Uniform convergence of finite-difference schemes for reaction-diffusion interface problems

Abstract

We consider a singularly perturbed reaction-diffusion equation in two dimensions (x,y) with concentrated source on a segment parallel to axis Oy. By means of an appropriate (including corner layer functions) decomposition, we describe the asymptotic behavior of the solution. Finite difference schemes for this problem of second and fourth order of local approximation on Shishkin mesh are constructed. We prove that the first scheme is almost second order uniformly convergent in the maximal norm. Numerical experiments illustrate the theoretical order of convergence of the first scheme and almost fourth order of convergence of the second scheme. © 2008 Springer.