Volume 11, Issue 4
A Modified Weak Galerkin Method for Stokes Equations

Adv. Appl. Math. Mech., 11 (2019), pp. 890-910.

Published online: 2019-06

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• Abstract

In this paper, we modify the weak Galerkin method introduced in [15] for Stokes equations. The modified method uses the $\mathbb{P}_k/\mathbb{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for velocity and pressure in the interior of elements. Especially, the numerical traces ${v}_{hb}$ which are defined in the interface of the elements belong to the space $C^0(\mathcal{E}_h)$, this change leads to less degree of freedom for the resultant linear system. The stability, priori error estimates and $L^2$ error estimates for velocity are proved in this paper. In addition, we prove that the modified method also yields globally divergence-free velocity approximations and has uniform error estimates with respect to the Reynolds number. Finally, numerical results illustrate the performance of the method, support the theoretical properties of the estimator and show the efficiency of the algorithm.

• Keywords

Weak Galerkin, Stokes equations, globally divergence-free, less degree of freedom, uniform error estimates.

65M60, 65N30

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@Article{AAMM-11-890, author = {Zhang , LiFeng , Minfu and Zhang , Jian}, title = {A Modified Weak Galerkin Method for Stokes Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {4}, pages = {890--910}, abstract = {

In this paper, we modify the weak Galerkin method introduced in [15] for Stokes equations. The modified method uses the $\mathbb{P}_k/\mathbb{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for velocity and pressure in the interior of elements. Especially, the numerical traces ${v}_{hb}$ which are defined in the interface of the elements belong to the space $C^0(\mathcal{E}_h)$, this change leads to less degree of freedom for the resultant linear system. The stability, priori error estimates and $L^2$ error estimates for velocity are proved in this paper. In addition, we prove that the modified method also yields globally divergence-free velocity approximations and has uniform error estimates with respect to the Reynolds number. Finally, numerical results illustrate the performance of the method, support the theoretical properties of the estimator and show the efficiency of the algorithm.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0138}, url = {http://global-sci.org/intro/article_detail/aamm/13193.html} }
TY - JOUR T1 - A Modified Weak Galerkin Method for Stokes Equations AU - Zhang , Li AU - Feng , Minfu AU - Zhang , Jian JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 890 EP - 910 PY - 2019 DA - 2019/06 SN - 11 DO - http://doi.org/10.4208/aamm.OA-2018-0138 UR - https://global-sci.org/intro/article_detail/aamm/13193.html KW - Weak Galerkin, Stokes equations, globally divergence-free, less degree of freedom, uniform error estimates. AB -

In this paper, we modify the weak Galerkin method introduced in [15] for Stokes equations. The modified method uses the $\mathbb{P}_k/\mathbb{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for velocity and pressure in the interior of elements. Especially, the numerical traces ${v}_{hb}$ which are defined in the interface of the elements belong to the space $C^0(\mathcal{E}_h)$, this change leads to less degree of freedom for the resultant linear system. The stability, priori error estimates and $L^2$ error estimates for velocity are proved in this paper. In addition, we prove that the modified method also yields globally divergence-free velocity approximations and has uniform error estimates with respect to the Reynolds number. Finally, numerical results illustrate the performance of the method, support the theoretical properties of the estimator and show the efficiency of the algorithm.

Li Zhang, Minfu Feng & Jian Zhang. (2019). A Modified Weak Galerkin Method for Stokes Equations. Advances in Applied Mathematics and Mechanics. 11 (4). 890-910. doi:10.4208/aamm.OA-2018-0138
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