Smooth Aggregation for Difficult Stretched Mesh and Coefficient Variation Problems

TL;DR

This paper introduces four adaptations of the smoothed aggregation algebraic multigrid method aimed at enhancing convergence and robustness when dealing with matrices featuring many weak connections, which can cause fill-in and smoothing issues.

Contribution

The paper proposes novel modifications to the SA-AMG method, including changes to the prolongator smoother, filtered matrix, and grid transfer operators, to improve performance on challenging problems.

Findings

Improved convergence in weak connection scenarios

Reduced fill-in in coarse matrices

Enhanced smoothing effectiveness

Abstract

Four adaptations of the smoothed aggregation algebraic multigrid (SA-AMG) method are proposed with an eye towards improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill-in within the coarse discretization matrices and can also give rise to sub-optimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi-like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.