A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers

The problem of constructing a parameter-uniform numerical method for a singularly perturbed self-adjoint ordinary differential equation is considered. It is shown that a suitably designed discrete Schwarz method, based on a standard finite difference operator with a uniform mesh on each subdomain, gives numerical approximations which converge in the maximum norm to the exact solution, uniformly with respect to the singular perturbation parameter. This parameter-uniform convergence is shown to be essentially second order. That this new discrete Schwarz method is efficient in practice is demonstrated by numerical experiments. (C) 2001 Elsevier Science B.V. All rights reserved.