A robust overlapping Schwarz domain decomposition algorithm for time-dependent singularly perturbed reaction-diffusion problems. (English) Zbl 1280.65105
Summary: 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 \(\epsilon\). To be more specific, we prove that, when \(\epsilon\) 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.
MSC:
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
| 35B25 | Singular perturbations in context of PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
singular perturbation; parabolic reaction-diffusion problems; parameter-robust; coupled system; error bounds; overlapping Schwarz domain decomposition method; convergence; numerical experimentsReferences:
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