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...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2015/460421 |
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| author | K. Muzhinji S. Shateyi S. S. Motsa |
| author_facet | K. Muzhinji S. Shateyi S. S. Motsa |
| author_sort | K. Muzhinji |
| collection | DOAJ |
| 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 Q2-Q1 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 | Article |
| id | doaj-art-e72f3d3d23be474b9ed9692c00114271 |
| institution | OA Journals |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-e72f3d3d23be474b9ed9692c001142712025-08-20T02:21:10ZengWileyThe Scientific World Journal2356-61401537-744X2015-01-01201510.1155/2015/460421460421The Mixed Finite Element Multigrid Method for Stokes EquationsK. Muzhinji0S. Shateyi1S. S. Motsa2Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South AfricaDepartment of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South AfricaDepartment of Mathematics, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South AfricaThe 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 Q2-Q1 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.http://dx.doi.org/10.1155/2015/460421 |
| spellingShingle | K. Muzhinji S. Shateyi S. S. Motsa The Mixed Finite Element Multigrid Method for Stokes Equations The Scientific World Journal |
| 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 |
| url | http://dx.doi.org/10.1155/2015/460421 |
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