A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

Huangxin Chen; Amiya K. Pani; Weifeng Qiu

doi:10.4208/cicp.OA-2021-0255

A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

Huangxin Chen, Amiya K. Pani, Weifeng Qiu^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

2 Citations (Scopus)

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+l element or vector valued Lagrange element with order k + 1, while the solution is approximated by Lagrange clement 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 H²-norm stability, which is useful not only in analysis of this scheme but also in C⁰ interior penalty methods and DG methods. Optimal convergences in both discrete H²-norm and L²-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.

Original language	English
Pages (from-to)	1434-1466
Journal	Communications in Computational Physics
Volume	31
Issue number	5
Online published	May 2022
DOIs	https://doi.org/10.4208/cicp.OA-2021-0255
Publication status	Published - May 2022

Funding

The work of Huangxin Chen was supported by the NSF of China (Grant No. 12122115, 11771363). The work of Amiya K. Pani is supported by IITB Chair Professor’s fund and also partly by a MATRIX Grant No. MTR/201S/000309 (SERB, DST, Govt. India). Weifeng Qiu is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302219). The third author is the corresponding author.

Research Keywords

Biharmonic equation
von Karman equations
mixed finite element methods
element-wise stabilization
discrete H2-stability
positive definite
ELLIPTIC-EQUATIONS
GALERKIN METHODS
PENALTY METHOD
APPROXIMATION
FAMILY

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title = "A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von K{\'a}rm{\'a}n Equations",

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 BDMk+l element or vector valued Lagrange element with order k + 1, while the solution is approximated by Lagrange clement 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 H2-norm stability, which is useful not only in analysis of this scheme but also in C0 interior penalty methods and DG methods. Optimal convergences in both discrete H2-norm and L2-norm are derived. This scheme with its analysis is further generalized to the von K{\'a}rm{\'a}n equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.",

keywords = "Biharmonic equation, von Karman equations, mixed finite element methods, element-wise stabilization, discrete H2-stability, positive definite, ELLIPTIC-EQUATIONS, GALERKIN METHODS, PENALTY METHOD, APPROXIMATION, FAMILY",

author = "Huangxin Chen and Pani, {Amiya K.} and Weifeng Qiu",

year = "2022",

month = may,

doi = "10.4208/cicp.OA-2021-0255",

language = "English",

volume = "31",

pages = "1434--1466",

journal = "Communications in Computational Physics",

issn = "1815-2406",

publisher = "Global Science Press",

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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

PY - 2022/5

Y1 - 2022/5

N2 - 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 BDMk+l element or vector valued Lagrange element with order k + 1, while the solution is approximated by Lagrange clement 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 H2-norm stability, which is useful not only in analysis of this scheme but also in C0 interior penalty methods and DG methods. Optimal convergences in both discrete H2-norm and L2-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.

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 BDMk+l element or vector valued Lagrange element with order k + 1, while the solution is approximated by Lagrange clement 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 H2-norm stability, which is useful not only in analysis of this scheme but also in C0 interior penalty methods and DG methods. Optimal convergences in both discrete H2-norm and L2-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.

KW - Biharmonic equation

KW - von Karman equations

KW - mixed finite element methods

KW - element-wise stabilization

KW - discrete H2-stability

KW - positive definite

KW - ELLIPTIC-EQUATIONS

KW - GALERKIN METHODS

KW - PENALTY METHOD

KW - APPROXIMATION

KW - FAMILY

UR - http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=LinksAMR&SrcApp=PARTNER_APP&DestLinkType=FullRecord&DestApp=WOS&KeyUT=000798494100005

U2 - 10.4208/cicp.OA-2021-0255

DO - 10.4208/cicp.OA-2021-0255

M3 - RGC 21 - Publication in refereed journal

SN - 1815-2406

VL - 31

SP - 1434

EP - 1466

JO - Communications in Computational Physics

JF - Communications in Computational Physics

IS - 5

ER -

A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

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GRF: Hybridizable Discontinuous Galerkin Approximation for Second Order Elliptic Operator in Non-divergence Form and Some Applications

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A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations

Abstract

Funding

Research Keywords

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GRF: Hybridizable Discontinuous Galerkin Approximation for Second Order Elliptic Operator in Non-divergence Form and Some Applications

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