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Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

Author: Zhimin Zhang
Journal: Math. Comp. 72 (2003), 1147-1177
MSC (2000): Primary 65N30, 65N15
Published electronically: February 3, 2003
MathSciNet review: 1972731
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Abstract: In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate $O(N^{-2}\ln^2 N + \epsilon N^{-1.5}\ln N)$ in a discrete $\epsilon$-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter $\epsilon$. Numerical tests indicate that the rate $O(N^{-2}\ln^2 N)$ is sharp for the boundary layer terms. As a by-product, an $\epsilon$-uniform convergence of the same order is obtained for the $L^2$-norm. Furthermore, under the same regularity assumption, an $\epsilon$-uniform convergence of order $N^{-3/2}\ln^{5/2} N + \epsilon N^{-1}\ln^{1/2} N$ in the $L^\infty$ norm is proved for some mesh points in the boundary layer region.

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Additional Information

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Keywords: Convection, diffusion, singularly perturbed, boundary layer, Shishkin mesh, finite element method.
Received by editor(s): July 19, 2000
Received by editor(s) in revised form: December 10, 2001
Published electronically: February 3, 2003
Additional Notes: This research was partially supported by the National Science Foundation grants DMS-0074301, DMS-0079743, and INT-0196139
Article copyright: © Copyright 2003 American Mathematical Society