Schwartz Methods for Singularly Perturbed Convection-Diffusion Problems

In this paper we consider singularly perturbed ordinary differential equations with convection terms. An ε-uniformly convergent finite difference scheme is constructed for such boundary value problems by using an upwind finite difference operator and a piecewise uniform mesh. Our primary interest is the design of a numerical method, for which a parallel technique will later be applicable. In this paper we consider the solution of the discrete problem using a domain decomposition method, based on the Schwartz alternating process with overlapping subdomains with essential and minimal overlapping width. We analyse and discuss the influence of various parameters, such as the perturbation parameter ε, the number of mesh points and the number of iterations, on the convergence of the iterative process. We give conditions under which the iterative difference schemes converge ε-uniformly. Illustrative numerical examples are presented.