Abstract: Based on the element energy projection (EEP) method for super-convergence calculation, this paper attempts to extend the recently proposed ‘fixed-end method’, a priori quantitative error estimate for one-dimensional finite element method (FEM), to two-dimensional (2D) FEM. In this approach, the EEP method is used to estimate the errors a priori on each individual 2D element and a desirable mesh can be accordingly generated immediately without the need for obtaining FE solutions in advance. Taking the Poisson equation as the model problem, some initial numerical results are given to show the validity and effectiveness of the proposed technique.