  The macro-element variant of the hybridized discontinuous Galerkin (HDG)
method combines advantages of continuous and discontinuous finite element
discretization. In this paper, we investigate the performance of the
macro-element HDG method for the analysis of compressible flow problems at
moderate Reynolds numbers. To efficiently handle the corresponding large
systems of equations, we explore several strategies at the solver level. On the
one hand, we devise a second-layer static condensation approach that reduces
the size of the local system matrix in each macro-element and hence the
factorization time of the local solver. On the other hand, we employ a
multi-level preconditioner based on the FGMRES solver for the global system
that integrates well within a matrix-free implementation. In addition, we
integrate a standard diagonally implicit Runge-Kutta scheme for time
integration. We test the matrix-free macro-element HDG method for compressible
flow benchmarks, including Couette flow, flow past a sphere, and the
Taylor-Green vortex. Our results show that unlike standard HDG, the
macro-element HDG method can operate efficiently for moderate polynomial
degrees, as the local computational load can be flexibly increased via mesh
refinement within a macro-element. Our results also show that due to the
balance of local and global operations, the reduction in degrees of freedom,
and the reduction of the global problem size and the number of iterations for
its solution, the macro-element HDG method can be a competitive option for the
analysis of compressible flow problems.
