A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations
Year: 2022
Author: Huangxin Chen, Amiya K. Pani, Weifeng Qiu
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0255
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1434–1466
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Biharmonic equation von Kármán equations mixed finite element methods element-wise stabilization discrete H2-stability positive definite.