New superconvergent structures are introduced by the finite volume element method, which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for k>=3. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in this talk.