American Institute of Mathematical Sciences

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August  2021, 26(8): 4131-4145. doi: 10.3934/dcdsb.2020277

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

Xiu Ye 1, and  Shangyou Zhang 2,,

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

* Corresponding author: Shangyou Zhang

Received  April 2020 Revised  August 2020 Published  August 2021 Early access  September 2020

Fund Project: The first author was supported in part by National Science Foundation Grant DMS-1620016

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $ H^1 $ norm and the standard $ L^2 $ norm. The numerical examples are tested on various meshes and confirm the theory.

Keywords: weak Galerkin, finite element methods, weak gradient, second-order elliptic problems, stabilizer free.
Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 35J50.
Citation: Xiu Ye, Shangyou Zhang. A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4131-4145. doi: 10.3934/dcdsb.2020277
References:
[1]

A. Al-Taweel and X. Wang, A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method, Applied Numerical Mathematics, 150 (2020), 444-451.  doi: 10.1016/j.apnum.2019.10.009.

[2]

M. Cui and S. Zhang, On the uniform convergence of the weak Galerkin finite element for a singularly-perturbed biharmonic equation, J. Sci. Comput., 82 (2020), Paper No. 5, 15 pp. doi: 10.1007/s10915-019-01120-z.

[3]

X. Hu, L. Mu and X. Ye, A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes, J. of Computational and Applied Mathematics, 362 (2019), 614-625.  doi: 10.1016/j.cam.2018.08.022.

[4]

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

[5]

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

[6]

L. Mu, J. Wang and X. Ye, weak Galerkin finite element method for the Helmholtz equation with large wave number on polytopal meshes, IMA J. Numer. Anal., 35 (2015), 1228-1255.  doi: 10.1093/imanum/dru026.

[7]

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

[8]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method for biharmonic equations on polytopal meshes, Numer. Meth. Partial Diff. Eq., 30 (2014), 1003-1029.  doi: 10.1002/num.21855.

[9]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.  doi: 10.1007/s10915-014-9964-4.

[10]

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.

[11]

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

[12]

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.

[13]

C. Wang and J. Wang, Discretization of div–curl systems by weak Galerkin finite element methods on polyhedral partitions, J. Sci. Comput., 68 (2016), 1144-1171.  doi: 10.1007/s10915-016-0176-y.

[14]

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.

[15]

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.

[16]

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

[17]

J. Wang, X. Ye and S. Zhang, Numerical investigation on weak Galerkin finite elements, Int. J. Numer. Anal. Model., 17 (2020), 517-531. 

[18]

X. Wang, Q. Zhai, R. Zhang and S. Zhang, The weak Galerkin finite element method for solving the time-dependent integro-differential equations, Adv. Appl. Math. Mech., 12 (2020), 164-188.  doi: 10.4208/aamm.OA-2019-0088.

[19]

X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math., 371 (2020), 112699, 9pp, arXiv: 1906.06634. doi: 10.1016/j.cam.2019.112699.

[20]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J of Numerical Analysis and Modeling, 17 (2020), 110–117, arXiv: 1904.03331.

[21]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part II, Int. J of Numerical Analysis and Modeling, 17 (2020), 281–296, arXiv: 1907.01397.

[22]

X. Ye, S. Zhang and Y. Zhu, Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs, Results in Applied Mathematics, 8 (2020), 100097. doi: 10.1016/j.rinam.2020.100097.

show all references

References:
[1]

A. Al-Taweel and X. Wang, A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method, Applied Numerical Mathematics, 150 (2020), 444-451.  doi: 10.1016/j.apnum.2019.10.009.

[2]

M. Cui and S. Zhang, On the uniform convergence of the weak Galerkin finite element for a singularly-perturbed biharmonic equation, J. Sci. Comput., 82 (2020), Paper No. 5, 15 pp. doi: 10.1007/s10915-019-01120-z.

[3]

X. Hu, L. Mu and X. Ye, A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes, J. of Computational and Applied Mathematics, 362 (2019), 614-625.  doi: 10.1016/j.cam.2018.08.022.

[4]

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

[5]

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

[6]

L. Mu, J. Wang and X. Ye, weak Galerkin finite element method for the Helmholtz equation with large wave number on polytopal meshes, IMA J. Numer. Anal., 35 (2015), 1228-1255.  doi: 10.1093/imanum/dru026.

[7]

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

[8]

L. Mu, J. Wang and X. Ye, A weak Galerkin finite element method for biharmonic equations on polytopal meshes, Numer. Meth. Partial Diff. Eq., 30 (2014), 1003-1029.  doi: 10.1002/num.21855.

[9]

L. Mu, J. Wang and X. Ye, Weak Galerkin finite element method for second-order elliptic problems on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), 31-53.  doi: 10.1007/s10915-014-9964-4.

[10]

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.

[11]

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

[12]

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.

[13]

C. Wang and J. Wang, Discretization of div–curl systems by weak Galerkin finite element methods on polyhedral partitions, J. Sci. Comput., 68 (2016), 1144-1171.  doi: 10.1007/s10915-016-0176-y.

[14]

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.

[15]

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.

[16]

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

[17]

J. Wang, X. Ye and S. Zhang, Numerical investigation on weak Galerkin finite elements, Int. J. Numer. Anal. Model., 17 (2020), 517-531. 

[18]

X. Wang, Q. Zhai, R. Zhang and S. Zhang, The weak Galerkin finite element method for solving the time-dependent integro-differential equations, Adv. Appl. Math. Mech., 12 (2020), 164-188.  doi: 10.4208/aamm.OA-2019-0088.

[19]

X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math., 371 (2020), 112699, 9pp, arXiv: 1906.06634. doi: 10.1016/j.cam.2019.112699.

[20]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J of Numerical Analysis and Modeling, 17 (2020), 110–117, arXiv: 1904.03331.

[21]

X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part II, Int. J of Numerical Analysis and Modeling, 17 (2020), 281–296, arXiv: 1907.01397.

[22]

X. Ye, S. Zhang and Y. Zhu, Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs, Results in Applied Mathematics, 8 (2020), 100097. doi: 10.1016/j.rinam.2020.100097.

Figure 5.1.  Example 1. The first three triangular grids
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Figure 5.2.  Example 2. The first three triangular grids
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Figure 5.3.  Example 3. The first three rectangular grids
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Figure 5.4.  Example 4. The first three polygonal grids
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Table 1.1.  Weak gradient calculated by (1.4), $ {{|||}}\cdot{{|||}} = O(h^{r_1}) $ and $ \|\cdot\| = O(h^{r_2}) $
$ P_{k}(T) $ $ P_{k-1}(e) $ $ [P_{k+1}(T)]^d $ $ r_1 $ $ r_2 $
$ P_1(T) $ $ P_0(e) $ $ [P_2(T)]^2 $ $ 0 $ $ 0 $
$ P_2(T) $ $ P_1(e) $ $ [P_3(T)]^2 $ $ 1 $ $ 2 $
$ P_3(T) $ $ P_2(e) $ $ [P_4(T)]^2 $ $ 2 $ $ 3 $
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$ P_{k}(T) $ $ P_{k-1}(e) $ $ [P_{k+1}(T)]^d $ $ r_1 $ $ r_2 $
$ P_1(T) $ $ P_0(e) $ $ [P_2(T)]^2 $ $ 0 $ $ 0 $
$ P_2(T) $ $ P_1(e) $ $ [P_3(T)]^2 $ $ 1 $ $ 2 $
$ P_3(T) $ $ P_2(e) $ $ [P_4(T)]^2 $ $ 2 $ $ 3 $
Table 1.2.  Weak gradient calculated by (2.3), $ {{|||}}\cdot{{|||}} = O(h^{r_1}) $ and $ \|\cdot\| = O(h^{r_2}) $
$ P_{k}(T) $ $ P_{k-1}(e) $ $ [P_{k+1}(T)]^d $ $ r_1 $ $ r_2 $
$ P_1(T) $ $ P_0(e) $ $ [P_2(T)]^2 $ $ 1 $ $ 2 $
$ P_2(T) $ $ P_1(e) $ $ [P_3(T)]^2 $ $ 2 $ $ 3 $
$ P_3(T) $ $ P_2(e) $ $ [P_4(T)]^2 $ $ 3 $ $ 4 $
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$ P_{k}(T) $ $ P_{k-1}(e) $ $ [P_{k+1}(T)]^d $ $ r_1 $ $ r_2 $
$ P_1(T) $ $ P_0(e) $ $ [P_2(T)]^2 $ $ 1 $ $ 2 $
$ P_2(T) $ $ P_1(e) $ $ [P_3(T)]^2 $ $ 2 $ $ 3 $
$ P_3(T) $ $ P_2(e) $ $ [P_4(T)]^2 $ $ 3 $ $ 4 $
Table 5.1.  Example 1. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on triangular grids shown in Figure 5.1
$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.3871E-01 1.00 0.3306E-03 1.99
1 7 0.1937E-01 1.00 0.8279E-04 2.00
8 0.9685E-02 1.00 0.2070E-04 2.00
6 0.4131E-03 1.98 0.1783E-05 2.95
2 7 0.1038E-03 1.99 0.2268E-06 2.97
8 0.2602E-04 2.00 0.2859E-07 2.99
5 0.2925E-04 2.99 0.1515E-06 3.98
3 6 0.3665E-05 3.00 0.9518E-08 3.99
7 0.4587E-06 3.00 0.5963E-09 4.00
5 0.4091E-06 3.99 0.1592E-08 4.97
4 6 0.2568E-07 3.99 0.5026E-10 4.99
7 0.1608E-08 4.00 0.1610E-11 4.96
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$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.3871E-01 1.00 0.3306E-03 1.99
1 7 0.1937E-01 1.00 0.8279E-04 2.00
8 0.9685E-02 1.00 0.2070E-04 2.00
6 0.4131E-03 1.98 0.1783E-05 2.95
2 7 0.1038E-03 1.99 0.2268E-06 2.97
8 0.2602E-04 2.00 0.2859E-07 2.99
5 0.2925E-04 2.99 0.1515E-06 3.98
3 6 0.3665E-05 3.00 0.9518E-08 3.99
7 0.4587E-06 3.00 0.5963E-09 4.00
5 0.4091E-06 3.99 0.1592E-08 4.97
4 6 0.2568E-07 3.99 0.5026E-10 4.99
7 0.1608E-08 4.00 0.1610E-11 4.96
Table 5.2.  Example 2. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on rectangular grids shown in Figure 5.2
$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.120E-02 2.01 0.141E-01 2.00
1 7 0.300E-03 2.00 0.354E-02 2.00
8 0.749E-04 2.00 0.885E-03 2.00
6 0.5734E-03 2.00 0.1482E-05 3.00
2 7 0.1434E-03 2.00 0.1852E-06 3.00
8 0.3584E-04 2.00 0.2314E-07 3.00
5 0.3645E-04 3.00 0.1360E-06 3.99
3 6 0.4559E-05 3.00 0.8517E-08 4.00
7 0.5699E-06 3.00 0.5326E-09 4.00
5 0.4222E-06 4.00 0.9119E-09 5.01
4 6 0.2639E-07 4.00 0.2846E-10 5.00
7 0.1650E-08 4.00 0.1110E-11 4.68
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$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.120E-02 2.01 0.141E-01 2.00
1 7 0.300E-03 2.00 0.354E-02 2.00
8 0.749E-04 2.00 0.885E-03 2.00
6 0.5734E-03 2.00 0.1482E-05 3.00
2 7 0.1434E-03 2.00 0.1852E-06 3.00
8 0.3584E-04 2.00 0.2314E-07 3.00
5 0.3645E-04 3.00 0.1360E-06 3.99
3 6 0.4559E-05 3.00 0.8517E-08 4.00
7 0.5699E-06 3.00 0.5326E-09 4.00
5 0.4222E-06 4.00 0.9119E-09 5.01
4 6 0.2639E-07 4.00 0.2846E-10 5.00
7 0.1650E-08 4.00 0.1110E-11 4.68
Table 5.3.  Example 3. The $ P_k-P_{k-1}-[P_{k+1}]^2 $ element, on rectangular grids shown in Figure 5.3
$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.141E-01 2.00 0.120E-02 2.01
1 7 0.354E-02 2.00 0.300E-03 2.00
8 0.885E-03 2.00 0.749E-04 2.00
6 0.158E-03 3.00 0.113E-05 4.00
2 7 0.197E-04 3.00 0.709E-07 4.00
8 0.246E-05 3.00 0.444E-08 4.00
4 0.251E-03 4.80 0.409E-05 5.28
3 5 0.143E-04 4.13 0.128E-06 4.99
6 0.889E-06 4.01 0.407E-08 4.98
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$ k $ $ {{\mathcal T}}_l $ $ {{|||}} Q_hu-u_h{{|||}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.141E-01 2.00 0.120E-02 2.01
1 7 0.354E-02 2.00 0.300E-03 2.00
8 0.885E-03 2.00 0.749E-04 2.00
6 0.158E-03 3.00 0.113E-05 4.00
2 7 0.197E-04 3.00 0.709E-07 4.00
8 0.246E-05 3.00 0.444E-08 4.00
4 0.251E-03 4.80 0.409E-05 5.28
3 5 0.143E-04 4.13 0.128E-06 4.99
6 0.889E-06 4.01 0.407E-08 4.98
Table 5.4.  Example 4. The $ P_k-P_{k-1}-[P_{k+2}]^2 $ element, on polygonal grids shown in Figure 5.4
$ k $ $ {{\mathcal T}}_l $ $ {{|\hspace{-.02in}|\hspace{-.02in}|}} Q_hu-u_h{{|\hspace{-.02in}|\hspace{-.02in}|}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.3735E-01 1.00 0.7856E-04 2.00
1 7 0.1868E-01 1.00 0.1966E-04 2.00
8 0.9339E-02 1.00 0.4916E-05 2.00
5 0.1504E-02 1.98 0.3242E-05 2.95
2 6 0.3782E-03 1.99 0.4131E-06 2.97
7 0.9482E-04 2.00 0.5216E-07 2.99
4 0.1267E-03 2.97 0.4106E-06 3.95
3 5 0.1600E-04 2.99 0.2636E-07 3.96
6 0.2010E-05 2.99 0.1673E-08 3.98
2 0.5517E-03 3.98 0.5905E-05 5.26
4 3 0.3518E-04 3.97 0.1699E-06 5.12
4 0.2234E-05 3.98 0.5253E-08 5.02
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$ k $ $ {{\mathcal T}}_l $ $ {{|\hspace{-.02in}|\hspace{-.02in}|}} Q_hu-u_h{{|\hspace{-.02in}|\hspace{-.02in}|}} $ Rate $ \|Q_hu-u_h\| $ Rate
6 0.3735E-01 1.00 0.7856E-04 2.00
1 7 0.1868E-01 1.00 0.1966E-04 2.00
8 0.9339E-02 1.00 0.4916E-05 2.00
5 0.1504E-02 1.98 0.3242E-05 2.95
2 6 0.3782E-03 1.99 0.4131E-06 2.97
7 0.9482E-04 2.00 0.5216E-07 2.99
4 0.1267E-03 2.97 0.4106E-06 3.95
3 5 0.1600E-04 2.99 0.2636E-07 3.96
6 0.2010E-05 2.99 0.1673E-08 3.98
2 0.5517E-03 3.98 0.5905E-05 5.26
4 3 0.3518E-04 3.97 0.1699E-06 5.12
4 0.2234E-05 3.98 0.5253E-08 5.02
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