Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

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This project continues the past research on domain decomposition methods for elliptic partial differential equations. These are iterative methods for solving large linear or nonlinear systems of algebraic equations which arise when elliptic problems are discretized by finite elements or finite differences, and they offer very considerable promise for efficient parallel solution. The methods considered in this investigation can be regarded as divide and conquer algorithms. In each step of an iteration, the original discrete elliptic problem is solved on the subregions into which the original region has been subdivided. The interaction between the different parts of the region is handled by transferring data of a suitable type across the interfaces which are created by the subdivision. In addition to the local problems, some global, coarse problem must be incorporated into the preconditioner in order to obtain a fast rate of convergence in the case of many subregions, the case of primary interest in parallel computing. Further work will be carried out to develop a general framework for the study of domain decomposition methods both those which use overlapping subregions and those which do not. There are two main aims of this research: (1) To develop new and improved methods and to extend them to the nonconforming case, mixed finite element models, the finite volume case, etc. (2) To develop tools and experience which will make it possible to provide specific recommendations for the solution of important problems of elasticity, fluid dynamics, problems with widely varying values of the coefficients, etc.