TY - JOUR

T1 - A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions

AU - Kopteva, N.

AU - Pickett, Maria

AU - Purtill, H.

N1 - Funders:
This research was supported by an Irish research council for science and technology (IRCSET) postdoctoral fellowship..

PY - 2009

Y1 - 2009

N2 - 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 sufficient 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.

AB - 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 sufficient 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.

M3 - Article

SN - 1705-5105

VL - 6

SP - 680

EP - 695

JO - International Journal of Numerical Analysis and Modeling

JF - International Journal of Numerical Analysis and Modeling

IS - 4

ER -