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 $O(\epsilon|\ln h|)$, 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 $\epsilon \in (0,1]$. It is shown, in particular, that when $\epsilon < C|\ln h|^{-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 $\epsilon$. Numerical results are presented to support our theoretical conclusions.