Optimization of the Multigrid-Convergence Rate on Semi-structured Meshes by Local Fourier Analysis

TL;DR

This paper develops a local Fourier analysis for multigrid methods on tetrahedral meshes, proposing efficient smoothers and a novel Fourier space partitioning to optimize convergence rates, validated through numerical experiments.

Contribution

It introduces a new local Fourier analysis framework for tetrahedral grids and designs a multigrid method with adaptive smoothers for improved convergence.

Findings

Four-color smoother is effective for regular tetrahedral grids.

Line and plane relaxations are better for poorly shaped tetrahedra.

Numerical tests confirm the theoretical convergence improvements.

Abstract

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.