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Viscosity robust weak Galerkin finite element methods for Stokes problems
A weak Galerkin finite element method for nonlinear conservation laws
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |

N | Rate | Rate | 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 |
N | Rate | Rate | 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 |
N | Rate | Rate | 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 |
N | Rate | Rate | 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 |
N | Rate | 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 |
N | Rate | 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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