Torsten Linß

Layer-adapted meshes for convection-diffusion problems

Abstract In the present paper, a classification and a survey are given of layer-adapted meshes for stationary convection-dominated convection–diffusion problems. For a number of standard numerical schemes, theoretical results are presented that demonstrate that the use of properly layer-adapted meshes yields robust methods, i.e., methods that perform equally well no matter how dominant the convection. We review a number of techniques used in the convergence analysis of these methods.

Numerical methods on Shishkin meshes for linear convection–diffusion problems

Abstract On the unit square, we consider a model singularly perturbed convection–diffusion problem whose solution contains exponential boundary and corner layers. Shishkin meshes are frequently used to solve such problems numerically. We compare and evaluate the performance of several numerical methods on these meshes and summarise the theoretical convergence results available in the literature.

Accurate Solution of a System of Coupled Singularly Perturbed Reaction-diffusion Equations

We study a system of coupled reaction-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them. Near each boundary, their solution exhibit two overlapping layers. A central difference scheme on layer-adapted piecewise uniform meshes is used to solve the system numerically. We show that the scheme is almost second-order convergent, uniformly in both perturbation parameters, thus improving previous results [5]. We present the results of numerical experiments to confirm our theoretical results.

An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem

Abstract We consider an upwind finite difference scheme on a novel layer-adapted mesh (a modification of Shishkins piecewise uniform mesh) for a model singularly perturbed convection–diffusion problem in two dimensions. We prove that the upwind scheme on the modified Shishkin mesh is first-order convergent in the discrete L∞ norm, independently of the diffusion parameter e, provided only that the perturbation parameter satisfies e⩽N−1, where O (N 2 ) mesh points are used. The new mesh yields more accurate results than simple upwinding on a standard Shishkin mesh, even though it requires essentially the same computational effort. Numerical experiments support these theoretical results.

The necessity of Shishkin decompositions

In the present paper, we study model singularly perturbed convection-diffusion problems with exponential boundary layers. It has been believed for some time that only a complete splitting of the exact solution into regular and layer parts provides the information necessary for the study of the uniform convergence properties of numerical methods for these problems on layer-adapted grids (such as Shishkin meshes). In the present paper, we give new proofs of uniform interpolation error estimates for linear and bilinear interpolation; these proofs are based on the older a priori bounds derived by Kellogg and Tsan [1].

Sufficient conditions for uniform convergence on layer-adapted grids

Abstract We study convergence properties of two upwind difference schemes for the solution of quasilinear convection–diffusion problems on layer-adapted grids. We derive conditions that are sufficient for convergence, uniformly in the perturbation parameter, of the methods. These conditions are easy to check and enable one to immediately deduce the rate of convergence. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

The sdfem on Shishkin meshes for linear convection-diffusion problems

Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local

Defect correction on Shishkin-type meshes

L_\infty

Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations

error estimates that hold true uniformly in the perturbation parameter

Uniform pointwise convergence of finite difference schemes using grid equidistribution

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