Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair (X-h, M-h) which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters epsilon, Delta t and h are sufficiently small.