American Institute of Mathematical Sciences

Advanced
August  2021, 29(3): 2489-2516. doi: 10.3934/era.2020126

Hybridized weak Galerkin finite element methods for Brinkman equations

Jiwei Jia 1,2, , Young-Ju Lee 3, , Yue Feng 1, , Zichan Wang 1,, and  Zhongshu Zhao 4,

1. 

School of Mathematics, Jilin University, Changchun, Jilin 130012, China
2. 

National Applied Mathematical Center (Jilin), Changchun, Jilin 130012, China
3. 

Department of Mathematics, Texas State University, San Marcos, TX 78666, USA
4. 

School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Zichan Wang

Received  August 2020 Revised  November 2020 Published  August 2021 Early access  December 2020

Fund Project: The first author is supported by NSFC grant 41704116 and Jilin Provincial Excellent Youth Talents Foundation 20180520093JH

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.

Keywords: Hybridized weak Galerkin, Brinkman equation.
Mathematics Subject Classification: Primary: 65N30, 65N15, 65N12; Secondary: 35J50.
Citation: Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126
References:
[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.  Google Scholar

[2]

Z. Chen, Finite Element Methods and Their Applications, Springer-Verlag Berlin, 2005.  Google Scholar

[3]

L. Chen, J. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511.  doi: 10.1007/s10915-013-9771-3.  Google Scholar

[4]

B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), 1319-1365.  doi: 10.1137/070706616.  Google Scholar

[5]

A. Hannukainen, M. Juntunen and R. Stenberg, Computations with finite element methods for the Brinkman problem, Comput. Geosci., 15 (2011), 155-166.   Google Scholar

[6]

M. Juntunen and R. Stenberg, Analysis of finite element methods for the Brinkman problem, Calcolo, 47 (2010), 129-147.  doi: 10.1007/s10092-009-0017-6.  Google Scholar

[7]

J. Könnö and R. Stenberg, Numerical computations with $H$(div)-finite elements for the Brinkman problem, Comput. Geosci., 16 (2012), 139-158.  doi: 10.1007/s10596-011-9259-x.  Google Scholar

[8]

K. A. Mardal, X.-C. Tai and R. Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal., 40 (2002), 1605-1631.  doi: 10.1137/S0036142901383910.  Google Scholar

[9]

L. Mu, J. Wang, Y. Wang and X. Ye, A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algorithms, 63 (2013), 753-777.  doi: 10.1007/s11075-012-9651-1.  Google Scholar

[10]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods PDE, 30 (2014), 1003-1029.  doi: 10.1002/num.21855.  Google Scholar

[11]

L. Mu, J. Wang and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273 (2014), 327-342.  doi: 10.1016/j.jcp.2014.04.017.  Google Scholar

[12]

L. Mu, J. Wang and X. Ye, A hybridized formulation for the weak Galerkin mixed finite element method, J. Comput. Appl. Math., 307 (2016), 335-345.  doi: 10.1016/j.cam.2016.01.004.  Google Scholar

[13]

L. Mu, J. Wang, X. Ye and S. Zhang, A $C^0$-weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59 (2014), 473-495.  doi: 10.1007/s10915-013-9770-4.  Google Scholar

[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.  Google Scholar

[15]

N. C. Nguyen, J. Peraire and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg., 199 (2010), 582-597.  doi: 10.1016/j.cma.2009.10.007.  Google Scholar

[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.  Google Scholar

[17]

J. Wang and X. Wang, Weak Galerkin finite element methods for elliptic PDEs(in Chinese), Sci. Sin. Math., 45 (2015), 1061-1092. Google Scholar

[18]

C. Wang, J. Wang, R. Wang and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307 (2016), 346-366.  doi: 10.1016/j.cam.2015.12.015.  Google Scholar

[19]

J. Wang, Y. Wang and X. Ye, Unified a posteriori error estimator for finite element methods for the Stokes equations, Int. J. Numer. Anal. Model., 10 (2013), 551-570.   Google Scholar

[20]

R. Wang, X. Wang, Q. Zhai and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), 171-185.  doi: 10.1016/j.cam.2016.01.025.  Google Scholar

[21]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[22]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.  Google Scholar

[23]

J. Wang and X. Ye, A weak Galerkin finite element method for the stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.  Google Scholar

[24]

J. Wang, X. Ye and R. Zhang, Basics of weak Garkin finite element methods(in Chinese), Math. Numer. Sin., 38 (2016), 289-308.  Google Scholar

[25]

X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24.  doi: 10.1016/j.cam.2016.04.031.  Google Scholar

[26]

H. Xie, Q. Zhai and R. Zhang, The weak galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015). Google Scholar

[27]

M. Yang, J. Liu and Y. Lin, Pressure recovery for weakly over-penalized discontinuous Galerkin methods for the Stokes problem, J. Sci. Comput., 63 (2015), 699-715.  doi: 10.1007/s10915-014-9911-4.  Google Scholar

[28]

Q. Zhai, R. Zhang and L. Mu, A new weak Galerkin finite element scheme for the Brinkman model, Commun. Comput. Phys., 19 (2016), 1409-1434.  doi: 10.4208/cicp.scpde14.44s.  Google Scholar

[29]

Q. Zhai, R. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472.  doi: 10.1007/s11425-015-5030-4.  Google Scholar

[30]

T. Zhang and L. Tang, A weak finite element method for elliptic problems in one space dimension, Appl. Math. Comput., 280 (2016), 1-10.  doi: 10.1016/j.amc.2016.01.018.  Google Scholar

[31]

R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585.  doi: 10.1007/s10915-014-9945-7.  Google Scholar

[32]

H. Zhang, Y. Zou, Y. Xu, Q. Zhai and H. Yue, Weak Galerkin finite element method for second order parabolic equations, Int. J. Numer. Anal. Model., 13 (2016), 525-544.   Google Scholar

show all references

References:
[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.  Google Scholar

[2]

Z. Chen, Finite Element Methods and Their Applications, Springer-Verlag Berlin, 2005.  Google Scholar

[3]

L. Chen, J. Wang and X. Ye, A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 59 (2014), 496-511.  doi: 10.1007/s10915-013-9771-3.  Google Scholar

[4]

B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), 1319-1365.  doi: 10.1137/070706616.  Google Scholar

[5]

A. Hannukainen, M. Juntunen and R. Stenberg, Computations with finite element methods for the Brinkman problem, Comput. Geosci., 15 (2011), 155-166.   Google Scholar

[6]

M. Juntunen and R. Stenberg, Analysis of finite element methods for the Brinkman problem, Calcolo, 47 (2010), 129-147.  doi: 10.1007/s10092-009-0017-6.  Google Scholar

[7]

J. Könnö and R. Stenberg, Numerical computations with $H$(div)-finite elements for the Brinkman problem, Comput. Geosci., 16 (2012), 139-158.  doi: 10.1007/s10596-011-9259-x.  Google Scholar

[8]

K. A. Mardal, X.-C. Tai and R. Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal., 40 (2002), 1605-1631.  doi: 10.1137/S0036142901383910.  Google Scholar

[9]

L. Mu, J. Wang, Y. Wang and X. Ye, A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algorithms, 63 (2013), 753-777.  doi: 10.1007/s11075-012-9651-1.  Google Scholar

[10]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Methods PDE, 30 (2014), 1003-1029.  doi: 10.1002/num.21855.  Google Scholar

[11]

L. Mu, J. Wang and X. Ye, A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods, J. Comput. Phys., 273 (2014), 327-342.  doi: 10.1016/j.jcp.2014.04.017.  Google Scholar

[12]

L. Mu, J. Wang and X. Ye, A hybridized formulation for the weak Galerkin mixed finite element method, J. Comput. Appl. Math., 307 (2016), 335-345.  doi: 10.1016/j.cam.2016.01.004.  Google Scholar

[13]

L. Mu, J. Wang, X. Ye and S. Zhang, A $C^0$-weak Galerkin finite element method for the biharmonic equation, J. Sci. Comput., 59 (2014), 473-495.  doi: 10.1007/s10915-013-9770-4.  Google Scholar

[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.  Google Scholar

[15]

N. C. Nguyen, J. Peraire and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg., 199 (2010), 582-597.  doi: 10.1016/j.cma.2009.10.007.  Google Scholar

[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.  Google Scholar

[17]

J. Wang and X. Wang, Weak Galerkin finite element methods for elliptic PDEs(in Chinese), Sci. Sin. Math., 45 (2015), 1061-1092. Google Scholar

[18]

C. Wang, J. Wang, R. Wang and R. Zhang, A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation, J. Comput. Appl. Math., 307 (2016), 346-366.  doi: 10.1016/j.cam.2015.12.015.  Google Scholar

[19]

J. Wang, Y. Wang and X. Ye, Unified a posteriori error estimator for finite element methods for the Stokes equations, Int. J. Numer. Anal. Model., 10 (2013), 551-570.   Google Scholar

[20]

R. Wang, X. Wang, Q. Zhai and R. Zhang, A weak Galerkin finite element scheme for solving the stationary Stokes equations, J. Comput. Appl. Math., 302 (2016), 171-185.  doi: 10.1016/j.cam.2016.01.025.  Google Scholar

[21]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115.  doi: 10.1016/j.cam.2012.10.003.  Google Scholar

[22]

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83 (2014), 2101-2126.  doi: 10.1090/S0025-5718-2014-02852-4.  Google Scholar

[23]

J. Wang and X. Ye, A weak Galerkin finite element method for the stokes equations, Adv. Comput. Math., 42 (2016), 155-174.  doi: 10.1007/s10444-015-9415-2.  Google Scholar

[24]

J. Wang, X. Ye and R. Zhang, Basics of weak Garkin finite element methods(in Chinese), Math. Numer. Sin., 38 (2016), 289-308.  Google Scholar

[25]

X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow, J. Comput. Appl. Math., 307 (2016), 13-24.  doi: 10.1016/j.cam.2016.04.031.  Google Scholar

[26]

H. Xie, Q. Zhai and R. Zhang, The weak galerkin method for eigenvalue problems, arXiv: 1508.05304, (2015). Google Scholar

[27]

M. Yang, J. Liu and Y. Lin, Pressure recovery for weakly over-penalized discontinuous Galerkin methods for the Stokes problem, J. Sci. Comput., 63 (2015), 699-715.  doi: 10.1007/s10915-014-9911-4.  Google Scholar

[28]

Q. Zhai, R. Zhang and L. Mu, A new weak Galerkin finite element scheme for the Brinkman model, Commun. Comput. Phys., 19 (2016), 1409-1434.  doi: 10.4208/cicp.scpde14.44s.  Google Scholar

[29]

Q. Zhai, R. Zhang and X. Wang, A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 58 (2015), 2455-2472.  doi: 10.1007/s11425-015-5030-4.  Google Scholar

[30]

T. Zhang and L. Tang, A weak finite element method for elliptic problems in one space dimension, Appl. Math. Comput., 280 (2016), 1-10.  doi: 10.1016/j.amc.2016.01.018.  Google Scholar

[31]

R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), 559-585.  doi: 10.1007/s10915-014-9945-7.  Google Scholar

[32]

H. Zhang, Y. Zou, Y. Xu, Q. Zhai and H. Yue, Weak Galerkin finite element method for second order parabolic equations, Int. J. Numer. Anal. Model., 13 (2016), 525-544.   Google Scholar

Table 1.  $ \mu = 1,a = 1 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 5.63 1.06 4.67e-01 1.85
1/8 2.87 0.97 1.81e-01 2.55 2.70e-01 0.79 1.09 0.76
1/16 1.43 1.00 3.30e-02 2.45 1.39e-01 0.95 5.75e-01 0.93
1/32 6.89e-01 1.00 7.17e-03 2.20 7.01e-02 0.99 2.93e-01 0.97
1/64 7.17e-01 1.00 1.71e-03 2.06 3.51e-02 1.00 1.47e-01 0.99
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 5.63 1.06 4.67e-01 1.85
1/8 2.87 0.97 1.81e-01 2.55 2.70e-01 0.79 1.09 0.76
1/16 1.43 1.00 3.30e-02 2.45 1.39e-01 0.95 5.75e-01 0.93
1/32 6.89e-01 1.00 7.17e-03 2.20 7.01e-02 0.99 2.93e-01 0.97
1/64 7.17e-01 1.00 1.71e-03 2.06 3.51e-02 1.00 1.47e-01 0.99
Table 2.  $ \mu = 1,a = 10^4 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 4.03 1.16e-01 9.05e-01 3.89
1/8 2.24 0.85 1.94e-02 2.59 7.38e-01 0.29 2.18 0.84
1/16 1.30 0.79 7.38e-03 1.39 4.46e-01 0.73 1.08 1.01
1/32 6.89e-01 0.91 3.07e-03 1.27 2.37e-01 0.91 5.36e-01 1.01
1/64 3.53e-01 0.96 1.06e-03 1.53 1.09e-01 1.13 2.50e-01 1.10
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 4.03 1.16e-01 9.05e-01 3.89
1/8 2.24 0.85 1.94e-02 2.59 7.38e-01 0.29 2.18 0.84
1/16 1.30 0.79 7.38e-03 1.39 4.46e-01 0.73 1.08 1.01
1/32 6.89e-01 0.91 3.07e-03 1.27 2.37e-01 0.91 5.36e-01 1.01
1/64 3.53e-01 0.96 1.06e-03 1.53 1.09e-01 1.13 2.50e-01 1.10
Table 3.  $ \mu = 0.01,a = 1 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 9.87e-01 6.02e-01 7.87e-02 1.82e-01
1/8 5.06e-01 0.96 1.63e-01 1.88 5.84e-02 0.43 1.25e-01 0.54
1/16 2.47e-01 1.03 3.63e-02 2.17 3.56e-02 0.71 7.54e-02 0.73
1/32 1.22e-01 1.02 8.08e-03 2.17 1.92e-02 0.89 4.04e-02 0.90
1/64 6.06e-02 1.01 1.92e-03 2.07 9.80e-03 0.97 2.07e-02 0.97
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 9.87e-01 6.02e-01 7.87e-02 1.82e-01
1/8 5.06e-01 0.96 1.63e-01 1.88 5.84e-02 0.43 1.25e-01 0.54
1/16 2.47e-01 1.03 3.63e-02 2.17 3.56e-02 0.71 7.54e-02 0.73
1/32 1.22e-01 1.02 8.08e-03 2.17 1.92e-02 0.89 4.04e-02 0.90
1/64 6.06e-02 1.01 1.92e-03 2.07 9.80e-03 0.97 2.07e-02 0.97
Table 4.  $ \mu = 0.01,a = 10^4 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 7.33e-01 8.32e-02 1.20e-01 4.07e-01
1/8 4.41e-01 0.73 3.38e-02 1.30 9.01e-02 0.42 2.07e-01 0.98
1/16 2.36e-01 0.90 1.02e-02 1.73 4.70e-02 0.94 9.96e-02 1.06
1/32 1.20e-01 0.97 2.63e-03 1.96 2.15e-02 1.13 4.52e-02 1.14
1/64 6.04e-02 0.99 6.56e-03 2.00 1.01e-02 1.09 2.14e-02 1.08
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 7.33e-01 8.32e-02 1.20e-01 4.07e-01
1/8 4.41e-01 0.73 3.38e-02 1.30 9.01e-02 0.42 2.07e-01 0.98
1/16 2.36e-01 0.90 1.02e-02 1.73 4.70e-02 0.94 9.96e-02 1.06
1/32 1.20e-01 0.97 2.63e-03 1.96 2.15e-02 1.13 4.52e-02 1.14
1/64 6.04e-02 0.99 6.56e-03 2.00 1.01e-02 1.09 2.14e-02 1.08
Table 5.  Comparison of the degrees of freedom between the weak Galerkin finite element method based on gradient divergence and Schur complement method
$ h $ dof dof schur
1/4 8.32e+02 6.40e+02
1/8 3.26e+03 2.50e+03
1/16 1.29e+03 9.86e+03
1/32 5.15e+04 3.92e+04
1/64 2.05e+05 1.56e+05
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$ h $ dof dof schur
1/4 8.32e+02 6.40e+02
1/8 3.26e+03 2.50e+03
1/16 1.29e+03 9.86e+03
1/32 5.15e+04 3.92e+04
1/64 2.05e+05 1.56e+05
Table 6.  $ \mu = 1,a = 1 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \| {\boldsymbol{\delta}}_h \| $ order
1/4 5.79 1.21 5.54e-01 8.08e-01
1/8 2.93 0.98 2.23e-01 2.44 3.00e-01 0.89 3.08e-01 1.39
1/16 1.46 1.40 4.74e-02 2.24 1.48e-01 1.02 9.44e-02 1.71
1/32 7.32e-01 1.00 1.13e-03 2.07 7.33e-02 1.02 2.64e-02 1.84
1/64 3.66e-01 1.00 2.80e-03 1.92 3.65e-02 1.01 7.19e-03 1.87
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \| {\boldsymbol{\delta}}_h \| $ order
1/4 5.79 1.21 5.54e-01 8.08e-01
1/8 2.93 0.98 2.23e-01 2.44 3.00e-01 0.89 3.08e-01 1.39
1/16 1.46 1.40 4.74e-02 2.24 1.48e-01 1.02 9.44e-02 1.71
1/32 7.32e-01 1.00 1.13e-03 2.07 7.33e-02 1.02 2.64e-02 1.84
1/64 3.66e-01 1.00 2.80e-03 1.92 3.65e-02 1.01 7.19e-03 1.87
Table 7.  $ \mu = 1,a = 10^3 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 3.51 1.90e-01 8.61e-01 3.14
1/8 2.36 0.57 6.20e-02 1.62 6.87e-01 0.33 1.70 0.89
1/16 1.35 0.80 2.45e-02 1.34 3.76e-01 0.87 7.73e-01 1.14
1/32 7.14e-01 0.92 8.46e-03 1.54 1.59e-01 1.24 2.97e-01 1.38
1/64 3.64e-01 0.97 2.44e-03 1.79 5.76e-02 1.47 9.30e-02 1.68
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 3.51 1.90e-01 8.61e-01 3.14
1/8 2.36 0.57 6.20e-02 1.62 6.87e-01 0.33 1.70 0.89
1/16 1.35 0.80 2.45e-02 1.34 3.76e-01 0.87 7.73e-01 1.14
1/32 7.14e-01 0.92 8.46e-03 1.54 1.59e-01 1.24 2.97e-01 1.38
1/64 3.64e-01 0.97 2.44e-03 1.79 5.76e-02 1.47 9.30e-02 1.68
Table 8.  $ \mu = 0.01,a = 1 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 1.14 6.87e-01 3.45e-02 1.02e-01
1/8 6.46e-01 0.82 2.31e-01 1.57 1.70e-02 1.02 5.39e-02 0.92
1/16 3.41e-01 0.92 7.23e-02 1.67 7.52-03 1.18 2.28e-02 1.24
1/32 1.75e-01 0.96 2.05e-02 1.82 2.85e-02 1.40 8.34e-03 1.45
1/64 8.83e-02 0.98 5.46e-03 1.91 1.01e-03 1.50 2.82e-03 1.57
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 1.14 6.87e-01 3.45e-02 1.02e-01
1/8 6.46e-01 0.82 2.31e-01 1.57 1.70e-02 1.02 5.39e-02 0.92
1/16 3.41e-01 0.92 7.23e-02 1.67 7.52-03 1.18 2.28e-02 1.24
1/32 1.75e-01 0.96 2.05e-02 1.82 2.85e-02 1.40 8.34e-03 1.45
1/64 8.83e-02 0.98 5.46e-03 1.91 1.01e-03 1.50 2.82e-03 1.57
Table 9.  $ \mu = 0.01,a = 10^3 $ Error and convergence order of velocity function $ {\boldsymbol{u}} $ and pressure function $ p $
$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 1.06 3.22e-01 7.91e-02 4.07e-01
1/8 6.21e-01 0.78 1.41e-02 1.19 5.52e-02 0.51 1.60e-01 0.44
1/16 3.33e-01 0.90 5.74e-02 1.30 2.97e-02 0.90 1.18e-01 0.89
1/32 1.73e-01 0.94 1.88e-02 1.61 1.14e-02 1.38 6.36e-02 1.37
1/64 8.81e-02 0.95 5.29e-03 1.91 3.47e-03 1.93 7.56e-02 1.51
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$ h $ $ ||| {\boldsymbol{e}}_h||| $ order $ \| {\boldsymbol{e}}_h\| $ order $ \|\varepsilon_h\| $ order $ \|{\boldsymbol{\delta}}_h\| $ order
1/4 1.06 3.22e-01 7.91e-02 4.07e-01
1/8 6.21e-01 0.78 1.41e-02 1.19 5.52e-02 0.51 1.60e-01 0.44
1/16 3.33e-01 0.90 5.74e-02 1.30 2.97e-02 0.90 1.18e-01 0.89
1/32 1.73e-01 0.94 1.88e-02 1.61 1.14e-02 1.38 6.36e-02 1.37
1/64 8.81e-02 0.95 5.29e-03 1.91 3.47e-03 1.93 7.56e-02 1.51
Table 10.  Comparison of the degrees of freedom between the weak Galerkin finite element method based on gradient divergence and Schur complement method
$ h $ dof dof Schur
1/4 7.20e+02 5.28e+02
1/8 2.85e+03 2.08e+03
1/16 1.33e+03 8.26e+03
1/32 4.52e+04 3.29e+04
1/64 1.80e+05 1.31e+05
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$ h $ dof dof Schur
1/4 7.20e+02 5.28e+02
1/8 2.85e+03 2.08e+03
1/16 1.33e+03 8.26e+03
1/32 4.52e+04 3.29e+04
1/64 1.80e+05 1.31e+05
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