A bounded numerical solution with a small mesh size implies existence of a smooth solution to the Navier–Stokes equations

We prove that for a given smooth initial value, if a finite element solution of the three-dimensional Navier–Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier–Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical solution.