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On the non-existence of $\varepsilon$-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems

Authors: Paul A. Farrell, John J. H. Miller, Eugene O’Riordan and Grigorii I. Shishkin
Journal: Math. Comp. 67 (1998), 603-617
MSC (1991): Primary 34B15, 65L12; Secondary 34L30, 65L10
MathSciNet review: 1451321
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Abstract: In this paper fitted finite difference methods on a uniform mesh with internodal spacing $h$, are considered for a singularly perturbed semilinear two-point boundary value problem. It is proved that a scheme of this type with a frozen fitting factor cannot converge $\varepsilon$-uniformly in the maximum norm to the solution of the differential equation as the mesh spacing $h$ goes to zero. Numerical experiments are presented which show that the same result is true for a number of schemes with variable fitting factors.

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Additional Information

Paul A. Farrell
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

John J. H. Miller
Affiliation: Department of Mathematics, Trinity College, Dublin 2, Ireland

Eugene O’Riordan
Affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

Grigorii I. Shishkin
Affiliation: Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

Keywords: Semilinear boundary value problem, singular perturbation, finite difference scheme, $\varepsilon$-uniform convergence, uniform mesh, frozen fitting factor
Received by editor(s): July 3, 1995
Received by editor(s) in revised form: February 9, 1996
Additional Notes: Supported in part under NSF grant DMS-9627244.
The first author was supported in part by the Research Council of Kent State University.
The fourth author was supported in part by the Russian Foundation for Basic Research under Grant N 95-01-00039.
Article copyright: © Copyright 1998 American Mathematical Society