Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

TL;DR

This paper extends the discontinuous Galerkin method to solve second-order elliptic problems on smooth surfaces, providing error estimates and numerical verification of convergence unaffected by surface discretization errors.

Contribution

It introduces an interior penalty DG method on surfaces, derives a-priori error estimates relating surface and discretized errors, and investigates conormal approximation choices.

Findings

Geometric errors do not affect convergence rates with linear basis functions.

Numerical tests confirm theoretical error estimates.

Implementation framework for surface PDE test problems.

Abstract

We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. An intricate issue is the approximation of the surface conormal required in the IP formulation, choices of which are investigated numerically. Furthermore, we present a generic implementation of test problems on surfaces.