On a Discontinuous Galerkin Method for PDEs on surfaces

Partial differential equations on surfaces have become an active area of research in recent years due to the fact that, in many applications, models have to be formulated not on a flat Euclidean domain but on a curved surface. For example, they arise naturally in fluid dynamics (e.g. surface active agents on the interface between two fluids ) and material science (e.g. diffusion of species along grain boundaries) but have also emerged in areas as diverse as image processing and cell biology (e.g. cell motility involving processes on the cell membrane, or phase separation on biomembranes).

Finite element methods (FEM) for elliptic problems and their error analysis have been successfully applied to problems on surfaces via the intrinsic approach in Dziuk (1988) based on interpolating the surface by a triangulated one.

However, as in the planar case there are a number of situations where FEM may not be the appropriate numerical method, for instance, advection dominated problems which lead to steep gradients or even discontinuities in the solution.

DG methods are a class of numerical methods that have been successfully applied to hyperbolic, elliptic and parabolic PDEs arising from a wide range of applications. Some of its main advantages compared to standard finite element methods include the ability of capturing discontinuities as arising in advection dominated problems, and less restriction on grid structure and refinement as well as on the choice of basis functions.

The main idea of DG methods is not to require continuity of the solution between elements. Instead, inter-element behaviour has to be prescribed carefully in such a way that the resulting scheme has adequate consistency, stability and accuracy properties.

In my presentation I will investigate the issues arising when attempting to apply DG methods to problems on surfaces. We restrict our analysis to a linear second-order elliptic PDE 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.