Volume 20, Issue 2
Multigrid Methods for the Generalized Stokes Equations Based on Mixed Finite Element Methods

J. Comp. Math., 20 (2002), pp. 129-152.

Published online: 2002-04

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Multigrid methods are developed and analyzed for the generalized stationary Stokes equations which are discretized by various mixed finite element methods. In this paper, the multigrid algorithm, the criterion for prolongation operators and the convergence analysis are all established in an abstract and element-independent fashion. It is proven that the multigrid algorithm converges optimally if the prolongation operator satisfies the criterion. To utilize the abstract result, more than ten well-known mixed finite elements for the Stokes problems are discussed in detail and examples of prolongation operators are constructed explicitly. For nonconforming elements, it is shown that the usual local averaging technique for constructing prolongation operators can be replaced by a computationally cheaper alternative, random choice technique. Moreover, since the algorithm and analysis allows using of nonnested meshes, the abstract result also applies to low order mixed finite elements, which are usually stable only for some special mesh structures.

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@Article{JCM-20-129, author = {Deng , Qing-Ping and Feng , Xiao-Ping}, title = {Multigrid Methods for the Generalized Stokes Equations Based on Mixed Finite Element Methods}, journal = {Journal of Computational Mathematics}, year = {2002}, volume = {20}, number = {2}, pages = {129--152}, abstract = {

Multigrid methods are developed and analyzed for the generalized stationary Stokes equations which are discretized by various mixed finite element methods. In this paper, the multigrid algorithm, the criterion for prolongation operators and the convergence analysis are all established in an abstract and element-independent fashion. It is proven that the multigrid algorithm converges optimally if the prolongation operator satisfies the criterion. To utilize the abstract result, more than ten well-known mixed finite elements for the Stokes problems are discussed in detail and examples of prolongation operators are constructed explicitly. For nonconforming elements, it is shown that the usual local averaging technique for constructing prolongation operators can be replaced by a computationally cheaper alternative, random choice technique. Moreover, since the algorithm and analysis allows using of nonnested meshes, the abstract result also applies to low order mixed finite elements, which are usually stable only for some special mesh structures.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8905.html} }
TY - JOUR T1 - Multigrid Methods for the Generalized Stokes Equations Based on Mixed Finite Element Methods AU - Deng , Qing-Ping AU - Feng , Xiao-Ping JO - Journal of Computational Mathematics VL - 2 SP - 129 EP - 152 PY - 2002 DA - 2002/04 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8905.html KW - Generalized Stokes equations, Mixed methods, Multigrid methods. AB -

Multigrid methods are developed and analyzed for the generalized stationary Stokes equations which are discretized by various mixed finite element methods. In this paper, the multigrid algorithm, the criterion for prolongation operators and the convergence analysis are all established in an abstract and element-independent fashion. It is proven that the multigrid algorithm converges optimally if the prolongation operator satisfies the criterion. To utilize the abstract result, more than ten well-known mixed finite elements for the Stokes problems are discussed in detail and examples of prolongation operators are constructed explicitly. For nonconforming elements, it is shown that the usual local averaging technique for constructing prolongation operators can be replaced by a computationally cheaper alternative, random choice technique. Moreover, since the algorithm and analysis allows using of nonnested meshes, the abstract result also applies to low order mixed finite elements, which are usually stable only for some special mesh structures.

Qing-Ping Deng & Xiao-Ping Feng. (1970). Multigrid Methods for the Generalized Stokes Equations Based on Mixed Finite Element Methods. Journal of Computational Mathematics. 20 (2). 129-152. doi:
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