A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem
In this paper we are considering a semilinear singular perturbation reac- tion - diffusion boundary value problem, which contains a small perturbation param- eter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is e-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.