Divergence-free stabilized virtual element method for the unsteady incompressible Navier–Stokes problem

In this paper, we extend the divergence-free virtual element method to the unsteady incompressible Navier–Stokes problem, and by enhancing the stabilizing term of the virtual element method, the method can not only exactly preserve the divergence-free constraint, but also control spurious oscillations in the velocity due to dominant convection. Both the continuous-in-time and the fully discrete schemes (Euler semi-implicit scheme) are proposed. The corresponding stability and error estimates are analyzed. Importantly, error estimates are derived in which the constants are independent of the Reynolds number, and error estimates do not explicitly depend on the pressure. Several numerical experiments verify the theoretical results and the favorable performance of the method.