For the Poisson equation with Robin boundary conditions, by using a few techniques such as or- thogonal expansion (M -type), separation of the main part and the finite element projection, we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3 )f oru ∈ H3. Based on the obtained asymptotic error expansions for linear finite elements, extrapolation cascadic multigrid method (EXCMG) can be used to solve Robin problems effectively. Furthermore, by virtue of Richardson ex- trapolation, not only the accuracy of the approximation is improved, but also a posteriori error estimation is obtained. Finally, some numerical experiments that confirm the theoretical analysis are presented.