A Robust Overlapping Schwarz Method for a Singularly Perturbed Semilinear Reaction-Diffusion Problem with Multiple Solutions
Year: 2009
International Journal of Numerical Analysis and Modeling, Vol. 6 (2009), Iss. 4 : pp. 680–695
Abstract
An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter $\varepsilon^2$ is arbitrarily small, which induces boundary layers. The Schwarz method invokes two boundary-layer subdomains and an interior subdomain, the narrow overlapping regions being of width $O(\varepsilon| \ln \varepsilon|)$. Constructing sub- and super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when $\varepsilon \leq CN^{-1}$ and layer-adapted meshes of Bakhvalov and Shishkin types are used, 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 $\varepsilon$, where $N$ is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2009-IJNAM-791
International Journal of Numerical Analysis and Modeling, Vol. 6 (2009), Iss. 4 : pp. 680–695
Published online: 2009-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Semilinear reaction-diffusion singularly perturbed boundary layers domain decomposition overlapping Schwarz method.