Error estimate of the P-1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations

We consider a finite element method for the penalty formulation of the time dependent Navier-Stokes equations. Usually the improper choice of the finite element space will lead that the error estimate (inversely) depends on the penalty parameter epsilon. We use the classical P-1 nonconforming finite element space for the spatial discretization. Optimal epsilon-uniform error estimations for both velocity and pressure are obtained.