December  2021, 29(6): 3609-3627. doi: 10.3934/era.2021053

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

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  November 2020 Revised  June 2021 Published  December 2021 Early access  July 2021

Fund Project: The first author is supported by National Science Foundation Grant DMS-1620016

A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the $ L^2 $ norm. Optimal order error estimate for pressure in the $ L^2 $ norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

Citation: Xiu Ye, Shangyou Zhang. A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh. Electronic Research Archive, 2021, 29 (6) : 3609-3627. doi: 10.3934/era.2021053
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, Appl. Numer. Math., 150 (2020), 444-451.  doi: 10.1016/j.apnum.2019.10.009.  Google Scholar

[2]

A. Al-Taweel and X. Wang, The lowest-order stabilizer free weak Galerkin finite element method, Appl. Numer. Math., 157 (2020), 434-445.  doi: 10.1016/j.apnum.2020.06.012.  Google Scholar

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J. Qin and S. Zhang, Stability and approximability of the P1-P0 element for Stokes equations, Internat. J. Numer. Methods Fluids, 54 (2007), 497-515.  doi: 10.1002/fld.1407.  Google Scholar

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J. Qin and S. Zhang, Stability of the finite elements $9/(4c+1)$ and $9/5c$ for stationary Stokes equations, Comput. $ & $ Structures, 84 (2005), 70-77.  doi: 10.1016/j.compstruc.2005.07.002.  Google Scholar

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[11]

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

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[15]

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[16]

M. Zhang and S. Zhang, A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations, Int. J. Numer. Anal. Model., 14 (2017), 730-743.   Google Scholar

[17]

S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., 74 (2005), 543-554.  doi: 10.1090/S0025-5718-04-01711-9.  Google Scholar

[18]

S. Zhang, On the P1 Powell-Sabin divergence-free finite element for the Stokes equations, J. Comput. Math., 26 (2008), 456-470.   Google Scholar

[19]

S. Zhang, Divergence-free finite elements on tetrahedral grids for $k\geq 6$, Math. Comp., 80 (2011), 669-695.  doi: 10.1090/S0025-5718-2010-02412-3.  Google Scholar

[20]

S. Zhang, Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids, Calcolo, 48 (2011), 211-244.  doi: 10.1007/s10092-010-0035-4.  Google Scholar

[21]

S. Zhang and S. Zhang, $C_0P_2$-$P_0$ Stokes finite element pair on sub-hexahedron tetrahedral grids, Calcolo, 54 (2017), 1403-1417.  doi: 10.1007/s10092-017-0235-2.  Google Scholar

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, Appl. Numer. Math., 150 (2020), 444-451.  doi: 10.1016/j.apnum.2019.10.009.  Google Scholar

[2]

A. Al-Taweel and X. Wang, The lowest-order stabilizer free weak Galerkin finite element method, Appl. Numer. Math., 157 (2020), 434-445.  doi: 10.1016/j.apnum.2020.06.012.  Google Scholar

[3]

D. LiY. Nie and C. Wang, Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions, Comput. Math. Appl., 78 (2019), 905-928.  doi: 10.1016/j.camwa.2019.03.010.  Google Scholar

[4]

D. LiC. Wang and J. Wang, Superconvergence of the gradient approximation for weak Galerkin finite element methods on rectangular partitions, Appl. Numer. Math., 150 (2020), 396-417.  doi: 10.1016/j.apnum.2019.10.013.  Google Scholar

[5]

M. LiS. Mao and S. Zhang, New error estimates of nonconforming mixed finite element methods for the Stokes problem, Math. Methods Appl. Sci., 37 (2014), 937-951.  doi: 10.1002/mma.2849.  Google Scholar

[6]

J. LiuS. Tavener and Z. Wang, Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes, SIAM J. Sci. Comput., 40 (2018), 1229-1252.  doi: 10.1137/17M1145677.  Google Scholar

[7]

L. Mu, Pressure robust weak Galerkin finite element methods for Stokes problems, SIAM J. Sci. Comput., 42 (2020), B608-B629. doi: 10.1137/19M1266320.  Google Scholar

[8]

J. Qin and S. Zhang, Stability and approximability of the P1-P0 element for Stokes equations, Internat. J. Numer. Methods Fluids, 54 (2007), 497-515.  doi: 10.1002/fld.1407.  Google Scholar

[9]

J. Qin and S. Zhang, Stability of the finite elements $9/(4c+1)$ and $9/5c$ for stationary Stokes equations, Comput. $ & $ Structures, 84 (2005), 70-77.  doi: 10.1016/j.compstruc.2005.07.002.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅱ, J. Comput. Appl. Math., 394 (2021), 113525, 11 pp. arXiv: 2008.13631. doi: 10.1016/j.cam.2021.113525.  Google Scholar

[14]

X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅲ, J. Comput. Appl. Math., 394 (2021), 113538, 9 pp. arXiv: 2009.08536. doi: 10.1016/j.cam.2021.113538.  Google Scholar

[15]

X. Ye and S. Zhang, A stabilizer-free pressure-robust finite element method for the Stokes equations, Adv. Comput. Math., 47 (2021), Paper No. 28, 17 pp. doi: 10.1007/s10444-021-09856-9.  Google Scholar

[16]

M. Zhang and S. Zhang, A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations, Int. J. Numer. Anal. Model., 14 (2017), 730-743.   Google Scholar

[17]

S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp., 74 (2005), 543-554.  doi: 10.1090/S0025-5718-04-01711-9.  Google Scholar

[18]

S. Zhang, On the P1 Powell-Sabin divergence-free finite element for the Stokes equations, J. Comput. Math., 26 (2008), 456-470.   Google Scholar

[19]

S. Zhang, Divergence-free finite elements on tetrahedral grids for $k\geq 6$, Math. Comp., 80 (2011), 669-695.  doi: 10.1090/S0025-5718-2010-02412-3.  Google Scholar

[20]

S. Zhang, Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids, Calcolo, 48 (2011), 211-244.  doi: 10.1007/s10092-010-0035-4.  Google Scholar

[21]

S. Zhang and S. Zhang, $C_0P_2$-$P_0$ Stokes finite element pair on sub-hexahedron tetrahedral grids, Calcolo, 54 (2017), 1403-1417.  doi: 10.1007/s10092-017-0235-2.  Google Scholar

Figure 1.  The first three quadrilateral grids for the computation of Table 1
Figure 2.  The $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element (65) solution $ ( {\bf u}_1)_h $ on the fifth grid of Figure 1 (on top), its error (in middle), and the error of the $ P_1^2 $-$ P_1^2 $-$ P_1 $ WG finite element (66) solution $ ( {\bf u}_1)_h $ on the fifth grid (at bottom). Both solutions are $ P_1 $ polynomials, but the latter error is 1000 times bigger
Figure 3.  The $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element (65) solution $ ( {\bf u}_2)_h $ on the fifth grid of Figure 1 (on top), its error (in middle), and the error of the $ P_1^2 $-$ P_1^2 $-$ P_1 $ WG finite element (66) solution $ ( {\bf u}_2)_h $ on the fifth grid (at bottom)
Figure 4.  The $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element (65) solution $ p_h $ on the fifth grid of Figure 1 (on top), its error (in middle), and the error of the $ P_1^2 $-$ P_1^2 $-$ P_1 $ WG finite element (66) solution $ p_h $ on the fifth grid (at bottom)
Figure 5.  The first three polygonal grids for the computation of Table 4
Figure 6.  The graded grid for the driven cavity computation
Figure 7.  The computed driven cavity velocity field on the whole domain
Figure 8.  The graded grid for the driven cavity computation
Figure 9.  The graded grid for the driven cavity computation
Figure 10.  The graded grid for the driven cavity computation
Figure 11.  The graded grid for the driven cavity computation
Figure 12.  The graded grid for the driven cavity computation
Figure 13.  The first three levels of wedge grids used in Table 5
Table 1.  Error profiles and convergence rates for solution (64) on quadrilateral grids shown in Figure 1
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element (65)
4 0.3051E-03 3.95 0.3440E-01 3.02 0.9223E-02 2.95
5 0.1964E-04 3.96 0.4313E-02 3.00 0.1209E-02 2.93
6 0.1248E-05 3.98 0.5421E-03 2.99 0.1555E-03 2.96
by the $ P_1^2 $-$ P_1^2 $-$ P_1 $ WG finite element (66)
4 0.5450E-01 1.88 0.2828E+01 0.94 0.1912E+01 0.89
5 0.1390E-01 1.97 0.1430E+01 0.98 0.9708E+00 0.98
6 0.3492E-02 1.99 0.7171E+00 1.00 0.4873E+00 0.99
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element (65)
4 0.3051E-03 3.95 0.3440E-01 3.02 0.9223E-02 2.95
5 0.1964E-04 3.96 0.4313E-02 3.00 0.1209E-02 2.93
6 0.1248E-05 3.98 0.5421E-03 2.99 0.1555E-03 2.96
by the $ P_1^2 $-$ P_1^2 $-$ P_1 $ WG finite element (66)
4 0.5450E-01 1.88 0.2828E+01 0.94 0.1912E+01 0.89
5 0.1390E-01 1.97 0.1430E+01 0.98 0.9708E+00 0.98
6 0.3492E-02 1.99 0.7171E+00 1.00 0.4873E+00 0.99
Table 2.  Error profiles and convergence rates for solution (64) on quadrilateral grids shown in Figure 1
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element (65)
3 0.8289E-03 5.12 0.8054E-01 4.12 0.5896E-02 4.32
4 0.2507E-04 5.05 0.4871E-02 4.05 0.3609E-03 4.03
5 0.7763E-06 5.01 0.3018E-03 4.01 0.2277E-04 3.99
by the $ P_2^2 $-$ P_2^2 $-$ P_2 $ WG finite element (66)
3 0.8848E-02 3.06 0.4878E+00 2.30 0.3476E+00 1.79
4 0.1128E-02 2.97 0.1213E+00 2.01 0.8938E-01 1.96
5 0.1416E-03 2.99 0.3035E-01 2.00 0.2234E-01 2.00
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element (65)
3 0.8289E-03 5.12 0.8054E-01 4.12 0.5896E-02 4.32
4 0.2507E-04 5.05 0.4871E-02 4.05 0.3609E-03 4.03
5 0.7763E-06 5.01 0.3018E-03 4.01 0.2277E-04 3.99
by the $ P_2^2 $-$ P_2^2 $-$ P_2 $ WG finite element (66)
3 0.8848E-02 3.06 0.4878E+00 2.30 0.3476E+00 1.79
4 0.1128E-02 2.97 0.1213E+00 2.01 0.8938E-01 1.96
5 0.1416E-03 2.99 0.3035E-01 2.00 0.2234E-01 2.00
Table 3.  Error profiles and convergence rates for solution (64) on quadrilateral grids shown in Figure 1
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_3^2 $-$ P_4^2 $-$ P_4 $ WG finite element (65)
2 0.6018E-02 6.29 0.3910E+00 5.28 0.1249E-01 5.72
3 0.8806E-04 6.09 0.1146E-01 5.09 0.2933E-03 5.41
4 0.1352E-05 6.03 0.3526E-03 5.02 0.8304E-05 5.14
by the $ P_3^2 $-$ P_3^2 $-$ P_3 $ WG finite element (66)
3 0.1595E-03 5.24 0.1700E-01 4.52 0.1625E-01 0.46
4 0.8572E-05 4.22 0.1641E-02 3.37 0.2105E-02 2.95
5 0.5330E-06 4.01 0.2041E-03 3.01 0.2636E-03 3.00
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_3^2 $-$ P_4^2 $-$ P_4 $ WG finite element (65)
2 0.6018E-02 6.29 0.3910E+00 5.28 0.1249E-01 5.72
3 0.8806E-04 6.09 0.1146E-01 5.09 0.2933E-03 5.41
4 0.1352E-05 6.03 0.3526E-03 5.02 0.8304E-05 5.14
by the $ P_3^2 $-$ P_3^2 $-$ P_3 $ WG finite element (66)
3 0.1595E-03 5.24 0.1700E-01 4.52 0.1625E-01 0.46
4 0.8572E-05 4.22 0.1641E-02 3.37 0.2105E-02 2.95
5 0.5330E-06 4.01 0.2041E-03 3.01 0.2636E-03 3.00
Table 4.  Error profiles for solution (64) on polygonal grids shown in Figure 5
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_0^2 $-$ P_1^2 $-$ P_1 $ WG finite element
4 0.2202E-01 1.80 0.2885E+00 1.91 0.2138E+00 1.97
5 0.5715E-02 1.95 0.7374E-01 1.97 0.5376E-01 1.99
6 0.1442E-02 1.99 0.1861E-01 1.99 0.1351E-01 1.99
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element
4 0.2512E-03 3.86 0.2922E-01 2.94 0.8673E-02 2.93
5 0.1661E-04 3.92 0.3737E-02 2.97 0.1147E-02 2.92
6 0.1066E-05 3.96 0.4735E-03 2.98 0.1481E-03 2.95
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element
3 0.5373E-03 5.10 0.5945E-01 4.16 0.5018E-02 4.11
4 0.1639E-04 5.04 0.3567E-02 4.06 0.3219E-03 3.96
5 0.5101E-06 5.01 0.2208E-03 4.01 0.2063E-04 3.96
by the $ P_3^2 $-$ P_4^2 $-$ P_4 $ WG finite element
2 0.3384E-02 6.26 0.2770E+00 5.30 0.8473E-02 5.54
3 0.4985E-04 6.08 0.8069E-02 5.10 0.2273E-03 5.22
4 0.7855E-06 5.99 0.2525E-03 5.00 0.6876E-05 5.05
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_0^2 $-$ P_1^2 $-$ P_1 $ WG finite element
4 0.2202E-01 1.80 0.2885E+00 1.91 0.2138E+00 1.97
5 0.5715E-02 1.95 0.7374E-01 1.97 0.5376E-01 1.99
6 0.1442E-02 1.99 0.1861E-01 1.99 0.1351E-01 1.99
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element
4 0.2512E-03 3.86 0.2922E-01 2.94 0.8673E-02 2.93
5 0.1661E-04 3.92 0.3737E-02 2.97 0.1147E-02 2.92
6 0.1066E-05 3.96 0.4735E-03 2.98 0.1481E-03 2.95
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element
3 0.5373E-03 5.10 0.5945E-01 4.16 0.5018E-02 4.11
4 0.1639E-04 5.04 0.3567E-02 4.06 0.3219E-03 3.96
5 0.5101E-06 5.01 0.2208E-03 4.01 0.2063E-04 3.96
by the $ P_3^2 $-$ P_4^2 $-$ P_4 $ WG finite element
2 0.3384E-02 6.26 0.2770E+00 5.30 0.8473E-02 5.54
3 0.4985E-04 6.08 0.8069E-02 5.10 0.2273E-03 5.22
4 0.7855E-06 5.99 0.2525E-03 5.00 0.6876E-05 5.05
Table 5.  Error profiles for solution (67) on wedge grids shown in Figure 13
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_0^2 $-$ P_1^2 $-$ P_1 $ WG finite element
4 0.8167E-01 1.59 0.1864E+01 1.83 0.5772E+00 1.78
5 0.2228E-01 1.87 0.4851E+00 1.94 0.1575E+00 1.87
6 0.5689E-02 1.97 0.1228E+00 1.98 0.3776E-01 2.06
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element
3 0.6428E-01 3.50 0.4486E+01 2.52 0.6305E+00 3.23
4 0.4636E-02 3.79 0.6105E+00 2.88 0.8163E-01 2.95
5 0.2856E-03 4.02 0.7796E-01 2.97 0.9492E-02 3.10
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element
2 0.7217E+00 3.28 0.3793E+02 1.89 0.2623E+01 5.54
3 0.2563E-01 4.82 0.2898E+01 3.71 0.2215E+00 3.57
4 0.8352E-03 4.94 0.1942E+00 3.90 0.1439E-01 3.94
Grid $ \|Q_h {\bf u}- {\bf u}_h \|_0 $ rate $ {{|||}} Q_h {\bf u}- {\bf u}_h {{|||}} $ rate $ \|p - p_h \|_0 $ rate
by the $ P_0^2 $-$ P_1^2 $-$ P_1 $ WG finite element
4 0.8167E-01 1.59 0.1864E+01 1.83 0.5772E+00 1.78
5 0.2228E-01 1.87 0.4851E+00 1.94 0.1575E+00 1.87
6 0.5689E-02 1.97 0.1228E+00 1.98 0.3776E-01 2.06
by the $ P_1^2 $-$ P_2^2 $-$ P_2 $ WG finite element
3 0.6428E-01 3.50 0.4486E+01 2.52 0.6305E+00 3.23
4 0.4636E-02 3.79 0.6105E+00 2.88 0.8163E-01 2.95
5 0.2856E-03 4.02 0.7796E-01 2.97 0.9492E-02 3.10
by the $ P_2^2 $-$ P_3^2 $-$ P_3 $ WG finite element
2 0.7217E+00 3.28 0.3793E+02 1.89 0.2623E+01 5.54
3 0.2563E-01 4.82 0.2898E+01 3.71 0.2215E+00 3.57
4 0.8352E-03 4.94 0.1942E+00 3.90 0.1439E-01 3.94
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