American Institute of Mathematical Sciences

Advanced
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  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, 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
[1]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096
[2]

Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097
[3]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[4]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[5]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[6]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[7]

Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054
[8]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[9]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[10]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[11]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[12]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[13]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[14]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331
[15]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[16]

Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083
[17]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327
[18]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[19]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248
[20]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (20)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences