On the non-existence of $\epsilon$-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

DOI:
https://doi.org/10.1090/S0025-5718-98-00922-3

MathSciNet review:
1451321

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Email:
farrell@mcs.kent.edu

**John J. H. Miller**

Affiliation:
Department of Mathematics, Trinity College, Dublin 2, Ireland

Email:
jmiller@tcd.ie

**Eugene O’Riordan**

Affiliation:
School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

Email:
oriordae@ccmail.dcu.ie

**Grigorii I. Shishkin**

Affiliation:
Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

Email:
grigorii@shishkin.ural.ru

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