  We present a superconvergent hybridizable discontinuous Galerkin (HDG) method
for the steady-state incompressible Navier-Stokes equations on general
polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials
of degree k+1, k, k to approximate the velocity, velocity gradient and
pressure, respectively. In contrast, we only use polynomials of degree k to
approximate the numerical trace of the velocity on the interfaces. Since the
numerical trace of the velocity field is the only globally coupled unknown,
this scheme allows a very efficient implementation of the method. For the
stationary case, and under the usual smallness condition for the source term,
we prove that the method is well defined and that the global L2-norm of the
error in each of the above-mentioned variables and the discrete H1-norm of the
error in the velocity converge with the order of k+1 for k>=0. We also show
that for k>=1, the global L2-norm of the error in velocity converges with the
order of k+2. From the point of view of degrees of freedom of the globally
coupled unknown: numerical trace, this method achieves optimal convergence for
all the above-mentioned variables in L2-norm for k>=0, superconvergence for the
velocity in the discrete H1-norm without postprocessing for k>=0, and
superconvergence for the velocity in L2-norm without postprocessing for k>=1.
