Optimization of the Multigrid-Convergence Rate on Semi-structured Meshes by Local Fourier Analysis
This work provides a theoretical and practical framework for optimizing multigrid convergence on semi-structured meshes, benefiting computational scientists solving PDEs on complex geometries.
The paper presents a local Fourier analysis for multigrid methods on tetrahedral grids, analyzing different smoothers for the Laplace operator. A four-color smoother is proposed for regular tetrahedral grids, and a block-wise multigrid method with adaptive smoothers is constructed, validated by numerical tests.
In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother is presented as an efficient choice for regular tetrahedral grids, whereas line and plane relaxations are needed for poorly shaped tetrahedra. A novel partitioning of the Fourier space is proposed to analyze the four-color smoother. Numerical test calculations validate the theoretical predictions. A multigrid method is constructed in a block-wise form, by using different smoothers and different numbers of pre- and post-smoothing steps in each tetrahedron of the coarsest grid of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.