Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.