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Analysis of the Streamline-Diffusion Finite Element Method on a Shishkin Mesh for a Convection-Diffusion Problem with Exponential Layers

by Stynes, Martin, Tobiska, Lutz.

Series: 1998-17, Preprints

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

On the unit square, we consider a singularly perturbed convection-diffusion
boundary value problem whose solution has two exponential boundary layers.
We apply the streamline-diffusion finite element method with piecewise
bilinear trial functions on a Shishkin mesh of O(N^2) points and show that
it is convergent, uniformly in the diffusion parameter $\varepsilon$,
of order $\varepsilon^{1/2} N^{-1}\ln^{3/2} N + N^{-3/2}$ in the usual
streamline-diffusion norm. As a corollary we prove that the method is
convergent of order $\varepsilon^{1/2} N^{-1/2}\ln^{2} N + N^{-1/2}\ln^{3/2}N$
(again uniformly in $\varepsilon$) in the local $L^\infty$ norm on the fine
part of the mesh (i.e., inside the boundary layers). This
local $L^\infty$ estimate within the layers can be improved to order
$\varepsilon^{1/2} N^{-1/2}\ln^{2} N + N^{-1}\ln^{1/2}N$, uniformly in
$\varepsilon$, away from the corner layer.

Streamline diffusion, finite element method, singular perturbation,convection-diffusion, Shishkin mesh