We consider finite element approximations of the stationary incompressible Navier-Stokes equations. An ideal preconditioner for the linear systems arising from these equations 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.

It has proven challenging to develop solvers that exhibit both properties; matrix factorisations are robust to Reynolds number but scale badly with dof count, whereas Schur complement based algorithms such as PCD and LSC scale linearly in the dof count but their performance decreases as the Reynolds number is increased

Building on the ideas of Schöberl, Benzi, and Olshanskii, we present an augmented Lagrangian based preconditioner with linear complexity and iteration counts that only grow mildly with respect to the Reynolds number. The key ingredient is a tailored multigrid scheme for the exactly divergence-free Scott-Vogelius discretisation consisting of custom smoothing and prolongation operators.

This work has been done in collaboration with Patrick Farrell, Lawrence Mitchell, and Ridgway Scott.