Two-grid algorithm for the solution of singularly perturbed two-parameter problem on Shishkin mesh

Abstract

A boundary value problem for a second-order semilinear singularly perturbed ordinary differential equation with two small parameters affecting the convection and diffusion terms is considered. We use Newton and Picard iterations for a linearization. To solve the problem at each iteration we apply the second order difference scheme on the Shishkin mesh which converges uniformly with respect to both singular perturbation parameters. To decrease the required number of arithmetical operations for resolving the difference scheme, a cascadic two-grid method is proposed. To increase the accuracy of difference scheme, we investigate the possibility to apply Richardson extrapolation using known solutions of the difference scheme on both meshes. The results of some numerical experiments are discussed.