A stabilized Crank-Nicolson virtual element method for the unsteady Navier-Stokes problems with high Reynolds number

This paper studies a stabilized virtual element method for the unsteady Navier-Stokes problems on polygonal meshes. Using “equal-order” virtual elements in space and the Crank-Nicolson scheme in time, we give a fully discrete formula. By introducing the local-projection type stabilizations, the method can not only circumvent the discrete inf-sup condition but also control spurious oscillations caused by high Reynolds numbers. Stability and error estimates for velocity and pressure are analyzed. Particularly, error estimates are derived in which the constants are independent of the negative powers of the viscosity. Several numerical experiments are simulated to support the theoretical results.