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Highly scalable parallel domain decomposition methods

Domain decomposition methods are an efficient approach for the solution of elliptic partial differential equations on parallel computers. Here, we understand by domain decomposition methods preconditioned iterative algorithms for the solution of large linear systems of equations obtained, either directly or by linearization, from the discretization of partial differential equations. In such methods, the domain, on which the partial differential equation has to be solved, is decomposed into a number of nonoverlapping subdomains. In each step of the iterative method and for each subdomain, a local problem is solved. This local problem is often an approximation of the partial differential equation restricted to the subdomain; here, we neglect for the moment that the boundary conditions are usually different for the local problem and the problem on the original domain. Depending on the particular domain decomposition method, the local problem is solved approximately itself or exactly, using a direct algorithm, e.g., a Gaussian elimination algorithm. For elliptic problems, also a small global problem is needed in order to obtain a parallel scalable algorithm, i.e., to exploit efficiently a growing number of processors of a parallel computer; in general one processor can obtain more than one subdomain. Robust and parallel scalable algorithms are developed for different problems including very demanding and difficult structural mechanics problems. Parallel scalability, for different algorithms, has been proven on several supercomputers and architectures, e.g., on up to 262 144 cores of the MIRA BG/Q supercomputer at Argonne National Laboratory, USA, on up to 131 072 cores of the Vulcan BG/Q at Lawrence Livermore National Laboratory, USA, on up to 65 536 cores of the JUGENE computer in Jülich, on up to 32 768 cores of SuperMuc at Leibniz Supercomputing Centre, Munich, and also on up to 4128 cores of the Cray XT6m at the Universität Duisburg-Essen.

A domain decomposition for parallel computing. Image from Klawonn and Rheinbach, SIAM J. Sci. Comput., Vol. 28, No. 5, pp. 1886-1906, 2006