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

Farrell, Patrick E.; Mitchell, Lawrence; Wechsung, Florian

doi:10.1137/18m1219370

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

Farrell, Patrick E.; Mitchell, Lawrence; Wechsung, Florian

Authors

Patrick E. Farrell

Lawrence Mitchell

Florian Wechsung

Abstract

In [M. Benzi and M. A. Olshanskii, SIAM J. Sci. Comput., 28 (2006), pp. 2095--2113] a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier--Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and for robustness requires the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from 4.5 at Re = 10 to 3 at Re = 1000 and 5 at Re = 5000.

Citation

Farrell, P. E., Mitchell, L., & Wechsung, F. (2019). An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds number. SIAM Journal on Scientific Computing, 41(5), A3073-A3096. https://doi.org/10.1137/18m1219370

Journal Article Type	Article
Acceptance Date	Jun 10, 2019
Online Publication Date	Oct 8, 2019
Publication Date	2019
Deposit Date	Oct 9, 2018
Publicly Available Date	Oct 10, 2019
Journal	SIAM Journal on Scientific Computing
Print ISSN	1064-8275
Electronic ISSN	1095-7197
Publisher	Society for Industrial and Applied Mathematics
Peer Reviewed	Peer Reviewed
Volume	41
Issue	5
Pages	A3073-A3096
DOI	https://doi.org/10.1137/18m1219370
Related Public URLs	https://arxiv.org/pdf/1810.03315.pdf