A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids

Yan Yang; Xiu Ye; Shangyou Zhang; Yan Yang; Xiu Ye; Shangyou Zhang

doi:10.3934/era.2024158

Electronic Research Archive

2024, Volume 32, Issue 5: 3413-3432. doi: 10.3934/era.2024158

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A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids

1.
School of Sciences, Southwest Petroleum University, Chengdu 610500, China
2.
Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
3.
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received: 13 April 2024 Revised: 10 May 2024 Accepted: 21 May 2024 Published: 27 May 2024

A pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the $ L^2 $ norm and the $ H^1 $ norm. The optimal-order error convergence has been proved for the pressure in the $ L^2 $ norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.
- stabilizer-free,
- weak Galerkin,
- finite element,
- Stokes equations,
- pressure robust,
- tetrahedral meshes
Citation: Yan Yang, Xiu Ye, Shangyou Zhang. A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids[J]. Electronic Research Archive, 2024, 32(5): 3413-3432. doi: 10.3934/era.2024158

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Abstract

A pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the $ L^2 $ norm and the $ H^1 $ norm. The optimal-order error convergence has been proved for the pressure in the $ L^2 $ norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.

References

[1]	A. Al-Twaeel, S. Hussian, X. Wang, A stabilizer free weak Galerkin finite element method for parabolic equation, J. Comput. Appl. Math., 392 (2021), 113373. https://doi.org/10.1016/j.cam.2020.113373 doi: 10.1016/j.cam.2020.113373
[2]	H. Zhang, Y. Zou, S. Chai, H. Yue, Weak Galerkin method with $(r, r-1, r-1)$-order finite elements for second order parabolic equations, Appl. Math. Comput., 275 (2016), 24–40. https://doi.org/10.1016/j.amc.2015.11.046 doi: 10.1016/j.amc.2015.11.046
[3]	S. Zhou, F. Gao, B. Li, Z. Sun, Weak galerkin finite element method with second-order accuracy in time for parabolic problems, Appl. Math. Lett., 90 (2019), 118–123. https://doi.org/10.1016/j.aml.2018.10.023 doi: 10.1016/j.aml.2018.10.023
[4]	A. AL-Taweel, X. Wang, X. Ye, S. Zhang, A stabilizer free weak Galerkin method with supercloseness of order two, Numer. Methods Partial Differ. Equations, 37 (2021), 1012–1029. https://doi.org/10.1002/num.22564 doi: 10.1002/num.22564
[5]	G. Chen, M. Feng, X. Xie, A robust WG finite element method for convection-diffusion-reaction equations, J. Comput. Appl. Math., 315 (2017), 107–125. https://doi.org/10.1016/j.cam.2016.10.029 doi: 10.1016/j.cam.2016.10.029
[6]	R. Lin, X. Ye, S. Zhang, P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), 1482–1497. https://doi.org/10.1137/17M1152528 doi: 10.1137/17M1152528
[7]	J. Burkardt, M. Gunzburger, W. Zhao, High-precision computation of the weak Galerkin methods for the fourth-order problem, Numer. Algorithms, 84 (2020), 181–205. https://doi.org/10.1007/s11075-019-00751-5 doi: 10.1007/s11075-019-00751-5
[8]	M. Cui, S. Zhang, On the uniform convergence of the weak Galerkin finite element method for a singularly-perturbed biharmonic equation, J. Sci. Comput., 82 (2020), 5–15.
[9]	Y. Feng, Y. Liu, R. Wang, S. Zhang, A conforming discontinuous Galerkin finite element method on rectangular partitions, Electron. Res. Arch., 29 (2021), 2375–2389. https://doi.org/10.3934/era.2020120 doi: 10.3934/era.2020120
[10]	X. Ye, S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. Model., 17 (2020), 110–117.
[11]	X. Ye, S. Zhang, A conforming discontinuous Galerkin finite element method: part Ⅱ, Int. J Numer. Anal. Model., 17 (2020), 281–296.
[12]	W. Zhao, Higher order weak Galerkin methods for the Navier-Stokes equations with large Reynolds number, Numer. Methods Partial Differ. Equations, 38 (2022), 1967–1992. https://doi.org/10.1002/num.22852 doi: 10.1002/num.22852
[13]	X. Hu, L. Mu, X. Ye, A weak Galerkin finite element method for the Navier-Stokes equations on polytopal meshes, J. Comput. Appl. Math., 362 (2019), 614–625. https://doi.org/10.1016/j.cam.2018.08.022 doi: 10.1016/j.cam.2018.08.022
[14]	J. Zhang, K. Zhang, J. Li, X. Wang, A weak Galerkin finite element method for the Navier-Stokes equations, Commun. Comput. Phys., 23 (2018), 706–746. https://doi.org/10.4208/cicp.OA-2016-0267 doi: 10.4208/cicp.OA-2016-0267
[15]	W. Huang, Y. Wang, Discrete maximum principle for the weak Galerkin method for anisotropic diffusion problems, Commun. Comput. Phys., 18 (2015), 65–90. https://doi.org/10.4208/cicp.180914.121214a doi: 10.4208/cicp.180914.121214a
[16]	G. Li, Y. Chen, Y. huang, A new weak Galerkin finite element scheme for general second-order elliptic problems, J. Comput. Appl. Math., 344 (2018), 701–715. https://doi.org/10.1016/j.cam.2018.05.021 doi: 10.1016/j.cam.2018.05.021
[17]	J. Wang, X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103–115. https://doi.org/10.1016/j.cam.2012.10.003 doi: 10.1016/j.cam.2012.10.003
[18]	H. Li, L. Mu, X. Ye, Interior energy estimates for the weak Galerkin finite element method, Numer. Math., 139 (2018), 447–478. https://doi.org/10.1007/s00211-017-0940-4 doi: 10.1007/s00211-017-0940-4
[19]	J. Liu, S. Tavener, Z. Wang, Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes, SIAM J. Sci. Comput., 40 (2018), 1229–1252. https://doi.org/10.1137/17M1145677 doi: 10.1137/17M1145677
[20]	J. Liu, S. Tavener, Z. Wang, The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes, J. Comput. Phys., 359 (2018), 312–330. https://doi.org/10.1016/j.jcp.2018.01.001 doi: 10.1016/j.jcp.2018.01.001
[21]	X. Liu, J. Li, Z. Chen, A weak Galerkin finite element method for the Oseen equations, Adv. Comput. Math., 42 (2016), 1473–1490. https://doi.org/10.1007/s10444-016-9471-2 doi: 10.1007/s10444-016-9471-2
[22]	L. Mu, X. Ye, S. Zhang, A stabilizer free, pressure robust and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh, SIAM J. Sci. Comput., 43 (2021), A2614–A2637. https://doi.org/10.1137/20M1380405 doi: 10.1137/20M1380405
[23]	L. Mu, X. Ye, S. Zhang, Development of pressure-robust discontinuous Galerkin finite element methods for the Stokes problem, J. Sci. Comput., 89 (2021), 26. https://doi.org/10.1007/s10915-021-01634-5 doi: 10.1007/s10915-021-01634-5
[24]	X. Ye, S. Zhang, A stabilizer-free pressure-robust finite element method for the Stokes equations, Adv. Comput. Math., 47 (2021), 28. https://doi.org/10.1007/s10444-021-09856-9 doi: 10.1007/s10444-021-09856-9
[25]	S. Shields, J. Li, E. A. Machorro, Weak Galerkin methods for time-dependent Maxwell's equations, Comput. Math. Appl., 74 (2017), 2106–2124. https://doi.org/10.1016/j.camwa.2017.07.047 doi: 10.1016/j.camwa.2017.07.047
[26]	J. Wang, Q. Zhai, R. Zhang, S. Zhang, A weak Galerkin finite element scheme for the Cahn-Hilliard equation, Math. Comput., 88 (2019), 211–235. https://doi.org/10.1090/mcom/3369 doi: 10.1090/mcom/3369
[27]	C. Wang, J. Wang, Discretization of div-curl systems by weak Galerkin finite element methods on polyhedral partitions, J. Sci. Comput., 68 (2016), 1144–1171. https://doi.org/10.1007/s10915-016-0176-y doi: 10.1007/s10915-016-0176-y
[28]	Y. Xie, L. Zhong, Convergence of adaptive weak Galerkin finite element methods for second order elliptic problems, J. Sci. Comput., 86 (2021), 17. https://doi.org/10.1007/s10915-020-01387-7 doi: 10.1007/s10915-020-01387-7
[29]	T. Zhang, T. Lin, A posteriori error estimate for a modified weak Galerkin method solving elliptic problems, Numer. Methods Partial Differ. Equations, 33 (2017), 381–398. https://doi.org/10.1002/num.22114 doi: 10.1002/num.22114
[30]	X. Ye, S. Zhang, A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes, SIAM J. Numer. Anal., 58 (2020), 2572–2588. https://doi.org/10.1137/19M1276601 doi: 10.1137/19M1276601
[31]	X. Ye, S. Zhang, A conforming DG method for the biharmonic equation on polytopal meshes, Int. J. Numer. Anal. Model., 20 (2023), 855–869. https://doi.org/10.4208/ijnam2023-1037 doi: 10.4208/ijnam2023-1037
[32]	X. Ye, S. Zhang, A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes, Appl. Numer. Math., 178 (2022), 155–165. https://doi.org/10.1016/j.apnum.2022.03.017 doi: 10.1016/j.apnum.2022.03.017
[33]	X. Ye, S. Zhang, A $C^0$-conforming DG finite element method for biharmonic equations on triangle/tetrahedron, J. Numer. Math., 30 (2022), 163–172. https://doi.org/10.1515/jnma-2021-0012 doi: 10.1515/jnma-2021-0012
[34]	X. Ye, S. Zhang, A numerical scheme with divergence free H-div triangular finite element for the Stokes equations, Appl. Numer. Math., 167 (2021), 211–217. https://doi.org/10.1016/j.apnum.2021.05.005 doi: 10.1016/j.apnum.2021.05.005
[35]	S. Zhang, P. Zhu, BDM H(div) weak Galerkin finite element methods for Stokes equations, Appl. Numer. Math., 197 (2024), 307–321. https://doi.org/10.1016/j.apnum.2023.11.021 doi: 10.1016/j.apnum.2023.11.021
[36]	X. Ye, S. Zhang, Achieving superconvergence by one-dimensional discontinuous finite elements: the CDG method, East Asian J. Appl. Math., 12 (2022), 781–790. https://doi.org/10.4208/eajam.121021.200122 doi: 10.4208/eajam.121021.200122
[37]	X. Ye, S. Zhang, Order two superconvergence of the CDG finite elements on triangular and tetrahedral meshes, CSIAM Trans. Appl. Math., 4 (2023), 256–274. https://doi.org/10.4208/csiam-am.SO-2021-0051 doi: 10.4208/csiam-am.SO-2021-0051
[38]	X. Ye, S. Zhang, Order two superconvergence of the CDG method for the Stokes equations on triangle/tetrahedron, J. Appl. Anal. Comput., 12 (2022), 2578–2592. https://doi.org/10.11948/20220112 doi: 10.11948/20220112
[39]	K. L. A. Kirk, S. Rhebergen, Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier-Stokes equations, J. Sci. Comput., 81 (2019), 881–897. https://doi.org/10.1007/s10915-019-01040-y doi: 10.1007/s10915-019-01040-y
[40]	J. Wang, X. Wang, X. Ye, Finite element methods for the Navier-Stokes equations by H(div) elements, J. Comput. Math., 26 (2008), 410–436.
[41]	X. Ye, S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: part Ⅲ, J. Comput. Appl. Math., 394 (2021), 113538. https://doi.org/10.1016/j.cam.2021.113538 doi: 10.1016/j.cam.2021.113538

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