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Mathematics of Computation

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A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions


Authors: R. Bruce Kellogg, Torsten Linss and Martin Stynes
Journal: Math. Comp. 77 (2008), 2085-2096
MSC (2000): Primary 65N06, 65N15, 65N50; Secondary 35J45
DOI: https://doi.org/10.1090/S0025-5718-08-02125-X
Published electronically: March 14, 2008
MathSciNet review: 2429875
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Abstract: An elliptic system of $ M (\ge 2)$ singularly perturbed linear reaction-diffusion equations, coupled through their zero-order terms, is considered on the unit square. This system does not in general satisfy a maximum principle. It is solved numerically using a standard difference scheme on tensor-product Bakhvalov and Shishkin meshes. An error analysis for these numerical methods shows that one obtains nodal $ O(N^{-2})$ convergence on the Bakhvalov mesh and $ O(N^{-2}\ln^2 N)$ convergence on the Shishkin mesh, where $ N$ mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter. The analysis is much simpler than previous analyses of similar problems, even in the case of a single reaction-diffusion equation, as it does not require the construction of an elaborate decomposition of the solution. Numerical results are presented to confirm our theoretical error estimates.


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Additional Information

R. Bruce Kellogg
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: rbmjk@windstream.net

Torsten Linss
Affiliation: Institut für Numerische Mathematik, Technische Univeristät, Dresden, Germany
Email: torsten.linss@tu-dresden.de

Martin Stynes
Affiliation: Department of Mathematics, National University of Ireland, Cork, Ireland
Email: m.stynes@ucc.ie

DOI: https://doi.org/10.1090/S0025-5718-08-02125-X
Received by editor(s): May 24, 2007
Published electronically: March 14, 2008
Additional Notes: The research of the first author was supported by the Boole Centre for Research in Informatics at the National University of Ireland, Cork, and by the Science Foundation Ireland under the Basic Research Grant Programme 2004 (Grants 04/BR/M0055, 04/BR/M0055s1)
The research of the second author was supported by the Boole Centre for Research in Informatics at the National University of Ireland, Cork and by the ZIH at TU Dresden
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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