Abstract: Based on the newly-developed Element Energy Projection (EEP) method for computation of super-convergent results in one-dimensional Finite Element Method (FEM), the task of self-adaptive FEM analysis is converted into the task of adaptive piecewise polynomial interpolation. As a result, an optimum FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce a FEM solution which satisfies the user specified error tolerance. This strategy has been found to be very simple, rapid, cheep and efficient. With an ordinary differential equation of the second order being the model problem, this paper gives a brief description of the new adaptive strategy and representative numerical examples are given to show the effectiveness and efficiency of the proposed approach.