We derive and analyze high order discontinuous Galerkin methods for second order elliptic problems on implicitly defined surfaces in R³. This is done by carefully adapting the unified discontinuous Galerkin framework of [D. N. Arnold et al., SIAM J. Numer. Anal., 39 (2002), pp. 1749--1779] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy and L² norms. Numerical results validating our theoretical estimates are also presented.
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