Int. J. Numer. Anal. Mod., 6 (2009), pp. 680-695. A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions Natalia Kopteva 1, Maria Pickett 1, Helen Purtill 11 Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland. Received by the editors August 9, 2008 and, in revised form, Febuary 9, 2009. 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. AMS subject classifications: 65L10, 65L12, 65L70 Key words: Semilinear reaction-di usion, singularly perturbed, boundary layers, domain decomposition, overlapping Schwarz method. Email: Natalia.Kopteva@ul.ie and Maria.Pickett@ul.ie, Helen.Purtill@ul.ie