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Volume 31, Issue 5
A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

Huangxin Chen, Amiya K. Pani & Weifeng Qiu

Commun. Comput. Phys., 31 (2022), pp. 1434-1466.

Published online: 2022-05

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  • Abstract

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

  • AMS Subject Headings

65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1434, author = {Chen , HuangxinPani , Amiya K. and Qiu , Weifeng}, title = {A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {1434--1466}, abstract = {

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0255}, url = {http://global-sci.org/intro/article_detail/cicp/20510.html} }
TY - JOUR T1 - A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations AU - Chen , Huangxin AU - Pani , Amiya K. AU - Qiu , Weifeng JO - Communications in Computational Physics VL - 5 SP - 1434 EP - 1466 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0255 UR - https://global-sci.org/intro/article_detail/cicp/20510.html KW - Biharmonic equation, von Kármán equations, mixed finite element methods, element-wise stabilization, discrete H2-stability, positive definite. AB -

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

Huangxin Chen, Amiya K. Pani & Weifeng Qiu. (2022). A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations. Communications in Computational Physics. 31 (5). 1434-1466. doi:10.4208/cicp.OA-2021-0255
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