Stability and convergence analysis of a Crank-Nicolson leap-frog scheme for the unsteady incompressible Navier-Stokes equations

A fully discrete Crank-Nicolson leap-frog (CNLF) scheme is presented and studied for the nonstationary incompressible Navier-Stokes equations. The proposed scheme deals with the spatial discretization by Galerkin finite element method (FEM), treats the temporal discretization by CNLF method for the linear term and the semi-implicit method for nonlinear term. The almost unconditional stability, i.e., the time step is no more than a constant, is proven. By a new negative norm technique, the L2-optimal error estimates with respect to temporal and spacial orientation for the velocity are derived. At last, some numerical results are provided to justify our theoretical analysis.