  We present an optimal preconditioner for the exactly divergence-free
discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and
Sch\"otzau [J. Sci. Comput., 31 (2007), pp. 61--73] and Wang and Ye [SIAM J.
Numer. Anal., 45 (2007), pp. 1269--1286] for the Stokes problem. This DG method
uses finite elements that use an $H({\rm div})$-conforming basis, thereby
significantly complicating its solution by iterative methods. Several
preconditioners for this Stokes discretization have been developed, but each is
based on specialized solvers or decompositions, and do not offer a clear
framework to generalize to Navier--Stokes. To avoid requiring custom solvers,
we hybridize the $H({\rm div})$-conforming finite element so that the velocity
lives in a standard $L^2$-DG space, and present a simple algebraic
preconditioner for the extended hybridized system. The proposed preconditioner
is optimal in $h$, super robust in element order (demonstrated up to 5th
order), effective in 2d and 3d, and only relies on standard relaxation and
algebraic multigrid methods available in many packages. The hybridization also
naturally extends to Navier--Stokes, providing a potential pathway to effective
black-box preconditioners for exactly divergence-free DG discretizations of
Navier--Stokes.

Numerical Analysis;Numerical Analysis