Schwarz domain decomposition methods for singularly perturbed differential equations

We study iterative numerical methods, based on Schwarz-iterative techniques and Shishkin meshes, for reaction-diffusion and convection-diffusion problems. We introduce the criteria of (e, N )-uniform convergent numerical approximations. We examine the convergence of the numerical approximations with respect to the dimension of the discrete problem and the number of iterations. It is shown that the techniques used to design an (e, Af)-uniform numerical method for reaction-diffusion problems are not applicable to convection-diffusion problems. A systematic analysis of several variants of Schwarz, including overlapping and non-overlapping methods using different boundary conditions, was undertaken for one dimensional convection-diffusion problems. The convergence behaviour and the iteration counts were examined. Unlike the reaction-diffusion problem, it is shown that the methods using uniform meshes in each subomain do not meet all the (s, N )—uniform convergence criteria. In the case of the convection-diffusion problems, it is demonstrated analytically and numerically that these iterative methods are convergent and have low computational cost for small values of the singular perturbation parameter e. We feel it is of importance that the methods can be extended to higher dimensions with sufficient ease. As an example of this, we extend a non-overlapping method to a two dimensional convection-diffusion problem. The analysis of this method illustrates an appropriate domain structure and the need for sharp bounds on the partial derivatives. Finally, it is shown that an overlapping Schwarz method, using uniform subdomains, can be used to produce (e, AQ-uniform convergence for a time dependent problem with parabolic boundary layers. Numerical results are presented for the methods studied