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 (https://doi.org/10.1090/S0025-5718-2011-02521-4)

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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.

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  Item type:	Article
  ID code:	44836
  Dates:	Date Event January 2012 Published 18 July 2011 Published Online
  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:	25 May 2023 09:17
  URI:	https://strathprints.strath.ac.uk/id/eprint/44836