Parameter-uniform convergence analysis of a domain decomposition method for singularly perturbed parabolic problems with Robin boundary conditions

Abstract

We construct and analyze a domain decomposition method to solve a class of singularly perturbed parabolic problems of reaction-diffusion type having Robin boundary conditions. The method considers three subdomains, of which two are finely meshed, and the other is coarsely meshed. The partial differential equation associated with the problem is discretized using the finite difference scheme on each subdomain, while the Robin boundary conditions associated with the problem are approximated using a special finite difference scheme to maintain the accuracy. Then, an iterative algorithm is introduced, where the transmission of information to the neighbours is done using a piecewise linear interpolation. It is proved that the resulting numerical approximations are parameter-uniform and, more interestingly, that the convergence of the iterates is optimal for small values of the perturbation parameters. The numerical results support the theoretical results about convergence.