A stabilized cut discontinuous Galerkin framework: II. Hyperbolic problems
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We present the second part of a stabilized cut discontinuous Galerkin (cutDG) framework initiated in [1] and develop a novel cutDG method for scalar hyperbolic problems. The domain of interest is embedded into a structured, unfitted background mesh in R^d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we employ and extend ghost penalty techniques from recently developed, continuous cut finite element methods instead of the cell merging technique commonly used in unfitted discontinuous Galerkin methods, thus allowing for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying a few abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and condition number estimates for the scalar hyperbolic problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two and three-dimensional test problems.
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