Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

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The development of numerical methods for large systems is central in the development of efficient codes for fluid dynamics, elasticity, and other core problems of continuum mechanics. Many other tasks in such codes parallelize relatively easily. The importance of the algebraic system solvers is therefore increasing with the appearance of parallel and distributed computing systems, with a substantial number of fast processors, each with relatively large memory. The development of improved methods will, together with more powerful computer systems, make it possible to carry out simulations in three dimensions, with quite high resolution, relatively easily. Work will continue in developing iterative substructuring and other domain decomposition methods for increasingly difficult elliptic problems. Domain decomposition algorithms are iterative methods, often of conjugate gradient type, for the parallel solution of the large linear, or nonlinear, systems of algebraic equations that arise when partial differential equations are discretized by finite elements, finite differences, or spectral methods. In each iteration step, local problems representing the restriction of the original problem to a potentially large number of subregions are solved approximately. The subregions, which can be allocated to individual processors of a parallel computer, form a decomposition of the entire domain of the problem. In addition, the inclusion of a coarse subproblem substantially increases the efficiency of the preconditioner. In this study, which combines mathematical analysis with the design and numerical testing of algorithms, a special emphasis is on the study of elements and other high order finite element methods, on nonconforming methods such as the mortar finite elements, and on the difficulties which are related to distorted subregions or elements.