Abstract: An algorithm of super-convergence for the Garlerkin FEM (finite element method) for the second order non-self-adjoint boundary-value problem is proposed based on the improved displacement mode. The new displacement mode is constructed by combining the displacement mode of the conventional finite element and the high-order displacement mode based on the Galerkin method, and the element equilibrium equation is derived using the integral form. A representative example is presented for the Hermite element in this paper, the accuracies of the displacements and the derivatives accuracy of nodes and elements have reached the order of h⁴.