Research on Aggregations for Algebraic Multigrid Preconditioning Methods

Abstract

Aggregation based algebraic multigrid is one of the most efficient methods to solve sparse linear systems. In this paper, a new method based on cliques and several classical aggregations are implemented based on sparse data structures, and compared in solving sparse linear systems from model partial differential equations. The results show that with Vanek scheme, the number of levels is often less than that with others, and when Jacobi smoothing is used, the number of preconditioned iterations and the time elapsed for iteration are both the least, while the time for setup is too long. From the view point of strong connections, it is beneficial to aggregate the points in a common clique, but the experiments have not shown this privilege. Instead, the two-point aggregation based on strong connection is effective in most cases, and the four-point and the eight-point aggregations which are based on loop of the two-point algorithm with two or three times respectively are effective too. In respect to total computation time, the best scheme is always one of these three schemes. In addition, the larger the average degree the related adjacent graph has, the less time required for the Vanek scheme to setup is and the more efficient it is, so is the aggregation scheme based on cliques.