In this manuscript we compare physical and reference frame discontinuous Galerkin (dG) discretizations with emphasis on the influence of reference-to-physical frame mappings on the discrete space properties. We assess the excellence of physical frame discrete spaces in terms of approximation capabilities as well as the increased flexibility compared to reference frame discretizations. As a matter of fact, whenever curved elements are considered, non-affine reference-to-physical frame mappings are able to spoil the convergence properties of reference frame discrete spaces. This poorly documented drawback does not affect basis functions defined directly in the physical frame.
The convergence degradation associated to reference frame discretizations is evaluated theoretically, providing error bounds for the approximation error of the L2-orthogonal projection operator, and the findings are justified by means of numerical test cases. In particular we exemplify by means of quadrilateral elements grids challenging grid configurations characterized by non-affine mappings and demonstrate the ability to predict the convergence rates without stringent assumptions on the element shapes.
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