Abstract
An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter ε2 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(ε| ln ε|). Constructing suband super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when ε ≤ 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 ε, where N is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.
Original language | English (Ireland) |
---|---|
Pages (from-to) | 680-695 |
Number of pages | 16 |
Journal | International Journal of Numerical Analysis and Modeling |
Volume | 6 |
Issue number | 4 |
Publication status | Published - 2009 |
Keywords
- Boundary layers
- Domain decomposition
- Overlapping Schwarz method
- Semilinear reaction-diffusion
- Singularly perturbed