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

Authors: Natalia Kopteva and Maria Pickett
Journal: Math. Comp. 81 (2012), 81-105
MSC (2010): Primary 65N06, 65N15, 65N30, 65N50, 65N55
Published electronically: July 18, 2011
MathSciNet review: 2833488
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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 $\varepsilon ^2$ 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 $O(\varepsilon |\ln h|)$, 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 $\varepsilon \in (0,1]$. It is shown, in particular, that when $\varepsilon \le C|\ln h|^{-1}$, 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 $\varepsilon$. Numerical results are presented to support our theoretical conclusions.

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Additional Information

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
MR Author ID: 610720
ORCID: 0000-0001-7477-6926

Maria Pickett
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
Address at time of publication: Department of Mathematics, Lion Gate Building, Lion Terrace, Portsmouth, Hampshire PO1 3HF, United Kingdom

Keywords: Semilinear reaction-diffusion, singular perturbation, domain decomposition, overlapping Schwarz, Bakhvalov mesh, Shishkin mesh, supra-convergence, lumped-mass finite elements
Received by editor(s): December 11, 2009
Received by editor(s) in revised form: November 4, 2010
Published electronically: July 18, 2011
Additional Notes: This research was supported by an Irish Research Council for Science and Technology (IRCSET) postdoctoral fellowship and a Science Foundation Ireland grant under the Research Frontiers Programme 2008.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.