High order discontinuous Galerkin methods on surfaces
  We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of Arnold et al. [2002] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy norm and the $L^2$ norm. 
Paola Antonietti, Andreas Dedner, Pravin Madhavan, Simone Stangalino,   Bj\"orn Stinner and Marco Verani
math.NA