A compact finite difference scheme for 2D reaction–diffusion singularly perturbed problems

Abstract

In this work we define a compact finite difference scheme of positive type to solve a class of 2D reaction–diffusion elliptic singularly perturbed problems. We prove that if the new scheme is constructed on a piecewise uniform mesh of Shishkin type, it provides better approximations than the classical central finite difference scheme. Moreover, the uniform parameter bound of the error shows that the scheme is third order convergent in the maximum norm when the singular perturbation parameter is sufficiently small. Some numerical experiments illustrate in practice the result of convergence proved theoretically.