Hybrid Multigrid/Schwarz algorithms for the spectral element method

James W. Lottes; Paul F. Fischer

doi:10.1007/s10915-004-4787-3

Hybrid Multigrid/Schwarz algorithms for the spectral element method

Research output: Contribution to journal › Article › peer-review

Abstract

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

Original language	English (US)
Pages (from-to)	613-646
Number of pages	34
Journal	Journal of Scientific Computing
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s10915-004-4787-3
State	Published - Jul 2005
Externally published	Yes

Keywords

Domain decomposition
Multigrid
P-version finite element
Schwarz methods
Spectral element methods

ASJC Scopus subject areas

Software
Theoretical Computer Science
Numerical Analysis
Engineering(all)
Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

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title = "Hybrid Multigrid/Schwarz algorithms for the spectral element method",

abstract = "We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.",

keywords = "Domain decomposition, Multigrid, P-version finite element, Schwarz methods, Spectral element methods",

author = "Lottes, {James W.} and Fischer, {Paul F.}",

note = "Funding Information: We thank Barry Smith for many useful discussions during the course of this work and the reviewers for many useful suggestions. This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng-38. We also thank the Division of Educational Programs at Argonne National Laboratory, which was instrumental in enabling the collaboration that resulted in this study.",

year = "2005",

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doi = "10.1007/s10915-004-4787-3",

language = "English (US)",

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journal = "Journal of Scientific Computing",

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T1 - Hybrid Multigrid/Schwarz algorithms for the spectral element method

AU - Lottes, James W.

AU - Fischer, Paul F.

N1 - Funding Information: We thank Barry Smith for many useful discussions during the course of this work and the reviewers for many useful suggestions. This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng-38. We also thank the Division of Educational Programs at Argonne National Laboratory, which was instrumental in enabling the collaboration that resulted in this study.

PY - 2005/7

Y1 - 2005/7

N2 - We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

AB - We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

KW - Domain decomposition

KW - Multigrid

KW - P-version finite element

KW - Schwarz methods

KW - Spectral element methods

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