  Augmented Lagrangian preconditioners have successfully yielded
Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes
equations, but only for specific discretizations. The discretizations for which
these preconditioners have been designed possess error estimates which depend
on the Reynolds number, with the discretization error deteriorating as the
Reynolds number is increased. In this paper we present an augmented Lagrangian
preconditioner for the Scott-Vogelius discretization on barycentrically-refined
meshes. This achieves both Reynolds-robust performance and Reynolds-robust
error estimates. A key consideration is the design of a suitable space
decomposition that captures the kernel of the grad-div term added to control
the Schur complement; the same barycentric refinement that guarantees inf-sup
stability also provides a local decomposition of the kernel of the divergence.
The robustness of the scheme is confirmed by numerical experiments in two and
three dimensions.
