Parallel algorithm for solving linear systems arising from PDE and integral equations

Abstract

In this paper, we consider solving matrix systems arising from the linear integral equations and PDE by two recent parallel techniques. The systems are indefinite due to linear constraints imposed on the fluid velocity. The first approach, known as the multilevel algorithm, employs a hierarchical technique to compute the constrained linear space for the unknowns, followed by the iterative solution of a positive definite reduced problem. The second approach exploits the banded structure of sparse matrices to obtain a different reduced system which is determined by the unknowns common to adjacent block rows. Although the reduced system in this approach may still be indefinite, the algorithm converges to the solution at an accelerated rate. These methods have two desirable characteristics, namely, robust numerical convergence and efficient parallelizability.