Finite Volume Difference Scheme for a Stiff Elliptic Reaction-Diffusion Problem with a Line Interface

Abstract

We consider a singularly perturbed elliptic problem in two dimensions with stiff discontinuous coefficients of order O(1) and O(ɛ) on the left and on the right of interface, respectively. The solution of this problem exhibits boundary and corner layers and is difficult to solve numerically. The FVM is implemented on condensed (Shishkin’s) mesh that resolves boundary and corners layers, and we prove that it yelds an accurate approximation of the solution both inside and outside these layers. We give error estimates in discrete energetic norm that hold true uniformly in the perturbation parameter ɛ. Numerical experiments confirm these theoretical results.