The Navier–Stokes equation is a non-linear partial differential equation that describes the motion of fluids, for example water and air. As one of the Millennium Prize Problems, the problem of existence and smoothness of the Navier-Stokes equation problem draws the attention of mathematican from the world. To solve the this problem, a breakthrough with new mathematical theory and new technique is expected. On one hand, the computer-assisted proof utilizing verified computing, which aims to give rigorous estimation for all error appearing in numerial compuations, provides a new approch to the solution existence proof to Navier-Stokes equation. In this talk, I will explain the basic idea of verified computing and show the latest progress in this field, which includes my recent research on rigorous eigenvalue bounds for dierential operators using finite element methods.