A Quadratic $C^0$ Interior Penalty Method for the von K\'{a}rm\'{a}n Obstacle Problem

Abstract

This article proposes and analyses a quadratic $C^0$ interior penalty method for the displacement obstacle problem of the von K\'arm\'an plate. The discrete space consists of Lagrange $P_2$ finite elements and the obstacle constraint is imposed at the vertices. The trilinear form of the von K\'arm\'an bracket is modified by terms on the edges so that it is bounded in the discrete energy norm. The well-posedness of the discrete problem, namely the existence of a discrete solution and its uniqueness under a smallness condition on the data, is established. The Sobolev and Friedrichs constants of the discrete energy norm are quantified with an explicit dependence on the mesh size. The main result is an error estimate of order $\mathcal{O}(h^{\alpha})$ in the discrete energy norm, where $1/2<\alpha\le1$ is the index of elliptic regularity of the biharmonic operator on the polygonal domain. Numerical experiments on a square and on an L-shaped domain confirm the predicted rates. The coincidence set has positive measure in one example and empty interior in another. The experiments also show how the penalty parameter affects the rates and identify a threshold in the size of the obstacle beyond which the iterative solver fails on fine meshes.