Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem
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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.References
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Additional Information
- Natalia Kopteva
- Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
- MR Author ID: 610720
- ORCID: 0000-0001-7477-6926
- Email: natalia.kopteva@ul.ie
- 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.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 631-646
- MSC (2000): Primary 65N06, 65N15, 65N30; Secondary 35B25
- DOI: https://doi.org/10.1090/S0025-5718-06-01938-7
- MathSciNet review: 2291831
Dedicated: Dedicated to Professor V. B. Andreev on the occasion of his 65th birthday