Electronic Research Archive

2021, Issue 3: 2489-2516. doi: 10.3934/era.2020126
Special Issues

Hybridized weak Galerkin finite element methods for Brinkman equations

• Received: 01 August 2020 Revised: 01 November 2020 Published: 14 December 2020
• Primary: 65N30, 65N15, 65N12; Secondary: 35J50

• This paper presents a hybridized weak Galerkin (HWG) finite element method for solving the Brinkman equations. Mathematically, Brinkman equations can model the Stokes and Darcy flows in a unified framework so as to describe the fluid motion in porous media with fractures. Numerical schemes for Brinkman equations, therefore, must be designed to tackle Stokes and Darcy flows at the same time. We demonstrate that HWG is capable of providing very accurate and stable numerical approximations for both Darcy and Stokes. The main features of HWG is that it approximates the differential operators by their weak forms as distributions and it introduces the Lagrange multipliers to relax certain constraints. We establish the optimal order error estimates for HWG solutions of Brinkman equations. We also present a Schur complement formulation of HWG, which reduces the systems' computational complexity significantly. A number of numerical experiments are provided to confirm the theoretical developments.

Citation: Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations[J]. Electronic Research Archive, 2021, 29(3): 2489-2516. doi: 10.3934/era.2020126

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• This paper presents a hybridized weak Galerkin (HWG) finite element method for solving the Brinkman equations. Mathematically, Brinkman equations can model the Stokes and Darcy flows in a unified framework so as to describe the fluid motion in porous media with fractures. Numerical schemes for Brinkman equations, therefore, must be designed to tackle Stokes and Darcy flows at the same time. We demonstrate that HWG is capable of providing very accurate and stable numerical approximations for both Darcy and Stokes. The main features of HWG is that it approximates the differential operators by their weak forms as distributions and it introduces the Lagrange multipliers to relax certain constraints. We establish the optimal order error estimates for HWG solutions of Brinkman equations. We also present a Schur complement formulation of HWG, which reduces the systems' computational complexity significantly. A number of numerical experiments are provided to confirm the theoretical developments.

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