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A parameter robust numerical method for a two dimensional reaction-diffusion problem


Authors: C. Clavero, J. L. Gracia and E. O'Riordan
Journal: Math. Comp. 74 (2005), 1743-1758
MSC (2000): Primary 65N06, 65N12, 65N15; Secondary 35J25
DOI: https://doi.org/10.1090/S0025-5718-05-01762-X
Published electronically: June 7, 2005
MathSciNet review: 2164094
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Abstract: In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.


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Additional Information

C. Clavero
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Zaragoza, Spain
Email: clavero@unizar.es

J. L. Gracia
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Teruel, Spain
Email: jlgracia@unizar.es

E. O'Riordan
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Email: eugene.oriordan@dcu.ie

DOI: https://doi.org/10.1090/S0025-5718-05-01762-X
Keywords: Reaction-diffusion, uniform convergence, Shihskin mesh, second order
Received by editor(s): May 19, 2004
Published electronically: June 7, 2005
Additional Notes: This research was partially supported by the Diputación General de Aragón and the project MCYT/FEDER BFM2001–2521
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society