Comparison of hybrid and standard discontinuous Galerkin methods in a mesh-optimisation setting

This paper assesses the benefits of hybridization on the accuracy and efficiency of high-order discontinuous Galerkin (DG) discretizations. Two hybridized methods are considered in addition to DG: hybridized DG (HDG) and embedded DG (EDG). These methods offer memory and computational time savings by introducing trace degrees of freedom on faces that become the only globally-coupled unknowns. To mitigate the effects of solution singularities on accuracy, the methods are compared in an adaptive setting on meshes optimised for the accurate prediction of chosen scalar outputs. Compressible flow results for the Euler and Reynolds-averaged Navier-Stokes equations demonstrate that the hybridized methods offer cost savings relative to DG in memory and computational time. In addition, for the cases tested, EDG yields the lowest error levels for a given number of degrees of freedom. These benefits disappear on uniformly-refined meshes, indicating the importance of using order-optimised meshes when comparing the discretizations.