  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.
