Shishkin meshes in the numerical solution of singularly perturbed differential equations.

*(English)*Zbl 1197.65094The 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.

Reviewer: Kai Diethelm (Braunschweig)

##### 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 |