This project will support the study of effective and robust strategies for the solution of large linear systems that arise from the discretization of elliptic boundary value problems. Major emphasis will be on the solution of non-self-adjoint, indefinite problems, e.g., the exterior Helmholtz problem. Both preconditioning strategies and computational strategies on parallel machines will be investigated. In the area of preconditioning strategies, the objective will be the derivation of estimates of condition numbers as well as the distribution of eigenvalues and singular values of the preconditioned systems. The preconditioning of pseudospectral discretizations by finite difference or finite element discretizations will also be studied. In the area of computational strategies, the effectiveness and implementation of various preconditioning strategies on SIMD and MIMD computer systems will be studied. In solving boundary value problems for elliptic equations numerically, it becomes necessary to solve the very large systems of linear equations that arise when the continuous problem is replaced with a discrete problem. The principal investigator will develop numerical methods that are both efficient and robust for solving such systems.
