Volume 36, Issue 6
An Over-Penalized Weak Galerkin Method for Second-Order Elliptic Problems

J. Comp. Math., 36 (2018), pp. 866-880.

Published online: 2018-08

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• Abstract

The weak Galerkin (WG) finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin (DG) methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak functions defined on the same edge. We introduce an over-penalized weak Galerkin (OPWG) method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in $H^1$ and $L^2$ norms are established for the finite element ($\mathbb{P}_k(K)$, $\mathbb{P}_k(e)$, $RT_k(K)$). In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CG, and BICGSTAB iterative methods.

• Keywords

Weak Galerkin, Over-penalized, Finite element methods, Second-order elliptic equation.

65N15, 65N30

liukf17@lzu.edu.cn (Kaifang Liu)

song@lzu.edu.cn (Lunji Song)

zhoushf14@lzu.edu.cn (Shuangfeng Zhou)

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@Article{JCM-36-866, author = {Kaifang and Liu and liukf17@lzu.edu.cn and 6747 and School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China and Kaifang Liu and Lunji and Song and song@lzu.edu.cn and 6748 and School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China and Lunji Song and Shuangfeng and Zhou and zhoushf14@lzu.edu.cn and 6746 and School of Mathematics and Statistics, and Key Laboratory of Applied Mathematics and Complex Systems (Gansu Province), Lanzhou University, Lanzhou 730000, China and Shuangfeng Zhou}, title = {An Over-Penalized Weak Galerkin Method for Second-Order Elliptic Problems}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {6}, pages = {866--880}, abstract = {

The weak Galerkin (WG) finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin (DG) methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak functions defined on the same edge. We introduce an over-penalized weak Galerkin (OPWG) method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in $H^1$ and $L^2$ norms are established for the finite element ($\mathbb{P}_k(K)$, $\mathbb{P}_k(e)$, $RT_k(K)$). In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CG, and BICGSTAB iterative methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1705-m2016-0744}, url = {http://global-sci.org/intro/article_detail/jcm/12606.html} }
TY - JOUR T1 - An Over-Penalized Weak Galerkin Method for Second-Order Elliptic Problems AU - Liu , Kaifang AU - Song , Lunji AU - Zhou , Shuangfeng JO - Journal of Computational Mathematics VL - 6 SP - 866 EP - 880 PY - 2018 DA - 2018/08 SN - 36 DO - http://doi.org/10.4208/jcm.1705-m2016-0744 UR - https://global-sci.org/intro/article_detail/jcm/12606.html KW - Weak Galerkin, Over-penalized, Finite element methods, Second-order elliptic equation. AB -

The weak Galerkin (WG) finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin (DG) methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak functions defined on the same edge. We introduce an over-penalized weak Galerkin (OPWG) method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in $H^1$ and $L^2$ norms are established for the finite element ($\mathbb{P}_k(K)$, $\mathbb{P}_k(e)$, $RT_k(K)$). In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CG, and BICGSTAB iterative methods.

Kaifang Liu, Lunji Song & Shuangfeng Zhou. (2020). An Over-Penalized Weak Galerkin Method for Second-Order Elliptic Problems. Journal of Computational Mathematics. 36 (6). 866-880. doi:10.4208/jcm.1705-m2016-0744
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