nach oben

2010 | Buch

Kapitel lesen Erstes Kapitel lesen

Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems

verfasst von: Torsten Linß

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Stationary linear reaction-convection-diffusion problems form the subject of this monograph:

$$ - \in u^ - bu^{'} + cu = f\text{ in (0,1), }u(0) = \gamma _0 ,\text{ }u(1) = \gamma _1 $$

and its two-dimensional analogue

$$ - \in \Delta u - b \cdot \nabla u + cu = f\text{ in }\Omega \subset \text{ }IR^2 \text{, }u|\partial \Omega = g$$

with a small positive parameter ε.

Such problems arise in various models of fluid flow [52,53,73]; they appear in the (linearised) Navier-Stokes and in the Oseen equations, in the equations modelling oil extraction from underground reservoirs [32], flows in chemical reactors [3] and convective heat transport with large Péclet number [56]. Other applications include the simulation of semiconductor devices [130].

Torsten Linß

Chapter 2. Layer-Adapted Meshes

Abstract

Before surveying a few of the most important ideas from the literature for constructing layer-adapted meshes, we shall introduce some basic concepts for describing layer-adapted meshes.

Throughout $\varpi :0\; = \;x_0 < x_1 < ... < x_N = 1$ denotes a generic mesh with N subintervals on [0, 1], while ω is the set of inner mesh nodes. Set $I_i : = \;[x_i - 1,x_i ].$. The local mesh sizes are $h_i : = x_i - x_{i - 1} ,i = 1,...,N$, while the maximum step size is $h: = \mathop {\max }\limits_{i = 1,...,N} hi.$.

Torsten Linß

One dimensional problems

Chapter 3. The Analytical Behaviour of Solutions

Abstract

In this chapter, we gather a number of analytical properties for singularly perturbed boundary-value problems for second-order ordinary differential equations of the general type

$$ - \in u^ - bu^{'} + cu = f\text{ in }(0,1),u(0) = \gamma _0 ,\text{ }u(1) = \gamma _1 ,$$

with a small positive parameter ε and functions b, c, f : [0, 1] → IR, and of its vector-valued counterpart

$$ - Eu^ - Bu^{'} + Au = f\text{ in }(0,1),u(0) = \gamma _0 ,\text{ }u(1) = \gamma _1 ,$$

with $ E = \text{diag}(\in ),\in = (\in _1 ,....,\in _\ell )^T $ and small positive constants $\in _i ,i = 1,...,\ell ,$ with matrix-valued functions A, B : $ [0,1] \to IR^{\ell ,\ell } ,$, and vector-valued functions $ f,u:[0,1] \to IR^\ell $.

Torsten Linß

Chapter 4. Finite Difference Schemes for Convection-Diffusion Problems

Abstract

This chapter is concerned with finite-difference discretisations of the stationary linear convection-diffusion problem

$$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1),\;\;u(0) = \gamma _0 ,\;u(1) = \gamma _1 ,$$

with b ≥ β > 0 on [0, 1]. For the sake of simplicity we shall assume that

$$c \ge 0\,\,{\rm and}\;\;{\rm b'} \ge 0\;\;\;\;\;{\rm on[0,1]}{\rm .}$$

Using (4.1) as a model problem, a general convergence theory for certain firstand second-order upwinded difference schemes on arbitrary and on layer-adapted meshes is derived. The close relationship between the differential operator and its upwinded discretisations is highlighted.

Torsten Linß

Chapter 5. Finite Element and Finite Volume Methods

Abstract

In this chapter we consider finite element and finite volume discretisations of

$$Lu\,: = - \varepsilon u'' - bu' + cu = f\,\,{\rm in}\,(0,1), \,\,\ u(0) = u (1) = 0,$$

with b ≥ β > 0. Its associated variational formulation is: Find $u \in H_0^1 (0,1)$ such that

$$a(u, v) = f(v)\,\,\, {\rm for\, all}\,\, v \in H_0^1 (0,1),$$

(5.2)

where

$$a(u,v): = \;\varepsilon (u',v') - (bu',v) + (cu,v)$$

and

$$f(v):=(f,v):= \int_0^1 {(fv)(x)dx.}$$

(5.3)

Throughout assume that

$$c + b' / 2 \ge \gamma > 0.$$

(5.4)

This condition guaranties the coercivity of the bilinear form in (5.2):

$$\||v |\|_\varepsilon^2 := _\varepsilon \|v' \|_0^2 + \gamma \|v \|^2_0 \le a(u,v)\,\,\,\, {\rm for\, all}\,\,\,\, v \in H^1_0 (0, 1).$$

This is verified using standard arguments, see e.g. [141]. If b ≥ β > 0 then (5.4) can always be ensured by a transformation $\bar u(x) = u(x)e^{\delta x}$ with δ chosen appropriately. We assume this transformation has been carried out.

Torsten Linß

Chapter 6. Discretisations of Reaction-Convection-Diffusion Problems

Abstract

This chapter is concerned with discretisations of the stationary linear reaction- 4 convection-diffusion problem

$$ - \varepsilon _d u^ - \varepsilon _c bu + cu = f\text{ in (0,1), }u(0) = \gamma _0 ,u(1) = \gamma _1 ,$$

with b ≥ 1 and c ≥ 1 on [0, 1].

In particular, we shall study the special case of scalar reaction-diffusion problems

$$ - \varepsilon _d u^ - \varepsilon _c bu + cu = f\text{ in (0,1), }u(0) = \gamma _0 ,u(1) = \gamma _1 ,$$

and its vector-valued counterpart

$$ - E^2 u'' + Au = f\;\;{\rm in}\;{\rm (0,1),}\;\;\;\;u(0) = \gamma _0 ,\;\;u(1) = \gamma _1 .$$

Torsten Linß

Two dimensional problems

Chapter 7. The Analytical Behaviour of Solutions

Abstract

In this chapter we gather a number of analytical properties for singularly perturbed elliptic boundary-value problems of the general type

$$ - \varepsilon \Delta u - b\nabla u + cu = f\text{ in }\Omega \subset \text{ }IR^2 ,u\backslash _{\partial \Omega } = g, $$

with a small positive parameter ε, the convection field b = (b ₁, b ₂)^T and functions b ₁, b ₂, c, f :

$$\bar \Omega \to IR\text{, and }g:\partial \Omega \to IR$$

Torsten Linß

Chapter 8. Reaction-Diffusion Problems

Abstract

In this chapter numerical methods for singularly perturbed reaction-diffusion equations on the square are studied. Find $ u \in C^2 (\Omega ) \cup C(\bar \Omega ) $ such that

$$ Lu: = - \varepsilon ^2 \Delta u + cu = f\text{ in }\Omega \text{ = (0,1)}^\text{2} ,\text{ }u|\partial \Omega = g, $$

(8.1)

where the parameter satisfies $ 0 < \varepsilon < < \text{ }1\text{ and }c \geqslant \gamma > 0\text{ on }\Omega $.

Torsten Linß

Chapter 9. Convection-Diffusion Problems

Abstract

This chapter is devoted to numerical methods for the convection-diffusion problem

$$- \varepsilon \Delta u - b\nabla u + cu = f\;in\;\Omega = (0,1)^2 ,\;u|_{\partial \Omega } = 0,$$

(9.1)

with b ₁ ≥ β₁ > 0, b ₂ ≥ β₂ > 0 on [0,1]², i.e., problems with regular boundary layers at the outflow boundary x = 0 and y = 0. The analytical behaviour of the solution of (9.1) was studied in Sect. 7.3.1.

Results for problems with characteristic layers will only be mentioned briefly.

Torsten Linß

Backmatter

Titel: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems
verfasst von: Torsten Linß
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-05134-0
Print ISBN: 978-3-642-05133-3
DOI: https://doi.org/10.1007/978-3-642-05134-0

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. Layer-Adapted Meshes

One dimensional problems

Chapter 3. The Analytical Behaviour of Solutions

Chapter 4. Finite Difference Schemes for Convection-Diffusion Problems

Chapter 5. Finite Element and Finite Volume Methods

Chapter 6. Discretisations of Reaction-Convection-Diffusion Problems

Two dimensional problems

Chapter 7. The Analytical Behaviour of Solutions

Chapter 8. Reaction-Diffusion Problems

Chapter 9. Convection-Diffusion Problems

Backmatter

Premium Partner