The estimation of distritization error in the finite element method is considered in this paper. The proposed method can estimate error of solutions obtained by the finite element method for 2-dimensional elastic problems with 4-node elements. Because quadratic term is expected to dominate the error field, an 8-node element is utilized in the estimation. Even though the 8-node element is used, the proposed method does not require a large memory or long computing time since error is estimated element by element. Not only rectangular elements but also any shape 4-node elements can be used in this method for estimating the error in the finite element computation with high accuracy. An error norm is computed in the paper which can indicate an element-wise relative error. The finite element solutions are also improved drastically by adding the estimated errors onto the original solutions. The proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used.