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1998-22

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..


Series: 1998-22, Preprints

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.

This paper was published in:
Analytical and Numerical