The Crank–Nicolson finite spectral element method and numerical simulations for 2D non‐stationary Navier–Stokes equations

In this paper, we first build a semi‐discretized Crank–Nicolson (CN) model about time for the two‐dimensional (2D) non‐stationary Navier–Stokes equations about vorticity–stream functions and discuss the existence, stability, and convergence of the time semi‐discretized CN solutions. And then, we build a fully discretized finite spectral element CN (FSECN) model based on the bilinear trigonometric basic functions on quadrilateral elements for the 2D non‐stationary Navier–Stokes equations about the vorticity–stream functions and discuss the existence, stability, and convergence of the FSECN solutions. Finally, we utilize two sets of numerical experiments to check out the correctness of theoretical consequences.