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

Abstract

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.

arXiv commentFixed sign of grad-div term in (3.7)

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• Recycling augmented Lagrangian preconditioner in an incompressible fluid solver
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• Monolithic Multigrid for Magnetohydrodynamics
• An augmented Lagrangian preconditioner for implicitly-constituted non-Newtonian incompressible flow
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• Analysis and computation of a pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations by using high-order finite elements
• Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes
• Code generation for productive portable scalable finite element simulation in Firedrake