A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presented for the two-dimensional time-dependent Navier-Stokes problem. The spatial discretization is based on a finite-element space pair (X h , M h ) for the approximation of the velocity and the pressure, constructed by using the Q 1 - P 0 quadrilateral element or the P 1 - P 0 triangular element; the time discretization is based on the Euler semi-implicit scheme. It is shown that the proposed fully discrete stabilized finite-element method results in the optimal order error bounds for the velocity and the pressure.