Iterative Substructuring Methods for Elliptic Problems & Related Algorithms

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This project continues research on iterative substructuring methods and other domain decomposition algorithms for partial differential equations. These algorithms are iterative methods for the parallel solution of very large systems of algebraic equations that arise in finite element/difference modeling of petroleum reservoirs, computational fluid dynamics, elasticity, etc. In each iteration, the restriction of the given problem to each of many subregions is solved approximately and data is exchanged between neighboring subdomains. A global problem, with few degrees of freedom per subdomain, is also required to obtain fast convergence when many subdomains are used. The research will focus on analyzing, improving and implementing several classes of domain decomposition algorithms, in particular, those which are suitable for emerging massively parallel systems. The ultimate goal is to solve very large, ill-conditioned problems of continuum mechanics for which traditional iterative methods are unsatisfactory and for which direct methods are too slow and require too much memory. The underlying theory, which has proven central in the recent development of these fast algorithms, will be further developed and the ideas tested in systematic numerical experiments.