TY - JOUR
T1 - Robust High Order Convergence of an Overlapping Schwarz Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems
AU - S. Chandra Sekhara Rao & Sunil Kumar
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 - <p style="text-align: justify;">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.</p>