An approximate method for solving incompressible Navier-Stokes problem with flow rate conditions

We consider the incompressible Navier–Stokes problem with flow rate boundary conditions. This problem has been investigated in [J. Heywood, R. Rannacher, S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations, Int. J. Numer. Methods Fluids 22 (1996) 325–352; L. Formaggia, J. F. Gerbeau, F. Nobile, A. Quarteroni, Numerical treatment of defective boundary conditions for the Navier–Stokes equation, SIAM J. Numer. Anal. 40(1) (2002) 376–401; A. Veneziani, C. Vergara, Flow rate defective boundary conditions in haemodynamics simulations, Int. J. Numer. Methods Fluids 47 (2005) 803–816]. In the latter two papers, a feasible numerical formulation of this problem, based on a Lagrange multiplier approach has been investigated. This approach has the drawback of high computational costs. In this paper, we propose an approximate formulation of this problem, yielding a strong reduction of the computational costs. The error analysis shows that the error introduced by this approximate formulation is confined in a small region of the boundary. This is confirmed by the numerical simulations.