98-22

Convergence Properties of the Streamline-Diffusion Finite Element Method on a Shishkin Mesh for Singularly Perturbed Elliptic Equations with Exponential Layers

by Tobiska, L.; Matthies, G.; Stynes, M.

 

Preprint series: 98-22, Preprints

The paper is published: Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, L.G. Vulkov, J.J.H. Miller, G.I. Shishkin eds., Nova Science Publishers, Inc. N.Y., 2000

MSC:
65N15 Error bounds
65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

 

Abstract: On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has exponential boundary layers along two sides of the square. We use the streamline-diffusion finite element method (SDFEM) with piecewise bilinear trial functions on a Shishkin mesh and state a recent result showing that the solution of the SDFEM converges, uniformly in the diffusion parameter, to its bilinear interpolant in the usual streamline-diffusion norm. As a corollary, the method converges pointwise on the fine part of the mesh (i.e., inside the boundary layers). We present numerical results to support these results and to examine the effect of replacing bilinear trials with linear trials in the SDFEM.

Keywords: Streamline-diffusion, Shishkin-mesh, singularly perturbed problem, finite elements.


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