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E-raamat: Difference Methods for Singular Perturbation Problems

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