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

Kopteva, Natalia and Pickett, Maria (2012) A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem. Mathematics of Computation, 81 (277). pp. 81-105. ISSN 0025-5718

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Abstract

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 ε2 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(ε| 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 ε ∈ (0, 1]. It is shown, in particular, that when ε ≤ 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 ε. Numerical results are presented to support our theoretical conclusions.