An analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems

In this paper, we present an analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems. For this purpose, we consider a model problem for coupled system of singularly perturbed reaction-diffusion equations with distinct small positive parameters, exhibiting overlapping boundary layers at both ends of the domain. A discrete overlapping Schwarz domain decomposition method is considered to solve the model problem numerically. The analysis is based on defining some auxiliary problems that allow to prove the uniform convergence of the method in two steps, splitting the discretization error and the iteration error. The numerical approximations generated from the method are shown to be almost second order uniformly convergent. The presented idea has a general significance and can be used for the analysis of overlapping domain decomposition methods for other classes of singularly perturbed reaction-diffusion problems.