Optimal Error Estimates of the Local Discontinuous Galerkin Method for Willmore Flow of Graphs on Cartesian Meshes

Abstract

In this paper, we analyze a local discontinuous Galerkin method for the willmore flow of graphs. We derive the optimal error estimates for this nonlinear equation in one-dimension and in multi-dimensions for Cartesian meshes using completely discontinuous piecewise polynomial space with degree $k \leq 1$.