Analysis of a penalty finite element approximation of the Navier Stokes equations

The analysis of a penalized formulation of the Navier-Stokes equations is considered. First, the existence of a unique solution (under certain sufficient conditions on the viscosity and body force) is proved. After similar considerations for the reduced integration penalty finite element problem, error estimates for the velocity and pressure approximations are derived. Next, the convergence of several standard iterative methods for the solution of the nonlinear algebraic system arising from the approximate problem is considered. Finally, the application of continuation methods for extending the range of flows at which convergence can be achieved is studied.