Discontinuous Galerkin methods for the $p$--biharmonic equation from a discrete variational perspective

Abstract

We study discontinuous Galerkin approximations of the $p$--biharmonic equation from a variational perspective. We propose a discrete variational formulation of the problem based on a appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a weak lower semicontinuity argument. We present numerical experiments aimed at testing the robustness of the method. We also note a superconvergence effect for some values of $p$.