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Volume 43, Issue 1
A Stabilizer Free Weak Galerkin Finite Element Method for Brinkman Equations

Haoning Dang, Qilong Zhai, Ran Zhang & Hui Peng

DOI: 10.4208/jcm.2307-m2022-0264

J. Comp. Math., 43 (2025), pp. 1-17.

Published online: 2024-11

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  • Abstract

We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and $L^2$ norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.

  • Keywords

Brinkman equations, Weak Galerkin method, Stabilizer free, Discrete weak differential operators.

  • AMS Subject Headings

65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-1, author = {Dang , HaoningZhai , QilongZhang , Ran and Peng , Hui}, title = {A Stabilizer Free Weak Galerkin Finite Element Method for Brinkman Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {1}, pages = {1--17}, abstract = {

We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and $L^2$ norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2307-m2022-0264}, url = {http://global-sci.org/intro/article_detail/jcm/23527.html} }
TY - JOUR T1 - A Stabilizer Free Weak Galerkin Finite Element Method for Brinkman Equations AU - Dang , Haoning AU - Zhai , Qilong AU - Zhang , Ran AU - Peng , Hui JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 17 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2307-m2022-0264 UR - https://global-sci.org/intro/article_detail/jcm/23527.html KW - Brinkman equations, Weak Galerkin method, Stabilizer free, Discrete weak differential operators. AB -

We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and $L^2$ norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.

Dang , HaoningZhai , QilongZhang , Ran and Peng , Hui. (2024). A Stabilizer Free Weak Galerkin Finite Element Method for Brinkman Equations. Journal of Computational Mathematics. 43 (1). 1-17. doi:10.4208/jcm.2307-m2022-0264
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