A second-order overlapping Schwarz method for a 2d singularly perturbed semilinear reaction-diffusion problem

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter epsilon(2) is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width O(epsilon vertical bar ln h vertical bar), where h is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed O(h(-2)). We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain. For this iterative method, we present maximum norm error estimates for epsilon is an element of (0, 1]. It is shown, in particular, that when epsilon <= C vertical bar ln h vertical bar(-1), one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in E. Numerical results are presented to support our theoretical conclusions.