  This work is concerned with implementing the hybridizable discontinuous
Galerkin (HDG) method to solve the linear anisotropic elastic equation in the
frequency domain. First-order formulation with the compliance tensor and Voigt
notation are employed to provide a compact description of the discretized
problem and flexibility with highly heterogeneous media. We further focus on
the question of optimal choices of stabilization in the definition of HDG
numerical traces. For this purpose, we construct a hybridized Godunov-upwind
flux for anisotropic elastic media possessing three distinct wavespeeds. This
stabilization removes the need to choose a scaling factor, contrary to the
identity and Kelvin-Christoffel based stabilizations which are popular choices
in the literature. We carry out comparisons among these families for isotropic
and anisotropic material, with constant background and highly heterogeneous
ones, in two and three dimensions. These experiments establish the optimality
of the Godunov stabilization which can be used as a reference choice for a
generic material in which different types of waves propagate.
