Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes

Chunmei Wang; Chunmei Wang

doi:10.3934/era.2025072

Electronic Research Archive

2025, Volume 33, Issue 3: 1523-1540. doi: 10.3934/era.2025072

Previous Article Next Article

Research article Special Issues

Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes

Chunmei Wang ^,

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Received: 17 December 2024 Revised: 23 February 2025 Accepted: 03 March 2025 Published: 19 March 2025

This paper presents a simplified weak Galerkin (WG) finite element method for solving biharmonic equations avoiding the use of traditional stabilizers. The proposed WG method supports both convex and non-convex polytopal elements in finite element partitions, utilizing bubble functions as a critical analytical tool. The simplified WG method is symmetric and positive definite. Optimal-order error estimates are established for WG approximations in both the discrete $ H^2 $ norm and the $ L^2 $ norm.
- weak Galerkin,
- weak second order partial derivative,
- stabilizer-free,
- bubble functions,
- non-convex, polytopal meshes,
- biharmonic equations
Citation: Chunmei Wang. Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes[J]. Electronic Research Archive, 2025, 33(3): 1523-1540. doi: 10.3934/era.2025072

Related Papers:

Abstract

This paper presents a simplified weak Galerkin (WG) finite element method for solving biharmonic equations avoiding the use of traditional stabilizers. The proposed WG method supports both convex and non-convex polytopal elements in finite element partitions, utilizing bubble functions as a critical analytical tool. The simplified WG method is symmetric and positive definite. Optimal-order error estimates are established for WG approximations in both the discrete $ H^2 $ norm and the $ L^2 $ norm.

References

[1]	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
[2]	S. Cao, C. Wang, J. Wang, A new numerical method for div-curl systems with low regularity assumptions, Comput. Math. Appl., 144 (2022), 47–59. https://doi.org/10.1016/j.camwa.2022.03.015 doi: 10.1016/j.camwa.2022.03.015
[3]	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
[4]	C. Wang, J. Wang, A hybridized formulation for weak Galerkin finite element methods for biharmonic equation on polygonal or polyhedral meshes, Int. J. Numer. Anal. Model., 12 (2015), 302–317.
[5]	C. Wang, J. Wang, An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68 (2014), 2314–2330. https://doi.org/10.1016/j.camwa.2014.03.021 doi: 10.1016/j.camwa.2014.03.021
[6]	C. Wang, J. Wang, S. Zhang, Weak Galerkin finite element methods for optimal control problems governed by second order elliptic partial differential equations, J. Comput. Math. Appl., 452 (2024), 115982. https://doi.org/10.1016/j.cam.2024.115982 doi: 10.1016/j.cam.2024.115982
[7]	X. Ye, S. Zhang, A stabilizer free weak Galerkin finite element 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
[8]	J. Wang, X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comput., 83 (2014), 2101–2126. https://doi.org/10.1090/S0025-5718-2014-02852-4 doi: 10.1090/S0025-5718-2014-02852-4
[9]	C. Wang, Auto-stabilized weak Galerkin finite element methods for biharmonic equations on polytopal meshes without convexity assumptions, preprint, arXiv: 2409.05887.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)