An analysis of a uniformly convergent finite difference/finite element scheme for a model singular-perturbation problem

Author:
Eugene C. Gartland

Journal:
Math. Comp. **51** (1988), 93-106

MSC:
Primary 65L10; Secondary 65L60

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942145-6

MathSciNet review:
942145

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Abstract: Uniform $\mathcal {O}({h^2})$ convergence is proved for the El-Mistikawy-Werle discretization of the problem $- \varepsilon u”+ au’+ bu = f$ on (0,1), $u(0) = A$, $u(1) = B$, subject only to the conditions $a,b,f \in {\mathcal {W}^{2,\infty }}[0,1]$ and $a(x) > 0, 0 \leq x \leq 1$. The principal tools used are a certain representation result for the solutions of such problems that is due to the author [*Math. Comp.*, v. 48, 1987, pp. 551-564] and the general stability results of Niederdrenk and Yserentant [*Numer. Math.*, v. 41, 1983, pp. 223-253]. Global uniform $\mathcal {O}(h)$ convergence is proved under slightly weaker assumptions for an equivalent Petrov-Galerkin formulation.

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*Strong Stability and a Representation Result for a Singular Perturbation Problem*, Technical Report AMS 87-1, Dept. of Mathematics, Southern Methodist University, January, 1987.

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Article copyright:
© Copyright 1988
American Mathematical Society