A new approach to numerical solution of defective boundary problems in incompressible fluid dynamics
Tuesday 17th October 2006
Formaggia, Luca; Veneziani, Alessandro; Vergara, Christian
We consider the incompressible Navier-Stokes equations with flow rate and mean pressure boundary conditions. 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, these approaches present some drawbacks. The former, for the flow rate problem, resorts to non standard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yealds exact solutions only in very specific 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 control reformulation of the problems at hand. This approach allows to treat the two problems successfully within the same framework. We carry out the well-posedness analysis of the problems obtained with this approach and we propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.