A Crank-Nicolson finite spectral element method for the 2D non-stationary Stokes equations about vorticity-stream functions.

In this article, we first develop a semi-discretized Crank-Nicolson format about time for the two-dimensional non-stationary Stokes equations about vorticity-stream functions and analyze the existence, uniqueness, stability, and convergence of the semi-discretized Crank-Nicolson solutions. Then we establish a fully discretized Crank-Nicolson finite spectral element format based on the quadrilateral elements for the two-dimensional non-stationary Stokes equations about vorticity-stream functions and analyze the existence, uniqueness, stability, and convergence of the Crank-Nicolson finite spectral element solutions. In the end, we use three numerical examples to confirm the validity of our theoretical conclusions.