Volume 31, Issue 5
Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems

J. Comp. Math., 31 (2013), pp. 509-521.

Published online: 2013-10

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• 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.

• Keywords

Singular perturbation, Semilinear reaction-diffusion, Overlapping Schwarz method, Robust convergence, Numerov scheme.

65L10, 65L11, 65L20.

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@Article{JCM-31-509, author = {}, title = {Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {5}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1307-m3787}, url = {http://global-sci.org/intro/article_detail/jcm/9750.html} }
TY - JOUR T1 - Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems JO - Journal of Computational Mathematics VL - 5 SP - 509 EP - 521 PY - 2013 DA - 2013/10 SN - 31 DO - http://doi.org/10.4208/jcm.1307-m3787 UR - https://global-sci.org/intro/article_detail/jcm/9750.html KW - Singular perturbation, Semilinear reaction-diffusion, Overlapping Schwarz method, Robust convergence, Numerov scheme. AB -

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.

S. Chandra Sekhara Rao & Sunil Kumar. (1970). Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems. Journal of Computational Mathematics. 31 (5). 509-521. doi:10.4208/jcm.1307-m3787
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