Iterative Substructuring Methods for Elliptic Problems

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The development of numerical methods for large algebraic systems is central in the development of efficient codes for computational 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. A very desirable feature of iterative substructuring and other domain decomposition algorithms is that they respect the memory hierarchy of modern parallel and distributed computing systems, which is essential for approaching peak floating point performance. The development of improved methods is, together with more powerful computer systems, making it possible to carry out simulations in three dimensions, with quite high resolution, relatively easily. This work is now supported by high quality software systems, such as Argonne's PETSC library, which will facilitate code development as well as the access to a variety of parallel and distributed computer systems. Work willcontinue in developing iterative substructuring and other domain decomposition methods for increasingly difficult partial differential equations. Domain decomposition algorithms are iterative methods, often of preconditioned 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 proble m often substantially increases the efficiency of the preconditioner. This study will combine mathematical analysis with the design and numerical testing of algorithms. A special emphasis will be placed on the study of spectral elements and other high order finite element methods, as well as on nonconforming methods such as the mortar and Nedelec finite element methods. The latter have been developed for Maxwell's equation. There will also be a focus on the often very ill-conditioned problems which arise in finite element approximations of elasticity. Iterative methods and production codes will also be developed for Helmholtz's equation and other time-harmonic models arising, e.g., from Maxwell's equations. These problems pose very real challenges for the development of iterative solvers and are also of great importance in a number of engineering applications.