High Order Discontinuous Galerkin Methods for Elliptic Problems on Surfaces

Abstract

We derive and analyze high order discontinuous Galerkin methods for second order elliptic problems on implicitly defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of [D. N. Arnold et al., SIAM J. Numer. Anal., 39 (2002), pp. 1749--1779] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy and $L^2$ norms. Numerical results validating our theoretical estimates are also presented.