Unconditional convergence and optimal L2 error estimates of the Crank-Nicolson extrapolation FEM for the nonstationary Navier-Stokes equations

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank-Nicolson extrapolation scheme for the nonstationary Navier-Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank-Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the L2, H1-unconditionally optimal error estimates for the velocity, and the L2-unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditionalL2-stability and convergent rates of this method are shown.