When approximating PDEs with the finite element method, large sparse
linear systems must be solved. The ideal preconditioner yields
convergence that is algorithmically optimal and parameter robust, i.e.
the number of Krylov iterations required to solve the linear system to a
given accuracy does not grow substantially as the mesh or problem
parameters are changed.

Achieving this for the stationary Navier-Stokes has proven challenging:
LU factorisation is Reynolds-robust but scales poorly with degree of
freedom count, while Schur complement approximations such as PCD and LSC
degrade as the Reynolds number is increased.

Building on the ideas of Schöberl, Benzi & Olshanskii, in this talk we
present the first preconditioner for the Newton linearisation of the
stationary Navier–Stokes equations in three dimensions that achieves
both optimal complexity and Reynolds-robustness. The scheme combines
augmented Lagrangian stabilisation to control the Schur complement, a
divergence-capturing additive Schwarz relaxation method on each level,
and a specialised prolongation operator involving non-overlapping local
Stokes solves on coarse cells. The properties of the preconditioner are
tailored to the divergence-free Scott–Vogelius discretisation.

We present 3D simulations with over one billion degrees of freedom with
robust performance from Reynolds numbers 10 to 5000.