A comparison of hybridized and standard DG methods for target-based hp-adaptive simulation of compressible flow

• We present a comparison between HDG and DG in the context of target-based hp-adaptation.
• h- or p-Refinement is chosen based on the smoothness of the solution.
• Results are shown for subsonic, transonic, and supersonic flow.
• hp-Adaptation shows superior results compared to pure h-adaptation.
• HDG is faster than DG from p = 2 on.

Objective: We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods.Method: Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators.Results: Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. $hp$-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p = 2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements.