The finite element method(FEM) is known as a powerful tool to give approximate solution for various partial differential equations. Recent research on the explicit error estimation of FEM, along with the interval arithmetic for floating-point numbers computation, can give mathematically correct estimation for the approximate solution of target problems. Thus, it can help to construct mathematical proof, especially for the non-linear equations. In this talk, I will give a review on the latest progress of computable error estimation for FEM and its application in solving non-linear equations.