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In Benzi & Olshanskii (SIAM J.~Sci.~Comput., 28(6) (2006)) a preconditionerof augmented Lagrangian type was presented for the two-dimensional stationaryincompressible Navier--Stokes equations that exhibits convergence almostindependent of Reynolds number. The algorithm relies on a highly specializedmultigrid method involving a custom prolongation operator and is tightlycoupled to the use of piecewise constant finite elements for the pressure.However, the prolongation operator and velocity element used do not directlyextend to three dimensions: the local solves necessary in the prolongationoperator do not satisfy the inf-sup condition. In this work we generalize thepreconditioner to three dimensions, proposing alternative finite elements forthe velocity and prolongation operators for which the preconditioner worksrobustly. The solver is effective at high Reynolds number: on athree-dimensional lid-driven cavity problem with approximately one billiondegrees of freedom, the average number of Krylov iterations per Newton stepvaries from $7$ at $\mathrm{Re} = 4.5$ to $3$ at $\mathrm{Re} = 1000$ and $5$at $\mathrm{Re} = 5000$.

An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number