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Augmented Lagrangian preconditioners have successfully yieldedReynolds-robust preconditioners for the stationary incompressible Navier-Stokesequations, but only for specific discretizations. The discretizations for whichthese preconditioners have been designed possess error estimates which dependon the Reynolds number, with the discretization error deteriorating as theReynolds number is increased. In this paper we present an augmented Lagrangianpreconditioner for the Scott-Vogelius discretization on barycentrically-refinedmeshes. This achieves both Reynolds-robust performance and Reynolds-robusterror estimates. A key consideration is the design of a suitable spacedecomposition that captures the kernel of the grad-div term added to controlthe Schur complement; the same barycentric refinement that guarantees inf-supstability also provides a local decomposition of the kernel of the divergence.The robustness of the scheme is confirmed by numerical experiments in two andthree dimensions.

A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations