Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems

1. Front Matter

   Pages 1-10

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2. Introduction

   • Torsten Linß

   Pages 1-4

3. Layer-Adapted Meshes

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   Pages 5-29

4. One Dimensional Problems

   1. Front Matter

      Pages 31-31

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5. One dimensional problems

   1. The Analytical Behaviour of Solutions

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      Pages 33-76

   2. Finite Difference Schemes for Convection-Diffusion Problems

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      Pages 77-149

   3. Finite Element and Finite Volume Methods

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      Pages 151-182

   4. Discretisations of Reaction-Convection-Diffusion Problems

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      Pages 183-231

6. Two Dimensional Problems

   1. Front Matter

      Pages 233-233

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7. Two dimensional problems

   1. The Analytical Behaviour of Solutions

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      Pages 235-246

   2. Reaction-Diffusion Problems

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      Pages 247-256

   3. Convection-Diffusion Problems

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      Pages 257-307

8. Back Matter

   Pages 1-18

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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.

• boundary layers
• convection-diffusion
• numerical analysis
• numerical methods for differential equations
• reaction-diffusion
• ordinary differential equations
• partial differential equations

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)

• Inst. Numerische Mathematik, TU Dresden, Dresden, Germany

  Torsten Linß

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