  In this study, we present an $hp$-multigrid preconditioner for a
divergence-conforming HDG scheme for the generalized Stokes and the
Navier-Stokes equations using an augmented Lagrangian formulation. Our method
relies on conforming simplicial meshes in two- and three-dimensions. The
$hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner
that employs the lowest-order space as the auxiliary space, and we developed a
geometric multigrid method as the auxiliary space solver. For the generalized
Stokes problem, the crucial ingredient of the geometric multigrid method is the
equivalence between the condensed lowest-order divergence-conforming HDG scheme
and a Crouzeix-Raviart discretization with a pressure-robust treatment as
introduced in Linke and Merdon (Comput. Methods Appl. Mech. Engrg., 311
(2016)), which allows for the direct application of geometric multigrid theory
on the Crouzeix-Raviart discretization. The numerical experiments demonstrate
the robustness of the proposed $hp$-multigrid preconditioner with respect to
mesh size and augmented Lagrangian parameter, with iteration counts
insensitivity to polynomial order increase. Inspired by the works by Benzi &
Olshanskii (SIAM J. Sci. Comput., 28(6) (2006)) and Farrell et al. (SIAM J.
Sci. Comput., 41(5) (2019)), we further test the proposed preconditioner on the
divergence-conforming HDG scheme for the Navier-Stokes equations. Numerical
experiments show a mild increase in the iteration counts of the preconditioned
GMRes solver with the rise in Reynolds number up to $10^3$.
