Pravin Madhavan

Andreas Dedner, Pravin Madhavan, Björn Stinner

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.

Andreas Dedner, Pravin Madhavan

We extend the discontinuous Galerkin (DG) framework to the analysis of first-order hyperbolic and advection-dominated problems posed on implicitly defined surfaces. The focus will be on the hyperbolic part, which is discretised using a "discrete surface" generalisation of the jump-stabilised upwind flux. A key issue arising in the analysis (which does not appear in the planar setting) is the treatment of the discrete velocity field, choices of which play an important role in the stability of the scheme. We then prove optimal error estimates in an appropriate norm given a number of assumptions on the discrete velocity field, which are then investigated and discussed in more detail. The theoretical results are verified numerically for a number of test problems exhibiting advection-dominated behaviour.