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Mathematics of Computation

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Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

Author: Natalia Kopteva
Journal: Math. Comp. 76 (2007), 631-646
MSC (2000): Primary 65N06, 65N15, 65N30; Secondary 35B25
Published electronically: December 27, 2006
MathSciNet review: 2291831
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Abstract: A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter $ \varepsilon^2$ is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in $ \varepsilon$ for $ \varepsilon\le Ch$. Here $ h>0$ is the maximum side length of mesh elements, while the number of mesh nodes does not exceed $ Ch^{-2}$. Numerical experiments are performed to support the theoretical results.

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Additional Information

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

Keywords: Semilinear reaction-diffusion, singular perturbation, maximum norm error estimate, $Z$-field, Bakhvalov mesh, Shishkin mesh, second order
Received by editor(s): October 8, 2005
Received by editor(s) in revised form: February 23, 2006
Published electronically: December 27, 2006
Additional Notes: This publication has emanated from research conducted with the financial support of Science Foundation Ireland under the Basic Research Grant Programme 2004; Grant 04/BR/M0055.
Dedicated: Dedicated to Professor V. B. Andreev on the occasion of his 65th birthday
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.