A Quadratic $C^0$ Interior Penalty Method for the Displacement Obstacle Problem of Clamped Kirchhoff Plates

Abstract

We study a quadratic $C^0$ interior penalty method for the displacement obstacle problem of Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. Under the conditions that the obstacles are sufficiently smooth and separated from each other and the boundary displacement, we prove that the magnitudes of the errors in the energy norm and the $L_\infty$ norm are $O(h^{\alpha})$, where $h$ is the mesh size and $\alpha>\frac12$ is determined by the interior angles of the polygonal domain. We also address the approximations of the coincidence set and the free boundary. The performance of the method is illustrated by numerical results.