Abstract: The adaptive finite element method (AFEM) based on the element energy projection (EEP) technique has succeeded in solving a wide range of problems, while few studies have been done on its adaptive characteristics. Taking the second-order ordinary differential equation as the model problem, the adaptive methods based on both the EEP and super-convergent patch recovery (SPR) solutions were studied. The meshes generated by the two adaptive methods were compared with the optimum mesh. Furthermore, an estimate formula reflecting the characteristics of AFEM was proposed with an adaptive convergence rate β. Numerical experiments show that, with the element of degree m, both SPR and EEP can be well applied for the smooth problem and the convergence rate achieved m+1. For the singular problem with a singularity factor α(<1), SPR failed while EEP-based adaptive procedure gained the convergence rate of m+α, a much higher rate than that of the conventional finite element method α.