A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction–diffusion problems

In this work, we consider a singularly perturbed parabolic problem of reaction–diffusion type. To solve this problem numerically we develop an overlapping Schwarz domain decomposition method, where we use the asymptotic behaviour of the exact solution for domain partitioning. We prove that the method gives uniform numerical approximations of first order in time and almost second order in space. Furthermore, we address the much faster convergence of the algorithm for small perturbation parameter εε. To be more specific, we prove that, when εε is small, just one iteration is required to achieve the desired accuracy. We then extend the method to a system of singularly perturbed parabolic problems of reaction–diffusion type. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.