Uniform Convergence of the Schwarz Alternating Method for Solving Singularly Perturbed Advection-Diffusion Equations

Abstract

Our purpose in this paper is to study the convergence rate of some variants of the Schwarz alternating method for solving two types of singularly perturbed advection-diffusion equations, one in which the advection term (first-order term) dominates, and the other in which the lowest-order term (zeroth-order term) dominates. For such problems, boundary or other layer regions may be present. For singular perturbation problems in which the advection dominates, we choose two overlapping subregions, one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region. Our main result for this problem shows that if the subdomains can be chosen to "follow the flow," i.e., if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow, then the Schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion. For the singular perturbation problem in which the lowest-order term dominates, we consider many overlapping subregions, with no special requirement on outflow or inflow boundaries. Our main result here, which is valid for two variants of the classical Schwarz algorithm, shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth-order term. For both problems, our proofs rely on the maximum principle and the construction of barrier functions.