Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques
Friday 20th March 2009
We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a pradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit them to set up an efficient solution strategy. In contrast to alternative discretization methods based for instance on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.