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

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.