High order discontinuous Galerkin methods on surfaces

TL;DR

This paper develops and analyzes high order discontinuous Galerkin methods for solving second-order elliptic problems on implicitly defined surfaces in three-dimensional space, providing optimal error estimates.

Contribution

It adapts the unified DG framework to surfaces and proves optimal error bounds in energy and L2 norms for the methods.

Findings

Optimal error estimates in energy norm

Optimal error estimates in L2 norm

Effective adaptation of DG methods to surface PDEs

Abstract

We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of Arnold et al. [2002] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy norm and the $L^2$ norm.