Finite element aproximation of steady Navier-Stokes equations with mixed boundary conditions

In the present paper, attention is given to the steady Navier-Stokes equations in a bounded domain with a smooth boundary Gamma. The boundary conditions involve the requirement that the normal velocity component and the tangential stress components vanish on Gamma. Problems of this type arise as subproblems in the study of fluid flows subject to surface tension. The continuous problem is discretized with the aid of a nonconforming mixed finite element method. Quadratic elements are used for the velocities, and linear elements for the pressure. Both the continuous and the discrete problem have unique solutions in cases in which the data are sufficiently small.