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.
Creators(s): | Kopteva, Natalia and Pickett, Maria; | Item type: | Article |
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ID code: | 44836 |
Keywords: | semilinear reaction-diffusion, singular perturbation, domain decomposition, overlapping Schwarz, Bakhvalov mesh, Shishkin mesh, supra-convergence, lumped-mass finite elements, Probabilities. Mathematical statistics, Statistics and Probability |
Subjects: | Science > Mathematics > Probabilities. Mathematical statistics |
Department: | Faculty of Science > Mathematics and Statistics |
Depositing user: | Pure Administrator |
Date deposited: | 13 Sep 2013 10:48 |
Last modified: | 26 Feb 2021 04:20 |
URI: | https://strathprints.strath.ac.uk/id/eprint/44836 |
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