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Shishkin meshes in the numerical solution of singularly perturbed differential equations. (English) Zbl 1197.65094
The paper provides a comprehensive review of G. I. Shishkin’s contributions to the development of numerical methods for singularly perturbed differential equations. This includes, in particular, the development of the so-called Shishkin meshes that allow to handle the effects caused by the boundary layers typically present in the solutions to such equations, and the analysis of finite difference methods based on these non-uniform meshes. Both ordinary and partial differential equations with singular perturbations can be handled in this way.

MSC:
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
34E15 Singular perturbations for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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