Scott P. MacLachlan

Yunhui He, Scott P. MacLachlan

In this paper, we develop a local Fourier analysis of multigrid methods based on block-structured relaxation schemes for stable and stabilized mixed finite-element discretizations of the Stokes equations, to analyze their convergence behavior. Three relaxation schemes are considered: distributive, Braess-Sarazin, and Uzawa relaxation. From this analysis, parameters that minimize the local Fourier analysis smoothing factor are proposed for the stabilized methods with distributive and Braess-Sarazin relaxation. Considering the failure of the local Fourier analysis smoothing factor in predicting the true two-grid convergence factor for the stable discretization, we numerically optimize the two-grid convergence predicted by local Fourier analysis in this case. We also compare the efficiency of the presented algorithms with variants using inexact solvers. Finally, some numerical experiments are presented to validate the two-grid and multigrid convergence factors.

James Brannick, Scott P. MacLachlan, Jacob B. Schroder et al.

Algebraic multigrid (AMG) methods are powerful solvers with linear or near-linear computational complexity for certain classes of linear systems, Ax=b. Broadening the scope of problems that AMG can effectively solve requires the development of improved interpolation operators. Such development is often based on AMG convergence theory. However, convergence theory in AMG tends to have a disconnect with AMG in practice due to the practical constraints of (i) maintaining matrix sparsity in transfer and coarse-grid operators, and (ii) retaining linear complexity in the setup and solve phase. This paper presents a review of fundamental results in AMG convergence theory, followed by a discussion on how these results can be used to motivate interpolation operators in practice. A general weighted energy minimization functional is then proposed to form interpolation operators, and a novel `diagonal' preconditioner for Sylvester- or Lyapunov-type equations developed simultaneously. Although results based on the weighted energy minimization typically underperform compared to a fully constrained energy minimization, numerical results provide new insight into the role of energy minimization and constraint vectors in AMG interpolation.