  We consider spectral mixed discontinuous Galerkin finite element
discretizations of the Lam\'e system of linear elasticity in polyhedral domains
in $\mathbb{R}^3$. In order to resolve possible corner, edge, and corner-edge
singularities, anisotropic geometric edge meshes consisting of hexahedral
elements are applied. We perform a computational study on the discrete inf-sup
stability of these methods, and especially focus on the robustness with respect
to the Poisson ratio close to the incompressible limit (i.e. the Stokes
system). Furthermore, under certain realistic assumptions (for analytic data)
on the regularity of the exact solution, we illustrate numerically that the
proposed mixed DG schemes converge exponentially in a natural DG norm.
