Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincare-Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert-Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of thehp-DGFEM for the biharmonic problem introduced by Mozolevski and Suli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.