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

TL;DR

This paper extends an augmented Lagrangian preconditioner for the 2D stationary incompressible Navier-Stokes equations to 3D, demonstrating robust performance at high Reynolds numbers with large-scale problems.

Contribution

The authors generalize a 2D preconditioner to 3D by developing new finite elements and prolongation operators, enabling efficient high Reynolds number simulations.

Findings

Effective at high Reynolds numbers up to 5000

Reduces Krylov iterations significantly at high Re

Scalable to problems with approximately one billion degrees of freedom

Abstract

In Benzi & Olshanskii (SIAM J.~Sci.~Comput., 28(6) (2006)) 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 is tightly coupled to 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 $7$ at $\mathrm{Re} = 4.5$ to $3$ at $\mathrm{Re} = 1000$ and $5$ at $\mathrm{Re} = 5000$.