Discontinuous Galerkin methods for the $p$--biharmonic equation from a discrete variational perspective
Provides a theoretical and numerical foundation for solving higher-order nonlinear PDEs with DG methods, but is incremental as it extends existing variational approaches to a specific equation.
The paper develops a discontinuous Galerkin method for the p-biharmonic equation using a discrete variational formulation, proving convergence without rates and demonstrating robustness via numerical experiments, with observed superconvergence for some p values.
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 $p$.