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An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at High Reynolds number. (English) Zbl 1448.65261
The paper deals with the numerical solution of algebraic systems arising from the discretization of the 3D incompressible Navier-Stokes equations by finite element methods. In [M. Benzi and M. A. Olshanskii, SIAM J. Sci. Comput. 28, No. 6, 2095–2113 (2006; Zbl 1126.76028)], the efficientl technique of the preconditioner of augmented Lagrangian type was presented for the two-dimensional problems. The algorithm relies on a specialized multigrid method involving a custom prolongation operator and for robustness requires the use of piecewise constant finite elements for the pressure. However, this technique can not be directly extended to 3D problems due to the violation of the inf-sup condition for the used finite element pair. In this paper, the authors generalize the preconditioner to 3D by 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 \(\operatorname{Re} = 10\) to 3 at \(\operatorname{Re} = 1000\) and 5 at \(\operatorname{Re} = 5000\).

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65H10 Numerical computation of solutions to systems of equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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