Parallel algorithms for indefinite linear systems

Abstract

Saddle-point problems give rise to indefinite linear systems that are challenging to solve via iterative methods. This paper surveys two recent techniques for solving such problems arising in computational fluid dynamics. 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. The paper presents the performance of these methods for incompressible particulate flow problems on a shared-memory parallel architecture.