Paper Info
Title
Two-level Fourier analysis of multigrid for higher-order finite-element discretizations of the Laplacian.
Abstract
In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher-order finite-element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse-grid correction is assumed to be an ideal operator that annihilates the low-frequency error components and leaves the high-frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two-grid convergence factor. While two-grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is "between" two-grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher-order finite element discretizations, including for Q(2)-Q(1) (Taylor-Hood) elements for the Stokes equations. Finally, we present two-grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two-grid and multigrid convergence factors.
Year
DOI
Venue
2020
10.1002/nla.2285
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Keywords
DocType
Volume
finite-element method,higher-order elements,Jacobi iteration,local Fourier analysis,multigrid
Journal
27.0
Issue
ISSN
Citations 
3.0
1070-5325
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Yunhui He111.71
S. P. MacLachlan29811.78