285
Improved finite difference scheme	p. 287
A finite difference scheme on a priori adapted grids for a singularly perturbed parabolic convection-diffusion equation	p. 289
Introduction	p. 289
Problem formulation. The aim of the research	p. 290
Grid approximations on locally refined grids that are uniform in subdomains	p. 293
Difference scheme on a priori adapted grid	p. 297
Convergence of the difference scheme on a priori adapted grid	p. 303
Appendix	p. 307
On conditioning of difference schemes and their matrices for singularly perturbed problems	p. 309
Introduction	p. 309
Conditioning of matrices to difference schemes on piecewise-uniform and uniform meshes. Model problem for ODE	p. 311
Conditioning of difference schemes on uniform and piecewise-uniform grids for the model problem	p. 316
On conditioning of difference schemes and their matrices for a parabolic problem	p. 323
Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters	p. 327
Introduction	p. 327
Problem formulation. The aim of the research	p. 328
Compatibility conditions. Some a priori estimates	p. 330
Derivation of a priori estimates for the problem (13.2) under the condition (13.5)	p. 333
A priori estimates for the problem (13.2) under the conditions (13.4), (13.6)	p. 341
The classical finite difference scheme	p. 343
The special finite difference scheme	p. 345
Generalizations	p. 348
Survey	p. 349
Application of special numerical methods to mathematical modeling problems	p. 349
Numerical methods for problems with piecewise-smooth and nonsmooth boundary functions	p. 351
On the approximation of solutions and derivatives	p. 352
On difference schemes on adaptive meshes	p. 354
On the design of constructive difference schemes for an elliptic convection-diffusion equation in an unbounded domain	p. 357
Problem formulation in an unbounded domain. The task of computing the solution in a bounded domain	p. 357
Domain of essential dependence for solutions of the boundary value problem	p. 359
Generalizations	p. 363
Compatibility-conditions for a boundary value problem on a rectangle for an elliptic convection-diffusion equation with a perturbation vector parameter	p. 364
Problem formulation	p. 365
Compatibility conditions	p. 366
References	p. 371
Index	p. 389
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