A stabilized finite element method for transient Navier-Stokes equations based on two local Gauss integrations

On the basis of two local Gauss integrations, a stabilized finite element method for transient Navier-Stokes equations is presented, which is defined by the lowest equal-order conforming finite element subspace (Xh,Mh) such as P1 -P1 (or Q1-Q1) elements. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. The diffusion term in these equations is discretized by using finite element method, and the temporal differentiation and advection terms are treated by characteristic schemes. Moreover, we present some numerical simulations to demonstrate the effectiveness, good stability, and accuracy properties of our method. Especially, the rate of convergence study tells us that the stability still keeps well when the Reynolds number is increasing.