American Institute of Mathematical Sciences

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doi: 10.3934/era.2020097

A weak Galerkin finite element method for nonlinear conservation laws

Xiu Ye 1,, , Shangyou Zhang 2, and  Peng Zhu 3,

1. 

Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
2. 

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
3. 

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

* Corresponding author: Xiu Ye

Received  April 2020 Revised  July 2020 Published  September 2020

Fund Project: The first author is supported in part by NSF grant DMS-1620016. The third author is supported in part by Zhejiang provincial NSF of China grant LY19A010008

A weak Galerkin (WG) finite element method is presented for nonlinear conservation laws. There are two built-in parameters in this WG framework. Different choices of the parameters will lead to different approaches for solving hyperbolic conservation laws. The convergence analysis is obtained for the forward Euler time discrete and the third order explicit TVDRK time discrete WG schemes respectively. The theoretical results are verified by numerical experiments.

Keywords: Weak Galerkin, finite element methods, weak derivative, conservation laws.
Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 35J50.
Citation: Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, doi: 10.3934/era.2020097
References:
[1]

B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. Ⅱ. General framework, Math. Comp., 52 (1989), 411-435.  doi: 10.2307/2008474.  Google Scholar

[2]

B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), 173-261.  doi: 10.1023/A:1012873910884.  Google Scholar

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S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

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M.-Y. Kim, A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions, Comput. Math. Appl., 70 (2015), 488-506.  doi: 10.1016/j.camwa.2015.05.003.  Google Scholar

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M.-Y. Kim, High order DG-DGLM method for hyperbolic conservation laws, Comput. Math. Appl., 75 (2018), 4458-4489.  doi: 10.1016/j.camwa.2018.03.043.  Google Scholar

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J. Li, X. Ye and S. Zhang, A weak Galerkin least-squares finite element method for div-curl systems, J. Comput. Phys., 363 (2018), 79-86.  doi: 10.1016/j.jcp.2018.02.036.  Google Scholar

[9]

G. Lin, J. Liu, L. Mu and X. Ye, Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity, J. Comput. Phys., 276 (2014), 422-437.  doi: 10.1016/j.jcp.2014.07.001.  Google Scholar

[10]

R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), 1482-1497.  doi: 10.1137/17M1152528.  Google Scholar

[11]

X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp., 85 (2016), 1225-1261.  doi: 10.1090/mcom/3022.  Google Scholar

[12]

L. Mu, J. Wang and X. Ye, A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35 (2015), 1228-1255.  doi: 10.1093/imanum/dru026.  Google Scholar

[13]

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

[14]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.   Google Scholar

[15]

L. Mu, J. Wang, X. Ye and S. Zhang, 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

[16]

L. Mu, J. Wang, X. Ye and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys., 325 (2016), 157-173.  doi: 10.1016/j.jcp.2016.08.024.  Google Scholar

[17]

S. Shields, J. Li and E. A. Machorro, Weak Galerkin methods for time-dependent Maxwell's equations, Comput. Math. Appl., 74 (2017), 2106-2124.  doi: 10.1016/j.camwa.2017.07.047.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

Q. Zhai, R. Zhang, N. Malluwawadu and S. Hussain, The weak Galerkin method for linear hyperbolic equation, Commun. Comput. Phys., 24 (2018), 152-166.  doi: 10.4208/cicp.oa-2017-0052.  Google Scholar

[22]

Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal., 42 (2004), 641-666.  doi: 10.1137/S0036142902404182.  Google Scholar

[23]

Q. Zhang and C.-W. Shu, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal., 48 (2010), 1038-1063.  doi: 10.1137/090771363.  Google Scholar

show all references

References:
[1]

B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. Ⅱ. General framework, Math. Comp., 52 (1989), 411-435.  doi: 10.2307/2008474.  Google Scholar

[2]

B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), 173-261.  doi: 10.1023/A:1012873910884.  Google Scholar

[3]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[4]

A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Uniformly high-order accurate essentially non-oscillatory schemes. Ⅲ, J. Comput. Phys., 71 (1987), 231-303.  doi: 10.1016/0021-9991(87)90031-3.  Google Scholar

[5]

G. S. Jiang and C.-W. Shu, On cell entropy inequality for discontinuous Galerkin methods, Math. Comp., 62 (1994), 531-538.  doi: 10.1090/S0025-5718-1994-1223232-7.  Google Scholar

[6]

M.-Y. Kim, A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions, Comput. Math. Appl., 70 (2015), 488-506.  doi: 10.1016/j.camwa.2015.05.003.  Google Scholar

[7]

M.-Y. Kim, High order DG-DGLM method for hyperbolic conservation laws, Comput. Math. Appl., 75 (2018), 4458-4489.  doi: 10.1016/j.camwa.2018.03.043.  Google Scholar

[8]

J. Li, X. Ye and S. Zhang, A weak Galerkin least-squares finite element method for div-curl systems, J. Comput. Phys., 363 (2018), 79-86.  doi: 10.1016/j.jcp.2018.02.036.  Google Scholar

[9]

G. Lin, J. Liu, L. Mu and X. Ye, Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity, J. Comput. Phys., 276 (2014), 422-437.  doi: 10.1016/j.jcp.2014.07.001.  Google Scholar

[10]

R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), 1482-1497.  doi: 10.1137/17M1152528.  Google Scholar

[11]

X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp., 85 (2016), 1225-1261.  doi: 10.1090/mcom/3022.  Google Scholar

[12]

L. Mu, J. Wang and X. Ye, A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35 (2015), 1228-1255.  doi: 10.1093/imanum/dru026.  Google Scholar

[13]

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

[14]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.   Google Scholar

[15]

L. Mu, J. Wang, X. Ye and S. Zhang, 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

[16]

L. Mu, J. Wang, X. Ye and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys., 325 (2016), 157-173.  doi: 10.1016/j.jcp.2016.08.024.  Google Scholar

[17]

S. Shields, J. Li and E. A. Machorro, Weak Galerkin methods for time-dependent Maxwell's equations, Comput. Math. Appl., 74 (2017), 2106-2124.  doi: 10.1016/j.camwa.2017.07.047.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

Q. Zhai, R. Zhang, N. Malluwawadu and S. Hussain, The weak Galerkin method for linear hyperbolic equation, Commun. Comput. Phys., 24 (2018), 152-166.  doi: 10.4208/cicp.oa-2017-0052.  Google Scholar

[22]

Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal., 42 (2004), 641-666.  doi: 10.1137/S0036142902404182.  Google Scholar

[23]

Q. Zhang and C.-W. Shu, Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal., 48 (2010), 1038-1063.  doi: 10.1137/090771363.  Google Scholar

Figure 1.  WG solution for Example 2, $ T = 1/\pi, \lambda_1 = 2,\lambda_2 = 1,N = 128 $
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Figure 2.  The $ P_1 $ WG solution and DG solution for Example 3, $ T = 2\pi, N = 512 $
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Table 1.  $ L^2 $ errors and corresponding convergence rates of Example 1. $ T = 2\pi $, $ \lambda_1 = \lambda_2 = 1 $
$ P_1 $ element $ P_2 $ element $ P_3 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 1.29E-01 3.36E-03 2.66E-04
16 3.02E-02 2.10 3.99E-04 3.08 1.94E-05 3.78
32 7.22E-03 2.06 4.93E-05 3.02 1.27E-06 3.93
64 1.78E-03 2.02 6.14E-06 3.00 8.06E-08 3.98
128 4.42E-04 2.01 7.67E-07 3.00 5.06E-09 4.00
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$ P_1 $ element $ P_2 $ element $ P_3 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 1.29E-01 3.36E-03 2.66E-04
16 3.02E-02 2.10 3.99E-04 3.08 1.94E-05 3.78
32 7.22E-03 2.06 4.93E-05 3.02 1.27E-06 3.93
64 1.78E-03 2.02 6.14E-06 3.00 8.06E-08 3.98
128 4.42E-04 2.01 7.67E-07 3.00 5.06E-09 4.00
Table 2.  $ L^2 $ errors and corresponding convergence rates of Example 2. $ T = 0.2 $, $ \lambda_1 = \lambda_2 = 2.5 $
$ P_1 $ element $ P_2 $ element $ P_3 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 1.68E-02 6.60E-03 1.89E-03
16 6.11E-03 1.46 7.86E-04 3.07 2.22E-04 3.09
32 1.42E-03 2.10 1.63E-04 2.27 9.96E-06 4.48
64 3.49E-04 2.03 2.85E-05 2.51 8.19E-07 3.60
128 8.67E-05 2.01 4.98E-06 2.51 5.81E-08 3.82
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$ P_1 $ element $ P_2 $ element $ P_3 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 1.68E-02 6.60E-03 1.89E-03
16 6.11E-03 1.46 7.86E-04 3.07 2.22E-04 3.09
32 1.42E-03 2.10 1.63E-04 2.27 9.96E-06 4.48
64 3.49E-04 2.03 2.85E-05 2.51 8.19E-07 3.60
128 8.67E-05 2.01 4.98E-06 2.51 5.81E-08 3.82
Table 3.  $ L^2 $ errors and corresponding convergence rates of Example 3. $ T = 2\pi $, $ \lambda_1 = 2,\lambda_2 = 1 $
$ P_1 $ element $ P_2 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 5.93E-01 4.23E-01
16 5.01E-01 0.24 3.25E-01 0.38
32 3.93E-01 0.35 2.52E-01 0.37
64 3.26E-01 0.27 1.98E-01 0.35
128 2.72E-01 0.26 1.58E-01 0.33
256 2.26E-01 0.27 1.27E-01 0.32
512 1.89E-01 0.26 1.03E-01 0.30
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$ P_1 $ element $ P_2 $ element
N $ \|u-u_0\| $ Rate $ \|u-u_0\| $ Rate
8 5.93E-01 4.23E-01
16 5.01E-01 0.24 3.25E-01 0.38
32 3.93E-01 0.35 2.52E-01 0.37
64 3.26E-01 0.27 1.98E-01 0.35
128 2.72E-01 0.26 1.58E-01 0.33
256 2.26E-01 0.27 1.27E-01 0.32
512 1.89E-01 0.26 1.03E-01 0.30
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