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doi: 10.3934/era.2020120

A conforming discontinuous Galerkin finite element method on rectangular partitions

Yue Feng 1, , Yujie Liu 2, , Ruishu Wang 1,, and  Shangyou Zhang 3,

1. 

Department of Mathematics, Jilin University, Changchun, China
2. 

Artificial Intelligence Research Center, Peng Cheng Laboratory, Shenzhen 518005, China
3. 

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

* Corresponding author: Ruishu Wang

Received  December 2019 Revised  September 2020 Published  November 2020

Fund Project: The research of the first author was supported in part by China Natural National Science Foundation (91630201, U1530116, 11726102, 11771179, 93K172018Z01, 11701210, JJKH20180113KJ, 20190103029JH), and by the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China.
The research of Liu was partially supported by China Natural National Science Foundation (No. 12001306), Guangdong Provincial Natural Science Foundation (No. 2017A030310285).
The research of Wang was partially supported by China Natural National Science Foundation (No. 12001230), Postdoctoral Research Fund (No. 2019M661199, BX20190142)

This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.

Keywords: Conforming discontinuous Galerkin finite element method, second order elliptic problem, rectangular partitions, error estimates.
Mathematics Subject Classification: Primary:65N15, 65N30;Secondary:35B45, 35J50.
Citation: 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
References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[2]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[3]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31 (1977), 45-59.  doi: 10.1090/S0025-5718-1977-0431742-5.  Google Scholar

[4]

L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.  Google Scholar

[5]

F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, 8 (1974), 129-151.   Google Scholar

[6]

P. Chatzipantelidis, A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions, Numer. Math., 82 (1999), 409-432.  doi: 10.1007/s002110050425.  Google Scholar

[7]

G. Chen, W. Hu, J. Shen, J. R. Singler, Y. Zhang and X. Zheng, An HDG method for distributed control of convection diffusion PDEs, J. Comput. Appl. Math., 343 (2018), 643-661.  doi: 10.1016/j.cam.2018.05.028.  Google Scholar

[8]

X. Feng and T. Lewis, Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations, J. Sci. Comput., 59 (2014), 129-157.  doi: 10.1007/s10915-013-9763-3.  Google Scholar

[9]

X. Feng, M. Neilan and S. Schenake, Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs, J. Sci. Comput., 74 (2018), 1651-1676.  doi: 10.1007/s10915-017-0519-3.  Google Scholar

[10]

K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes, J. Comput. Phys., 230 (2011), 305-328.  doi: 10.1016/j.jcp.2010.09.007.  Google Scholar

[11]

Y. Liu and J. Wang, Simplified weak Galerkin and new finite difference schemes for the Stokes equation, J. Comput. Appl. Math., 361 (2019), 176-206.  doi: 10.1016/j.cam.2019.04.024.  Google Scholar

[12]

Y. Liu and J. Wang, A locking-free $ P_0 $ finite element method for linear elasticity equations on polytopal partitions, preprint, arXiv: 1911.08728, 2019. Google Scholar

[13]

L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, Numerical Solution of Partial Differential Equations: Theory, Algorithms, and their Applications, in: Springer Proceedings in Mathematics and Statistics, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13.  Google Scholar

[14]

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

[15]

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., vol. 606, Springer-Verlag, New York, 1977. Technical Report LA-UR-73-0479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.  Google Scholar

[16]

M. Stynes, Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements, Math. Comp., 83 (2014), 2675-2689. doi: 10.1090/S0025-5718-2014-02826-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. and Model., 17 (2020), 110-117. arXiv: 1904.03331.  Google Scholar

[22]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part Ⅱ, arXiv: 1907.01397. Google Scholar

[23]

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]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2001/02), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[2]

I. Babuška, The finite element method with penalty, Math. Comp., 27 (1973), 221-228.  doi: 10.1090/S0025-5718-1973-0351118-5.  Google Scholar

[3]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31 (1977), 45-59.  doi: 10.1090/S0025-5718-1977-0431742-5.  Google Scholar

[4]

L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), 199-214.  doi: 10.1142/S0218202512500492.  Google Scholar

[5]

F. Brezzi, On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers, RAIRO, 8 (1974), 129-151.   Google Scholar

[6]

P. Chatzipantelidis, A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions, Numer. Math., 82 (1999), 409-432.  doi: 10.1007/s002110050425.  Google Scholar

[7]

G. Chen, W. Hu, J. Shen, J. R. Singler, Y. Zhang and X. Zheng, An HDG method for distributed control of convection diffusion PDEs, J. Comput. Appl. Math., 343 (2018), 643-661.  doi: 10.1016/j.cam.2018.05.028.  Google Scholar

[8]

X. Feng and T. Lewis, Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations, J. Sci. Comput., 59 (2014), 129-157.  doi: 10.1007/s10915-013-9763-3.  Google Scholar

[9]

X. Feng, M. Neilan and S. Schenake, Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs, J. Sci. Comput., 74 (2018), 1651-1676.  doi: 10.1007/s10915-017-0519-3.  Google Scholar

[10]

K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes, J. Comput. Phys., 230 (2011), 305-328.  doi: 10.1016/j.jcp.2010.09.007.  Google Scholar

[11]

Y. Liu and J. Wang, Simplified weak Galerkin and new finite difference schemes for the Stokes equation, J. Comput. Appl. Math., 361 (2019), 176-206.  doi: 10.1016/j.cam.2019.04.024.  Google Scholar

[12]

Y. Liu and J. Wang, A locking-free $ P_0 $ finite element method for linear elasticity equations on polytopal partitions, preprint, arXiv: 1911.08728, 2019. Google Scholar

[13]

L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, Numerical Solution of Partial Differential Equations: Theory, Algorithms, and their Applications, in: Springer Proceedings in Mathematics and Statistics, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13.  Google Scholar

[14]

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

[15]

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., vol. 606, Springer-Verlag, New York, 1977. Technical Report LA-UR-73-0479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.  Google Scholar

[16]

M. Stynes, Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements, Math. Comp., 83 (2014), 2675-2689. doi: 10.1090/S0025-5718-2014-02826-3.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. and Model., 17 (2020), 110-117. arXiv: 1904.03331.  Google Scholar

[22]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part Ⅱ, arXiv: 1907.01397. Google Scholar

[23]

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

Figure 1.  The first three grids used in the computation
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Table 1.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and $ P_k $ conforming DG spaces
level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
6 0.1996E-02 1.97 0.8887E-02 1.98 1024
7 0.5013E-03 1.99 0.2228E-02 2.00 4096
8 0.1255E-03 2.00 0.5574E-03 2.00 16384
by $ P_1 $ conforming discontinuous Galerkin elements
6 0.2427E-02 1.97 0.1027E+00 1.02 3072
7 0.6100E-03 1.99 0.5105E-01 1.01 12288
8 0.1527E-03 2.00 0.2546E-01 1.00 49152
by $ P_2 $ conforming discontinuous Galerkin elements
5 0.1533E-03 3.00 0.2042E-01 2.03 1536
6 0.1915E-04 3.00 0.5061E-02 2.01 6144
7 0.2394E-05 3.00 0.1260E-02 2.01 24576
by $ P_3 $ conforming discontinuous Galerkin elements
5 0.7959E-05 4.00 0.1965E-02 3.00 2560
6 0.4971E-06 4.00 0.2451E-03 3.00 10240
7 0.3140E-07 3.98 0.3059E-04 3.00 40960
by $ P_4 $ conforming discontinuous Galerkin elements
4 0.1055E-04 4.97 0.1421E-02 4.05 960
5 0.3314E-06 4.99 0.8735E-04 4.02 3840
6 0.1057E-07 4.97 0.5417E-05 4.01 15360
by $ P_5 $ conforming discontinuous Galerkin elements
2 0.2835E-02 6.24 0.1450E+00 5.49 84
3 0.4532E-04 5.97 0.4718E-02 4.94 336
4 0.7115E-06 5.99 0.1478E-03 5.00 1344
Table Options

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level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
6 0.1996E-02 1.97 0.8887E-02 1.98 1024
7 0.5013E-03 1.99 0.2228E-02 2.00 4096
8 0.1255E-03 2.00 0.5574E-03 2.00 16384
by $ P_1 $ conforming discontinuous Galerkin elements
6 0.2427E-02 1.97 0.1027E+00 1.02 3072
7 0.6100E-03 1.99 0.5105E-01 1.01 12288
8 0.1527E-03 2.00 0.2546E-01 1.00 49152
by $ P_2 $ conforming discontinuous Galerkin elements
5 0.1533E-03 3.00 0.2042E-01 2.03 1536
6 0.1915E-04 3.00 0.5061E-02 2.01 6144
7 0.2394E-05 3.00 0.1260E-02 2.01 24576
by $ P_3 $ conforming discontinuous Galerkin elements
5 0.7959E-05 4.00 0.1965E-02 3.00 2560
6 0.4971E-06 4.00 0.2451E-03 3.00 10240
7 0.3140E-07 3.98 0.3059E-04 3.00 40960
by $ P_4 $ conforming discontinuous Galerkin elements
4 0.1055E-04 4.97 0.1421E-02 4.05 960
5 0.3314E-06 4.99 0.8735E-04 4.02 3840
6 0.1057E-07 4.97 0.5417E-05 4.01 15360
by $ P_5 $ conforming discontinuous Galerkin elements
2 0.2835E-02 6.24 0.1450E+00 5.49 84
3 0.4532E-04 5.97 0.4718E-02 4.94 336
4 0.7115E-06 5.99 0.1478E-03 5.00 1344
Table 2.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and $ Q_k $ conforming DG spaces
level $ \|u_h- Q_hu\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ Q_1 $ conforming discontinuous Galerkin elements
6 0.4006E-03 1.99 0.2389E-02 1.99 4096
7 0.1003E-03 2.00 0.5982E-03 2.00 16384
8 0.2510E-04 2.00 0.1496E-03 2.00 65536
by $ Q_2 $ conforming discontinuous Galerkin elements
6 0.2360E-04 2.99 0.3186E-02 1.99 9216
7 0.2953E-05 3.00 0.7976E-03 2.00 36864
8 0.3692E-06 3.00 0.1995E-03 2.00 147456
by $ Q_3 $ conforming discontinuous Galerkin elements
5 0.1413E-04 4.08 0.1650E-02 2.97 4096
6 0.8676E-06 4.03 0.2072E-03 2.99 16384
7 0.5398E-07 4.01 0.2593E-04 3.00 65536
by $ Q_4 $ conforming discontinuous Galerkin elements
3 0.2226E-02 4.59 0.5414E-01 3.52 400
4 0.9610E-04 4.53 0.3723E-02 3.86 1600
5 0.3279E-05 4.87 0.2392E-03 3.96 6400
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level $ \|u_h- Q_hu\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ Q_1 $ conforming discontinuous Galerkin elements
6 0.4006E-03 1.99 0.2389E-02 1.99 4096
7 0.1003E-03 2.00 0.5982E-03 2.00 16384
8 0.2510E-04 2.00 0.1496E-03 2.00 65536
by $ Q_2 $ conforming discontinuous Galerkin elements
6 0.2360E-04 2.99 0.3186E-02 1.99 9216
7 0.2953E-05 3.00 0.7976E-03 2.00 36864
8 0.3692E-06 3.00 0.1995E-03 2.00 147456
by $ Q_3 $ conforming discontinuous Galerkin elements
5 0.1413E-04 4.08 0.1650E-02 2.97 4096
6 0.8676E-06 4.03 0.2072E-03 2.99 16384
7 0.5398E-07 4.01 0.2593E-04 3.00 65536
by $ Q_4 $ conforming discontinuous Galerkin elements
3 0.2226E-02 4.59 0.5414E-01 3.52 400
4 0.9610E-04 4.53 0.3723E-02 3.86 1600
5 0.3279E-05 4.87 0.2392E-03 3.96 6400
Table 3.  Error profiles and convergence rates for test case (46) obtained with uniform grids and $ P_0 $ conforming DG spaces
level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
3 0.8265E-02 1.06 0.4577E-01 1.14 16
4 0.2772E-02 1.58 0.1732E-01 1.40 64
5 0.7965E-03 1.80 0.6331E-02 1.45 256
6 0.2142E-03 1.90 0.2290E-02 1.47 1024
7 0.5564E-04 1.94 0.8213E-03 1.48 4096
8 0.1419E-04 1.97 0.2928E-03 1.49 16384
Table Options

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level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
3 0.8265E-02 1.06 0.4577E-01 1.14 16
4 0.2772E-02 1.58 0.1732E-01 1.40 64
5 0.7965E-03 1.80 0.6331E-02 1.45 256
6 0.2142E-03 1.90 0.2290E-02 1.47 1024
7 0.5564E-04 1.94 0.8213E-03 1.48 4096
8 0.1419E-04 1.97 0.2928E-03 1.49 16384
Table 4.  Error profiles and convergence rates for test case (47) obtained with uniform grids and $ P_0 $ conforming DG spaces
level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
3 0.4929E-02 0.97 0.5371E-01 0.80 16
4 0.1917E-02 1.36 0.2401E-01 1.16 64
5 0.6004E-03 1.67 0.9407E-02 1.35 256
6 0.1682E-03 1.84 0.3507E-02 1.42 1024
7 0.4457E-04 1.92 0.1275E-02 1.46 4096
8 0.1148E-04 1.96 0.4576E-03 1.48 16384
Table Options

Download as excel

level $ \|u_h- Q_h u\|_0 $ rate $ {|\!|\!|} u_h- Q_h u{|\!|\!|} $ rate $ \#Dof $
by $ P_0 $ conforming discontinuous Galerkin elements
3 0.4929E-02 0.97 0.5371E-01 0.80 16
4 0.1917E-02 1.36 0.2401E-01 1.16 64
5 0.6004E-03 1.67 0.9407E-02 1.35 256
6 0.1682E-03 1.84 0.3507E-02 1.42 1024
7 0.4457E-04 1.92 0.1275E-02 1.46 4096
8 0.1148E-04 1.96 0.4576E-03 1.48 16384
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