A fast preconditioner for the incompressible Navier Stokes Equations

The pressure matrix method is a well known scheme for the solution of the incompressible Navier–Stokes equations by splitting the computation of the velocity and the pressure fields (see, e.g., [17]). However, the set-up of effective preconditioners for the pressure matrix is mandatory in order to have an acceptable computational cost. Different strategies can be pursued (see, e.g., [6, 22 , 4, 7, 9]). Inexact block LU factorizations of the matrix obtained after the discretization and linearization of the problem, originally proposed as fractional step solvers, provide also a strategy for building effective preconditioners of the pressure matrix (see [23]). In this paper, we present numerical results about a new preconditioner, based on an inexact factorization. The new preconditioner applies to the case of the generalized Stokes problem and to the Navier–Stokes one, as well. In the former case, it improves the performances of the well known Cahouet–Chabard preconditioner (see [2]). In the latter one, numerical results presented here show an almost optimal behavior (with respect to the space discretization) and suggest that the new preconditioner is well suited also for flexible or inexact strategies, in which the systems for the preconditioner are solved inaccurately.