The Modified Weak Galerkin Finite Element Method for Solving Brinkman Equations
Received:September 17, 2019  Revised:October 25, 2019
Key Words: the Brinkman equations   the modified weak Galerkin finite element method   discrete weak gradient
Fund Project:Supported by the Natural National Science Foundation of China (Grant Nos.91630201; U1530116; 11726102; 11771179; 93K172018Z01; 11701210; JJKH20180113KJ; 20190103029JH), the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University.
 Author Name Affiliation Li-na SUN Department of Mathematics, Jilin University, Jilin 130012, P. R. China Yue FENG Department of Mathematics, Jilin University, Jilin 130012, P. R. China Yuanyuan LIU Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100083, P. R. China Ran ZHANG Department of Mathematics, Jilin University, Jilin 130012, P. R. China
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A modified weak Galerkin (MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree $k$ and $k-1$ are used to approximate the velocity $\textbf{\textit{u}}$ and the pressure $p$, respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than $k-1$. The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal order error estimates are established for the velocity and pressure approximations in $H^1$ and $L^2$ norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.