Domain Decomposition Methods: Algorithms and Theory

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This research project aims primarily to further develop fast and reliable methods for large scale computations to support the solution of complicated engineering problems. Examples are provided by oil platforms, complicated antenna systems, and flow of oil, gas, and contaminants in porous media. Successful computer simulations of such problems require careful modeling as well as efficient methods to obtain timely solutions of the often very large systems of equations that will arise in any attempt to provide reliable support for the design and optimization of complicated engineering structures. The need for such work can be illustrated by the very costly failures that have happened to oil platforms already installed or in the process of being built. This work involves the development of accurate mathematical models as well as the design of fast solvers; the current project will focus on developing such solvers. Large scale computational models require access to modern computer technology, in particular to computer systems with many processors. The principal investigator will continue to work actively with software engineers to develop improved solvers for a variety of problem classes. The algorithms developed in this project will all be based on domain decomposition. Domain decomposition algorithms respect the memory hierarchies of modern parallel computing systems, and experiments clearly illustrate that they scale very well up to the full set of processors and billions of degrees of freedom. Domain decomposition methods provide iterative solvers based on a conjugate gradient algorithm combined with a preconditioner. A preconditioner provides an approximate inverse of the stiffness matrix of the partial differential equation formulated variationally and approximated using a Galerkin method. Any successful domain decomposition algorithm works with solvers on often very many subdomains into which the domain of the given partial differential equation has been subdivided. In addition, to obtain a scalable algorithm, i.e., an algorithm with a convergence rate that does not deteriorate when the number of subdomains and processors are increased, a coarse global part of the preconditioner must be introduced; for large problems a third even coarser level is also introduced. Firmly rooted in mathematical theory, these algorithms are now developing rapidly in a way not foreseen just a few years ago. Powerful ideas are now developing that provide much improved design of the coarse components at the expense of solving relatively small generalized eigenvalue problems in the set-up phase of the computation. Recent experiments show that these devices greatly improve the robustness of the algorithms even of problems with greatly varying material properties. This project aims to contribute to these developments.