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

Farrell, Patrick E.; Mitchell, Lawrence; Scott, L. Ridgway; Wechsung, Florian

doi:10.5802/smai-jcm.72

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

Farrell, Patrick E.; Mitchell, Lawrence; Scott, L. Ridgway; Wechsung, Florian

Authors

Patrick E. Farrell

Lawrence Mitchell

L. Ridgway Scott

Florian Wechsung

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.

Citation

Farrell, P. E., Mitchell, L., Scott, L. R., & Wechsung, F. (2021). A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations. The SMAI journal of computational mathematics, 7, 75-96. https://doi.org/10.5802/smai-jcm.72

Journal Article Type	Article
Acceptance Date	Feb 2, 2021
Online Publication Date	Mar 24, 2021
Publication Date	2021
Deposit Date	May 20, 2020
Publicly Available Date	Feb 18, 2021
Journal	SMAI Journal of Computational Mathematics
Publisher	Centre Mersenne
Peer Reviewed	Peer Reviewed
Volume	7
Pages	75-96
DOI	https://doi.org/10.5802/smai-jcm.72
Related Public URLs	https://arxiv.org/pdf/2004.09398.pdf

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