Cargando…

The Mixed Finite Element Multigrid Method for Stokes Equations

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates...

Descripción completa

Detalles Bibliográficos
Autores principales: Muzhinji, K., Shateyi, S., Motsa, S. S.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Hindawi Publishing Corporation 2015
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4405300/
https://www.ncbi.nlm.nih.gov/pubmed/25945361
http://dx.doi.org/10.1155/2015/460421
_version_ 1782367619649634304
author Muzhinji, K.
Shateyi, S.
Motsa, S. S.
author_facet Muzhinji, K.
Shateyi, S.
Motsa, S. S.
author_sort Muzhinji, K.
collection PubMed
description The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q (2)-Q (1) pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.
format Online
Article
Text
id pubmed-4405300
institution National Center for Biotechnology Information
language English
publishDate 2015
publisher Hindawi Publishing Corporation
record_format MEDLINE/PubMed
spelling pubmed-44053002015-05-05 The Mixed Finite Element Multigrid Method for Stokes Equations Muzhinji, K. Shateyi, S. Motsa, S. S. ScientificWorldJournal Research Article The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q (2)-Q (1) pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results. Hindawi Publishing Corporation 2015 2015-04-07 /pmc/articles/PMC4405300/ /pubmed/25945361 http://dx.doi.org/10.1155/2015/460421 Text en Copyright © 2015 K. Muzhinji et al. https://creativecommons.org/licenses/by/3.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
spellingShingle Research Article
Muzhinji, K.
Shateyi, S.
Motsa, S. S.
The Mixed Finite Element Multigrid Method for Stokes Equations
title The Mixed Finite Element Multigrid Method for Stokes Equations
title_full The Mixed Finite Element Multigrid Method for Stokes Equations
title_fullStr The Mixed Finite Element Multigrid Method for Stokes Equations
title_full_unstemmed The Mixed Finite Element Multigrid Method for Stokes Equations
title_short The Mixed Finite Element Multigrid Method for Stokes Equations
title_sort mixed finite element multigrid method for stokes equations
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4405300/
https://www.ncbi.nlm.nih.gov/pubmed/25945361
http://dx.doi.org/10.1155/2015/460421
work_keys_str_mv AT muzhinjik themixedfiniteelementmultigridmethodforstokesequations
AT shateyis themixedfiniteelementmultigridmethodforstokesequations
AT motsass themixedfiniteelementmultigridmethodforstokesequations
AT muzhinjik mixedfiniteelementmultigridmethodforstokesequations
AT shateyis mixedfiniteelementmultigridmethodforstokesequations
AT motsass mixedfiniteelementmultigridmethodforstokesequations