Acceleration of a Schwarz waveform relaxation method for parabolic problems

In this paper we generalize the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems to the additive Schwarz waveform relaxation for parabolic problems. The domain decomposition is in space and time. The standard Schwarz waveform relaxation algorithm has a linear rate of convergence and low numerical efficiency. This algorithm is, however, friendly to cache use and scales with the memory in parallel environments. It also minimizes the number of messages sent in a parallel implementation and is therefore very insensitive to delays due to a high latency network. M. Gander and co-workers have shown that the convergence of this algorithm can be speed up by optimizing the transmission conditions. Our Aitken-like acceleration is an alternative method that consists of postprocessing the sequence of interfaces generated by the domain decomposition solver and might be combined to the method of Gander et al. We show that our technique (1) is a direct solver that requires at most four solves per sub-domain in the case of a one space dimension linear parabolic problem with time independent coefficients, (2) can be applied easily to multi-dimensional problems, provided that the operator is separable in space (3) is an efficient iterative procedure for parabolic problems that are weak nonlinear perturbations of linear operators with time independent coefficients, (4) provides a rigorous framework to optimize the parallel implementation on a slow network of computers.