A Uniformly Convergent Method on Arbitrary Meshes for a Semilinear Convection-Diffusion Problem with Discontinuous Data

Abstract

This paper deals with a uniform (in a perturbation parameter) convergent difference scheme for solving a nonlinear singularly perturbed two-point boundary value problem with discontinuous data of a convection-diffusion type. Construction of the difference scheme is based on locally exact schemes or on local Green's functions. Uniform convergence with first order of the proposed difference scheme on arbitrary meshes is proven. A monotone iterative method, which is based on the method of upper and lower solutions, is applied to computing the nonlinear difference scheme. Numerical experiments are presented.