Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems

ISBN-13: 9783642051333 / Angielski / Miękka / 2009 / 326 str.

Torsten Lina

Opis

This is a book on numerical methods for singular perturbation problems - in part- ular, stationary reaction-convection-diffusion problems exhibiting layer behaviour. More precisely, it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. Numerical methods for singularly perturbed differential equations have been studied since the early 1970s and the research frontier has been constantly - panding since. A comprehensive exposition of the state of the art in the analysis of numerical methods for singular perturbation problems is 141] which was p- lished in 2008. As that monograph covers a big variety of numerical methods, it only contains a rather short introduction to layer-adapted meshes, while the present book is exclusively dedicated to that subject. An early important contribution towards the optimisation of numerical methods by means of special meshes was made by N.S. Bakhvalov 18] in 1969. His paper spawned a lively discussion in the literature with a number of further meshes - ing proposed and applied to various singular perturbation problems. However, in the mid 1980s, this development stalled, but was enlivened again by G.I. Shishkin's proposal of piecewise-equidistant meshes in the early 1990s 121,150]. Because of their very simple structure, they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on c- peting meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue.

Recenzje

From the reviews:

"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. ... 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." (Thomas Sonar, Zentralblatt MATH, Vol. 1202, 2011)

Spis treści

One dimensional problems.- The Analytical Behaviour of Solutions.- Finite Difference Schemes for Convection-Diffusion Problems.- Finite Element and Finite Volume Methods.- Discretisations of Reaction-Convection-Diffusion Problems.- Layer-Adapted Meshes.- Two dimensional problems.- The Analytical Behaviour of Solutions.- Reaction-Diffusion Problems.- Convection-Diffusion Problems.

Blurb

This book on numerical methods for singular perturbation problems - in particular, stationary reaction-convection-diffusion problems exhibiting layer behaviour is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. A classification and a survey of layer-adapted meshes for reaction-convection-diffusion problems are included.

This structured and comprehensive account of current ideas in the numerical analysis for various methods on layer-adapted meshes is addressed to researchers in finite element theory and perturbation problems. Finite differences, finite elements and finite volumes are all covered.