Volume 7, Issue 3
A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

East Asian J. Appl. Math., 7 (2017), pp. 508-529.

Published online: 2018-02

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• Abstract

A robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the $L^2$-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

• Keywords

The Stokes equations, weak Galerkin method, a posteriori error estimator.

• AMS Subject Headings

65N15, 65N30, 76D07

• BibTex
• RIS
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@Article{EAJAM-7-508, author = {}, title = {A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {3}, pages = {508--529}, abstract = {

A robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the $L^2$-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.221216.250417a}, url = {http://global-sci.org/intro/article_detail/eajam/10762.html} }
TY - JOUR T1 - A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem JO - East Asian Journal on Applied Mathematics VL - 3 SP - 508 EP - 529 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.221216.250417a UR - https://global-sci.org/intro/article_detail/eajam/10762.html KW - The Stokes equations, weak Galerkin method, a posteriori error estimator. AB -

A robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the $L^2$-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

Xiaobo Zheng & Xiaoping Xie. (2020). A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem. East Asian Journal on Applied Mathematics. 7 (3). 508-529. doi:10.4208/eajam.221216.250417a
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