TY - JOUR
T1 - A Robust Overlapping Schwarz Method for a Singularly Perturbed Semilinear Reaction-Diffusion Problem with Multiple Solutions
AU - N. Kopteva, M. Pickett & H. Purtill
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 680
EP - 695
PY - 2009
DA - 2009/06
SN - 6
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/791.html
KW - Semilinear reaction-diffusion, singularly perturbed, boundary layers, domain decomposition, overlapping Schwarz method.
AB - <p style="text-align: justify;">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.</p>