A parallel Aitken-additive Schwarz waveform relaxation suitable for the grid

Abstract

The objective of this paper is to describe a grid-efficient parallel implementation of the Aitken-Schwarz waveform relaxation method for the heat equation problem. This new parallel domain decomposition algorithm, introduced by Garbey [M. Garbey, A direct solver for the heat equation with domain decomposition in space and time, in: Springer Ulrich Langer et al. (Ed.), Domain Decomposition in Science and Engineering XVII, vol. 60, 2007, pp. 501-508], generalizes the Aitken-like acceleration method of the additive Schwarz algorithm for elliptic problems. Although the standard Schwarz waveform relaxation algorithm has a linear rate of convergence and low numerical efficiency, it can be easily optimized with respect to cache memory access and it scales well on a parallel system as the number of subdomains increases. The Aitken-like acceleration method transforms the Schwarz algorithm into a direct solver for the parabolic problem when one knows a priori the eigenvectors of the trace transfer operator. A standard example is the linear three dimensional heat equation problem discretized with a seven point scheme on a regular Cartesian grid. The core idea of the method is to postprocess the sequence of interfaces generated by the additive Schwarz wave relaxation solver. The parallel implementation of the domain decomposition algorithm presented here is capable of achieving robustness and scalability in heterogeneous distributed computing environments and it is also naturally fault tolerant. All these features make such a numerical solver ideal for computational grid environments. This paper presents experimental results with a few loosely coupled parallel systems, remotely connected through the internet, located in Europe, Russia and the USA.