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We prove that for a given smooth initial value, if the 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.

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