Preconditioning for the Steady-State Navier--Stokes Equations with Low Viscosity

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Abstract
We introduce a preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to 1, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.