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Volume 37, Issue 2
Stabilized Continuous Linear Element Method for the Biharmonic Problems

Ying Cai, Hailong Guo & Zhimin Zhang

DOI: 10.4208/cicp.OA-2023-0302

Commun. Comput. Phys., 37 (2025), pp. 498-520.

Published online: 2025-02

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  • Abstract

In this paper, we introduce a new stabilized continuous linear element method for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. By adding a stability term to the discrete bilinear form, we bypass the need for the discrete Poincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimates in energy and $L^2$ norms. Numerical results are provided to validate our theoretical findings.

  • Keywords

Biharmonic problems, gradient recovery, superconvergence, linear finite element.

  • AMS Subject Headings

65N30, 65N12, 35J15, 35D35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-37-498, author = {Cai , YingGuo , Hailong and Zhang , Zhimin}, title = {Stabilized Continuous Linear Element Method for the Biharmonic Problems}, journal = {Communications in Computational Physics}, year = {2025}, volume = {37}, number = {2}, pages = {498--520}, abstract = {

In this paper, we introduce a new stabilized continuous linear element method for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. By adding a stability term to the discrete bilinear form, we bypass the need for the discrete Poincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimates in energy and $L^2$ norms. Numerical results are provided to validate our theoretical findings.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0302}, url = {http://global-sci.org/intro/article_detail/cicp/23871.html} }
TY - JOUR T1 - Stabilized Continuous Linear Element Method for the Biharmonic Problems AU - Cai , Ying AU - Guo , Hailong AU - Zhang , Zhimin JO - Communications in Computational Physics VL - 2 SP - 498 EP - 520 PY - 2025 DA - 2025/02 SN - 37 DO - http://doi.org/10.4208/cicp.OA-2023-0302 UR - https://global-sci.org/intro/article_detail/cicp/23871.html KW - Biharmonic problems, gradient recovery, superconvergence, linear finite element. AB -

In this paper, we introduce a new stabilized continuous linear element method for solving biharmonic problems. Leveraging the gradient recovery operator, we reconstruct the discrete Hessian for piecewise continuous linear functions. By adding a stability term to the discrete bilinear form, we bypass the need for the discrete Poincaré inequality. We employ Nitsche's method for weakly enforcing boundary conditions. We establish well-posedness of the solution and derive optimal error estimates in energy and $L^2$ norms. Numerical results are provided to validate our theoretical findings.

Cai , YingGuo , Hailong and Zhang , Zhimin. (2025). Stabilized Continuous Linear Element Method for the Biharmonic Problems. Communications in Computational Physics. 37 (2). 498-520. doi:10.4208/cicp.OA-2023-0302
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