A note on the convergence rate of the finite element method for singularly perturbed problems using the Shishkin mesh

Authors:
Christos Xenophontos
Affiliations:
Department of Mathematical Sciences, Loyola College, 4501 N. Charles Street, Baltimore, MD
Venue:
Applied Mathematics and Computation
Year:
2003

Citing 3
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Abstract

We consider the numerical approximation of singularly perturbed problems by the h version of the finite element method on a piecewise uniform, Shishkin mesh. It is well known that this method yields uniform approximations, with respect to the perturbation parameter, at a quasi-optimal algebraic convergence rate. In this note we show that the known rate is asymptotically sharp (as the meshwidth h → 0), by obtaining lower and upper bounds on the error. Numerical results that agree with the analysis are presented, as are comparisons with certain other non-uniform mesh strategies.