Existence of smooth solutions to the 3D Navier–Stokes equations based on numerical solutions by the Crank–Nicolson finite element method

A Crank–Nicolson finite element method is proposed to solve the time-dependent Navier–Stokes equations. We prove that for a given smooth initial value, if a fully discrete finite element solution of the three-dimensional Navier–Stokes equations is bounded in a certain norm, then the strong solution of the Navier–Stokes equations satisfies uniqueness and smoothness. Further, the fully discrete solution converges to the strong solution as temporal step size \(\tau \rightarrow 0\) $\tau \to 0$ and spatial step size \(h\rightarrow 0\) $h\to 0$ .