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

Abstract

A singularly perturbed reaction-diffusion one-dimension problemis solved numerically by collocation method with quadratic$C^1-$splines. Using appropriately recursively graded mesh the 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 emphasize the importance of using the recursively graded mesh. The former results obtained by mentioned method on the smoothed Shishkin mesh are improved. Numerical experimentssupport the theoretical results. Numerical results for the reaction-diffusion problem in two-dimension show the same order of the convergence as well as in one-dimensional case, but theoretical analysis presents an ongoing work.