A collocation method on a Gartland-type mesh for a singularly perturbed reaction-diffusion problem

Abstract

© 2019 Department of Mathematics, University of Osijek. A singularly perturbed reaction-diffusion problem in one dimension is solved numerically by a collocation method with quadratic C1-splines. Using an appropriately graded mesh of Gartland type, second order of convergence is obtained in the supremum norm uniformly, up to a logarithmic factor, in the singular perturbation parameter. The aim of this paper is to establish the advantage of using the Gartland-type mesh. The method presented here generates results that are superior to those obtained on the so-called ”smoothed Shishkin mesh”. Results of numerical experiments which illustrate our theoretical findings are presented. Furthermore, numerical results for a two-dimensional problem reveal the same order of the convergence as in the one-dimensional case, though efforts to establish its theoretical foundation are still ongoing.