Volume 38, Issue 1
Recovery Based Finite Element Method for Biharmonic Equation in 2D

Yunqing Huang, Huayi Wei, Wei Yang & Nianyu Yi

J. Comp. Math., 38 (2020), pp. 84-102.

Published online: 2020-02

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  • Abstract

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $\Delta$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore, the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.

  • Keywords

Biharmonic equation, Linear finite element, Recovery, Adaptive.

  • AMS Subject Headings

65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

huangyq@xtu.edu.cn (Yunqing Huang)

weihuayi@xtu.edu.cn (Huayi Wei)

yangwei@xtu.edu.cn (Wei Yang)

yinianyu@xtu.edu.cn (Nianyu Yi)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-84, author = {Huang , Yunqing and Wei , Huayi and Yang , Wei and Yi , Nianyu }, title = {Recovery Based Finite Element Method for Biharmonic Equation in 2D}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {1}, pages = {84--102}, abstract = {

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $\Delta$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore, the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1902-m2018-0187}, url = {http://global-sci.org/intro/article_detail/jcm/13686.html} }
TY - JOUR T1 - Recovery Based Finite Element Method for Biharmonic Equation in 2D AU - Huang , Yunqing AU - Wei , Huayi AU - Yang , Wei AU - Yi , Nianyu JO - Journal of Computational Mathematics VL - 1 SP - 84 EP - 102 PY - 2020 DA - 2020/02 SN - 38 DO - http://doi.org/10.4208/jcm.1902-m2018-0187 UR - https://global-sci.org/intro/article_detail/jcm/13686.html KW - Biharmonic equation, Linear finite element, Recovery, Adaptive. AB -

We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation. The main idea is to replace the gradient operator $\nabla$ on linear finite element space by $G(\nabla)$ in the weak formulation of the biharmonic equation, where $G$ is the recovery operator which recovers the piecewise constant function into the linear finite element space. By operator $G$, Laplace operator $\Delta$ is replaced by $\nabla\cdot G(\nabla)$. Furthermore, the boundary condition on normal derivative $\nabla u\cdot \pmb{n}$ is treated by the boundary penalty method. The explicit matrix expression of the proposed method is also introduced. Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method.

Yunqing Huang, Huayi Wei, Wei Yang & Nianyu Yi. (2020). Recovery Based Finite Element Method for Biharmonic Equation in 2D. Journal of Computational Mathematics. 38 (1). 84-102. doi:10.4208/jcm.1902-m2018-0187
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