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

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posted on 2017-05-04, 12:02 authored by Natalia KoptevaNatalia Kopteva, Maria PickettAn 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.

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Mathematics of Computation;81, pp. 81-105## Publisher

American Mathematical Society## Note

peer-reviewed## Other Funding information

IRC, SFI## Rights

First published in Mathematics of Computation, 81, pp. 81-105, 2012, published by the American Mathematical Society## Language

English## External identifier

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