An approximate method for solving incompressible Navier–Stokes problems with flow rate conditions Original Research Article

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.