The solution of singularly perturbed parabolic problems by parallel algorithms combining Crank-Nicholson scheme and overlapping Schwarz methods

Parallel algorithms combining a time discretization and overlapping domain decomposition methods are applied to the solution of singularly perturbed parabolic problems. Two domain decomposition methods based on the Schwarz alternating procedure are proposed: the method with two-colour ordering of subdomains and the method with additional “correcting” problems. Moreover, modifications of the methods are considered using time extrapolation on subdomain interfaces. Convergence properties of the algorithms are established at the differential level. Numerical results for a test singularly perturbed problem are presented to confirm the convergence theory and to evaluate the parallel arithmetic complexity of the algorithms.