Scientific and High-Performance Computing
Martin Gander, Mathematics, University of Geneva
Riemann, Schwarz, Schur, Picard: From a Gap in a Proof to Modern Domain Decomposition Methods
Domain decomposition methods have been developed in various contexts, and with very different goals in mind. The first domain decomposition method was invented by Schwarz in 1869 to close an important gap in the proof of the Riemann mapping theorem. It was only a century later when the so called alternating Schwarz method became useful as a computational tool, through the work of Lions, and Dryja and Widlund. Schwarz methods work because the subdomains are overlapping, and have strong roots in functional analysis. A very different type of domain decomposition methods was invented in the engineering community in 1963 by Przemieniecki, who was working for Boeing. These methods are based on non-overlapping domain decompositions, and were initially non-iterative. Their formulation is based on Schur complements, and their modern variants are the FETI and balancing Neumann-Neumann methods. All these methods are for steady problems, but there is a third type of domain decomposition methods that was formulated for evolution problems: the waveform relaxation methods. Like Schwarz methods, these methods have their roots in analysis, namely the existence proof for solutions of ordinary differential equations by Picard in 1893, but they were reinvented in the engineering community at IBM by Ruehli, Lelarasmee and Sangiovanni-Vincentelli in 1982. We will show for simple model problems how all these domain decomposition methods function, give precise convergence results for the model problem, and also explain the most general convergence results available currently for these methods.