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

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

doi:10.5802/smai-jcm.72

Patrick E. Farrell ¹; Lawrence Mitchell ²; L. Ridgway Scott ³; Florian Wechsung ⁴

¹ Mathematical Institute, University of Oxford, Oxford, UK
² Department of Computer Science, Durham University, Durham, UK
³ Department of Computer Science, University of Chicago, Chicago, USA
⁴ Courant Institute of Mathematical Sciences, New York University, New York, USA

The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 75-96.

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.

Published online: 2021-03-24

Zbl

DOI: 10.5802/smai-jcm.72

Classification: 65N55, 65F08, 65N30
Keywords: Navier–Stokes, Scott–Vogelius element, exactly divergence-free, multigrid, preconditioning, Reynolds-robust solvers

Author's affiliations:

Patrick E. Farrell ¹; Lawrence Mitchell ²; L. Ridgway Scott ³; Florian Wechsung ⁴

¹ Mathematical Institute, University of Oxford, Oxford, UK
² Department of Computer Science, Durham University, Durham, UK
³ Department of Computer Science, University of Chicago, Chicago, USA
⁴ Courant Institute of Mathematical Sciences, New York University, New York, USA

License:

CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Patrick~E. Farrell and Lawrence Mitchell and L.~Ridgway Scott and Florian Wechsung},
     title = {A {Reynolds-robust} preconditioner for the {Scott-Vogelius} discretization of the stationary incompressible {Navier-Stokes} equations},
     journal = {The SMAI Journal of computational mathematics},
     pages = {75--96},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Patrick E. Farrell; Lawrence Mitchell; L. Ridgway Scott; Florian Wechsung. A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations. The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 75-96. doi : 10.5802/smai-jcm.72. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.72/

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