Stabilized low order finite elements for Stokes equations with damping

In this paper, we consider the stationary Stokes equations with damping. Low order mixed finite element spaces are used to approximate the velocity and the pressure, and a local pressure projection stabilization method is used for the pairs to overcome the lack of the inf-sup condition. The stability of this method is proved, and the optimal order error estimates are derived by some nonlinear analysis techniques. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method.