A new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics

We consider the incompressible Navier-Stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, S. Turek, Int J Num Meth Fluids 22 (1996), pp. 325-352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, A. Quarteroni, SIAM J Num Anal, 40-1 (2002), pp. 376--401, and A. Veneziani, C. Vergara, Int J Num Meth Fluids, 47 (2005), pp. 803--816). However, both these approaches present some drawbacks. For the flow rate problem, the former resorts to non standard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yields exact solutions only in very special cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions. In this paper, we propose a new strategy, based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional. This approach allows to treat the two kind of problems (flow rate and mean pressure) successfully within the same framework, that can be useful in view of mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries. Moreover, it is more versatile, being prone to be extended to other kind of defective conditions. We analyze the problems obtained with this approach and propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.