Before revision:  In this paper, the latest progress about the solution verification for the stationary Navier--Stokes equationover a non-convex 3D domain is reported .
After revision:  In this paper, the latest progress about the solution verification for the stationary Navier--Stokes equationover a non-convex 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation .