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


Preprint series: 98-17, Preprints

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 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.

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

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