Layer-adapted meshes for reaction-convection-diffusion problems.

*(English)*Zbl 1202.65120
Lecture Notes in Mathematics 1985. Berlin: Springer (ISBN 978-3-642-05133-3/pbk; 978-3-642-05134-0/ebook). xi, 320 p. (2010).

The book gives a thorough analysis of layer-adapted meshes used in the numerical treatment of reaction-convection-diffusion equations and hence is a book on numerical methods for singular perturbation problems.

After introducing layer-adapted meshes the first part of the book opens with one dimensional problems. After discussing the analytical behaviour of solutions of different types of equations finite difference methods are discussed in some detail. A shorter chapter on finite volume and finite element methods follows and the first part is concluded by the description of some particular discretisations.

In the second part, two dimensional problems are considered where after a discussion of the analytical properties of solutions reaction-diffusion as well as convection-diffusion problems are treated. The book is a welcome enrichement of the classical literature in that it shows clearly what can be done on meshes which are not of Shishkin-type.

After introducing layer-adapted meshes the first part of the book opens with one dimensional problems. After discussing the analytical behaviour of solutions of different types of equations finite difference methods are discussed in some detail. A shorter chapter on finite volume and finite element methods follows and the first part is concluded by the description of some particular discretisations.

In the second part, two dimensional problems are considered where after a discussion of the analytical properties of solutions reaction-diffusion as well as convection-diffusion problems are treated. The book is a welcome enrichement of the classical literature in that it shows clearly what can be done on meshes which are not of Shishkin-type.

Reviewer: Thomas Sonar (Braunschweig)

##### MSC:

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |