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Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $ L$-shaped domain

Authors: Vladimir B. Andreev and Natalia Kopteva
Journal: Math. Comp. 77 (2008), 2125-2139
MSC (2000): Primary 65N06, 65N15, 65N50; Secondary 35B25
Published electronically: February 19, 2008
MathSciNet review: 2429877
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Abstract: A singularly perturbed reaction-diffusion equation is posed in a two-dimensional $ L$-shaped domain $ \Omega$ subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space $ C^{2/3}(\bar\Omega)$ and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle $ 3\pi/2$. We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

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Additional Information

Vladimir B. Andreev
Affiliation: Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie Gory, 119991, Moscow, Russia

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

Keywords: Reaction-diffusion, singular perturbation, corner singularity, $L$-shaped domain, pointwise error estimate, Shishkin mesh, second order
Received by editor(s): April 27, 2007
Received by editor(s) in revised form: August 31, 2007
Published electronically: February 19, 2008
Additional Notes: This research was supported by Enterprise Ireland International Collaboration Programme grant IC/2006/8.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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