An Augmented Lagrangian Preconditioner for the 3D Stationary Incompressible Navier--Stokes Equations at High Reynolds Number
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.
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Published In
SIAM Journal on Scientific Computing
Pages: A3073 - A3096
DOI: 10.1137/18M1219370
ISSN (online): 1095-7197
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© 2019, Society for Industrial and Applied Mathematics.
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Submitted: 8 October 2018
Accepted: 10 June 2019
Published online: 8 October 2019
Keywords
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Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/K030930/1, EP/M011151/1, EP/M011054/1, EP/L000407/1, EP/L015803/1
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/N032861/1
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