Abstract: Based on the direct derivation and analysis of the error terms of Element Energy Projection (EEP) method, the omitted error terms in the EEP solution of the simplified form are calculated, and the subsequent accuracy recovery results in an EEP enhanced form with convergence accuracy of one order higher than that of the EEP simplified form. Taking the one-dimensional Galerkin finite element as an example, the formulas and mathematical proofs of the EEP enhanced form are given. Theoretical analysis and numerical verification show that using elements with degree m \geqslant 1, the element interior displacements and derivatives calculated by the EEP enhanced form can both gain a convergence order of h^\min (m + 3,2m), and for problems with special coefficients, the convergence accuracy can even reach to the orders of h^\min (m + 5,2m) and h^\min (m + 4,2m). Potential further extension and development of this method are also addressed in this paper.