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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A uniformly convergent method for a singularly perturbed semilinear reaction--diffusion problem with multiple solutions


Authors: Guangfu Sun and Martin Stynes
Journal: Math. Comp. 65 (1996), 1085-1109
MSC (1991): Primary 34E15, 65L10, 65L12, 65L50
MathSciNet review: 1351205
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Abstract: This paper considers a simple central difference scheme for a singularly perturbed semilinear reaction--diffusion problem, which may have multiple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains $O(N)$ points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order $N^{-2}\ln ^2 N$, in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter $\varepsilon $ and $N$. Numerical results are presented that verify this rate of convergence.


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Additional Information

Guangfu Sun
Affiliation: Department of Mathematics, University College, Cork, Ireland

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

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00753-3
PII: S 0025-5718(96)00753-3
Received by editor(s): December 16, 1993
Received by editor(s) in revised form: April 3, 1995
Article copyright: © Copyright 1996 American Mathematical Society