A mixed finite element scheme for biharmonic equation with variable coefficient and von K\'arm\'an equations

TL;DR

This paper introduces a new mixed finite element scheme for biharmonic equations with variable coefficients, providing stability, optimal convergence, and extension to von Kármán equations, supported by numerical verification.

Contribution

A novel element-wise stabilized mixed finite element scheme for biharmonic equations that avoids interface integration and extends to von Kármán equations.

Findings

Scheme is easy to implement and yields positive definite linear systems.

Achieves optimal convergence in discrete H^2 and L^2 norms.

Numerical results confirm theoretical error estimates.

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 $H({\rm div})$-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 $k\geq 0$.This scheme can be easily implemented and produces 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 $C^{0}$ 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 K\'arm\'an equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.