Stabilized mixed finite element methods for the Navier-Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier-Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed-point theorem. Then, optimal error estimates for the H-1-norm and L-2-norm of the velocity and the L-2-norm of the pressure are derived. Moreover, on the basis of the optimal L-2-norm error estimate of the velocity, a stabilized two-step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.