Local Fourier analysis for mixed finite-element methods for the Stokes equations
For researchers working on multigrid solvers for Stokes equations, this provides a theoretical analysis and parameter optimization for block-structured relaxations, though it is an incremental extension of existing local Fourier analysis techniques.
This paper develops a local Fourier analysis for multigrid methods with block-structured relaxation schemes (distributive, Braess-Sarazin, Uzawa) for Stokes equations, proposing optimal parameters for stabilized methods and numerically optimizing two-grid convergence for stable discretizations. Numerical experiments validate the predicted convergence factors.
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.