Volume 6, Issue 4
A Robust Overlapping Schwarz Method for a Singularly Perturbed Semilinear Reaction-diffusion Problem with Multiple Solutions
DOI:

Int. J. Numer. Anal. Mod., 6 (2009), pp. 680-695

Published online: 2009-06

Preview Full PDF 0 453
Export citation
• Abstract

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter $\epsilon^2$ 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 $O(\epsilon| \ln \epsilon|)$. Constructing sub- and super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when $\epsilon \leq CN^{-1}$ and layer-adapted meshes of Bakhvalov and Shishkin types are used, one iteration is suffcient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $\epsilon$, where N is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.

• Keywords

Semilinear reaction-di usion singularly perturbed boundary layers domain decomposition overlapping Schwarz method

65L10 65L12 65L70

@Article{IJNAM-6-680, author = {N. Kopteva, M. Pickett and H. Purtill}, title = {A Robust Overlapping Schwarz Method for a Singularly Perturbed Semilinear Reaction-diffusion Problem with Multiple Solutions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2009}, volume = {6}, number = {4}, pages = {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 $\epsilon^2$ 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 $O(\epsilon| \ln \epsilon|)$. Constructing sub- and super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when $\epsilon \leq CN^{-1}$ and layer-adapted meshes of Bakhvalov and Shishkin types are used, one iteration is suffcient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $\epsilon$, where N is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/791.html} }