Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems
Year: 2013
Journal of Computational Mathematics, Vol. 31 (2013), Iss. 5 : pp. 509–521
Abstract
In this article we propose an overlapping Schwarz domain decomposition method for solving a singularly perturbed semilinear reaction-diffusion problem. The solution to this problem exhibits boundary layers of width $O(\sqrt{\varepsilon}ln(1/\sqrt{\varepsilon}))$ at both ends of the domain due to the presence of singular perturbation parameter $\varepsilon$. The method splits the domain into three overlapping subdomains, and uses the Numerov or Hermite scheme with a uniform mesh on two boundary layer subdomains and a hybrid scheme with a uniform mesh on the interior subdomain. The numerical approximations obtained from this method are proved to be almost fourth order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Furthermore, it is proved that, for small $\varepsilon$, one iteration is sufficient to achieve almost fourth order uniform convergence. Numerical experiments are given to illustrate the theoretical order of convergence established for the method.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1307-m3787
Journal of Computational Mathematics, Vol. 31 (2013), Iss. 5 : pp. 509–521
Published online: 2013-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Singular perturbation Semilinear reaction-diffusion Overlapping Schwarz method Robust convergence Numerov scheme.