Comparison results of $P_2$-finite elements for fourth-order semilinear von Karman equations

TL;DR

This paper compares various $P_2$-finite element methods for fourth-order von Karman equations, establishing error equivalences under a unified norm and supporting findings with numerical experiments.

Contribution

It demonstrates the error equivalence of different $P_2$ finite element methods for von Karman equations under a common norm, clarifying their comparative performance.

Findings

Errors for different methods are equivalent up to higher-order oscillation.

Numerical experiments support the theoretical error comparison.

Unified norm effectively compares various finite element approaches.

Abstract

Lower-order $P_2$ finite elements are popular for solving fourth-order elliptic PDEs when the solution has limited regularity. A priori and a posteriori error estimates for von Karman equations are considered in Carstensen et al. (2019, 2020) with respect to different mesh dependent norms which involve different jump and penalization terms. This paper addresses the question, whether they are comparable with respect to a common norm. This article establishes that the errors for the quadratic symmetric interior discontinuous Galerkin, $C^0$ interior penalty and nonconforming Morley finite element methods are equivalent upto some higher-order oscillation term with respect to a unified norm. Numerical experiments are performed to substantiate the comparison results.