Analysis of finite element method using subspaces with approximate coundary condition for stationary navier-stokes equations

The approximation of the solution for the stationary Navier-Stokes equations in two dimensional domain using certain finite element method is considered in this paper. The formulation is based on different approximate subspaces for the velocity field and the pressure. These spaces contain the ordinary Galerkin space based on approximate subspaees with functions vanishing on the boundary of the domain and also some spaces without such restrictions. Optimal rate of convergence estimates are derived. Moreover, two kinds of iterative procedures for solving the system of nonlinear equations associated with the finite element scheme are constructed. They are shown to be superquadratically and locally convergent.