Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
Provides a theoretical and numerical foundation for applying DG methods to surface PDEs, which is incremental for computational mathematics.
The paper extends the discontinuous Galerkin method to elliptic problems on surfaces, proving that geometric errors from surface discretization do not reduce the convergence rate for linear ansatz functions, and verifying this numerically.
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.