Convergence analysis of V-Cycle multigrid methods for anisotropic elliptic equations

Yongke Wu; Long Chen; Xiaoping Xie; Jinchao Xu

doi:10.1093/imanum/drr043

Convergence analysis of V-Cycle multigrid methods for anisotropic elliptic equations

Yongke Wu
, Long Chen
, Xiaoping Xie
, Jinchao Xu

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

9 Scopus citations

Abstract

Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the 'regularity assumption' is not used in the analysis, the results can be extended to general domains consisting of rectangles.

Original language	English (US)
Pages (from-to)	1329-1347
Number of pages	19
Journal	IMA Journal of Numerical Analysis
Volume	32
Issue number	4
DOIs	https://doi.org/10.1093/imanum/drr043
State	Published - Oct 2012

All Science Journal Classification (ASJC) codes

General Mathematics
Computational Mathematics
Applied Mathematics

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10.1093/imanum/drr043

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@article{9395c17749e1421189b926cc85a1bb0e,

title = "Convergence analysis of V-Cycle multigrid methods for anisotropic elliptic equations",

abstract = "Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the 'regularity assumption' is not used in the analysis, the results can be extended to general domains consisting of rectangles.",

author = "Yongke Wu and Long Chen and Xiaoping Xie and Jinchao Xu",

note = "Funding Information: The National Natural Science Foundation of China (11171239 to Y.W. and X.X.); National Science Foundation (NSF) (DMS-0811272 and DMS-1115961 to L.C.), UC Irvine 2009-2010 Academic Senate Council on Research, Computing and Libraries (CORCL) to L.C.; NSF (DMS-0749202 and DMS-0915153 to J.X.). Funding Information: We would like to thank the reviewers for their helpful comments and suggestions, which led to the removal of | log h| factor in the semicoarsening case. We would also like to acknowledge the support from Beijing International Center for Mathematical Research (BICMR) and the Center for Computational Mathematics and Applications (CCMA) in the Department of Mathematics at Penn State. Part of collaboration is carried through the visiting of these two places.",

year = "2012",

month = oct,

doi = "10.1093/imanum/drr043",

language = "English (US)",

volume = "32",

pages = "1329--1347",

journal = "IMA Journal of Numerical Analysis",

issn = "0272-4979",

publisher = "Oxford University Press",

number = "4",

}

TY - JOUR

T1 - Convergence analysis of V-Cycle multigrid methods for anisotropic elliptic equations

AU - Wu, Yongke

AU - Chen, Long

AU - Xie, Xiaoping

AU - Xu, Jinchao

N1 - Funding Information: The National Natural Science Foundation of China (11171239 to Y.W. and X.X.); National Science Foundation (NSF) (DMS-0811272 and DMS-1115961 to L.C.), UC Irvine 2009-2010 Academic Senate Council on Research, Computing and Libraries (CORCL) to L.C.; NSF (DMS-0749202 and DMS-0915153 to J.X.). Funding Information: We would like to thank the reviewers for their helpful comments and suggestions, which led to the removal of | log h| factor in the semicoarsening case. We would also like to acknowledge the support from Beijing International Center for Mathematical Research (BICMR) and the Center for Computational Mathematics and Applications (CCMA) in the Department of Mathematics at Penn State. Part of collaboration is carried through the visiting of these two places.

PY - 2012/10

Y1 - 2012/10

N2 - Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the 'regularity assumption' is not used in the analysis, the results can be extended to general domains consisting of rectangles.

AB - Fast multigrid solvers are considered for the linear systems arising from the bilinear finite element discretizations of second-order elliptic equations with anisotropic diffusion. Optimal convergence of Vcycle multigrid methods in the semicoarsening case and nearly optimal convergence of V-cycle multigrid method with line smoothing in the uniformly-coarsening case are established using the Xu-Zikatanov identity. Since the 'regularity assumption' is not used in the analysis, the results can be extended to general domains consisting of rectangles.

UR - https://www.scopus.com/pages/publications/84867507394

UR - https://www.scopus.com/pages/publications/84867507394#tab=citedBy

U2 - 10.1093/imanum/drr043

DO - 10.1093/imanum/drr043

M3 - Article

AN - SCOPUS:84867507394

SN - 0272-4979

VL - 32

SP - 1329

EP - 1347

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

IS - 4

ER -

Convergence analysis of V-Cycle multigrid methods for anisotropic elliptic equations

Abstract

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