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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

    Related Papers:

  • 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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    [1] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0
    [2] Z. Chen, Finite Element Methods and Their Applications, Springer-Verlag Berlin, 2005.
    [3] A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput. (2014) 59: 496-511.
    [4] Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. (2009) 47: 1319-1365.
    [5] Computations with finite element methods for the Brinkman problem. Comput. Geosci. (2011) 15: 155-166.
    [6] Analysis of finite element methods for the Brinkman problem. Calcolo (2010) 47: 129-147.
    [7] Numerical computations with $H$(div)-finite elements for the Brinkman problem. Comput. Geosci. (2012) 16: 139-158.
    [8] A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. (2002) 40: 1605-1631.
    [9] A computational study of the weak Galerkin method for second-order elliptic equations. Numer. Algorithms (2013) 63: 753-777.
    [10] Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods PDE (2014) 30: 1003-1029.
    [11] A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J. Comput. Phys. (2014) 273: 327-342.
    [12] A hybridized formulation for the weak Galerkin mixed finite element method. J. Comput. Appl. Math. (2016) 307: 335-345.
    [13] A $C^0$-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. (2014) 59: 473-495.
    [14] L. Mu, J. Wang, X. Ye et al, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363-386. doi: 10.1007/s10915-014-9964-4
    [15] A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg. (2010) 199: 582-597.
    [16] P.-A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, in: Lecture Notes in Math., Springer, Berlin, 606 (1977), 292-315. Technical Report LA-UR-73-0479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.
    [17] J. Wang and X. Wang, Weak Galerkin finite element methods for elliptic PDEs(in Chinese), Sci. Sin. Math., 45 (2015), 1061-1092.
    [18] A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J. Comput. Appl. Math. (2016) 307: 346-366.
    [19] Unified a posteriori error estimator for finite element methods for the Stokes equations. Int. J. Numer. Anal. Model. (2013) 10: 551-570.
    [20] A weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. (2016) 302: 171-185.
    [21] A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. (2013) 241: 103-115.
    [22] A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. (2014) 83: 2101-2126.
    [23] A weak Galerkin finite element method for the stokes equations. Adv. Comput. Math. (2016) 42: 155-174.
    [24] J. Wang, X. Ye and R. Zhang, Basics of weak Garkin finite element methods(in Chinese), Math. Numer. Sin., 38 (2016), 289-308.
    [25] The weak Galerkin method for solving the incompressible Brinkman flow. J. Comput. Appl. Math. (2016) 307: 13-24.
    [26] H. Xie, Q. Zhai and R. Zhang, The weak galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015).
    [27] Pressure recovery for weakly over-penalized discontinuous Galerkin methods for the Stokes problem. J. Sci. Comput. (2015) 63: 699-715.
    [28] A new weak Galerkin finite element scheme for the Brinkman model. Commun. Comput. Phys. (2016) 19: 1409-1434.
    [29] A hybridized weak Galerkin finite element scheme for the Stokes equations. Sci. China Math. (2015) 58: 2455-2472.
    [30] A weak finite element method for elliptic problems in one space dimension. Appl. Math. Comput. (2016) 280: 1-10.
    [31] A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput. (2015) 64: 559-585.
    [32] Weak Galerkin finite element method for second order parabolic equations. Int. J. Numer. Anal. Model. (2016) 13: 525-544.
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