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Mathematics of Computation

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On the non-existence of $\epsilon$-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems
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by Paul A. Farrell, John J. H. Miller, Eugene O’Riordan and Grigorii I. Shishkin PDF
Math. Comp. 67 (1998), 603-617 Request permission

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
  • 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.
  • © Copyright 1998 American Mathematical Society
  • 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