Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

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This project continues to work on iterative substructuring methods. These are iterative methods for large linear systems of algebraic equations which arise when elliptic problems are discretized by finite elements or finite differences. The region of the problem is divided into subregions (substructures) and the problems on the subregions are solved repeatedly while using a preconditioned conjugate gradient algorithm to satisfy the continuity requirements across the curves or surfaces that divide the region. Main current research issues are the extension of these algorithms to nonconforming finite elements, mixed methods, etc., and the development of new algorithms, particularly in three dimensions. This program will combine theoretical work with systematic numerical experiments using both model problems and industrial finite element codes. The best of these algorithms often represent an improvement over previously known methods on the uni processors and they hold considerable promise for multi processor systems. In addition, new algorithms for certain queuing network problems will be developed. Kolmogorov's balance equations are generated and the goal is to compute the steady state probability distribution in terms of the null element of the corresponding very large, sparse, nonsymmetric matrix. The basic idea, which has proven quite successful in a number of cases, is to use suitably chosen nearby decomposable problems as preconditioners. These algorithms will be further developed, the theory will be refined to try to explain the interesting distribution of the singular values of the iteration matrices and an attempt will be made to extend the applicability of methods to new classes of problems. This research addresses new and faster methods for solving large systems of linear equations, a computational task that is fundamental to many scientific and engineering problems.