Abstract: The finite element method of lines (FEMOL) is a general and powerful semi-discretized method for BVPs. By viewing it as a generalized one-dimensional method, the well-developed Element Energy Projection (EEP) method for super-convergence computation in one-dimensional FEM can readily be extended to the case of two-dimensional FEMOL. In addition, the successful self-adaptive strategy in one-dimensional FEM can also be extended to the two-dimensional FEMOL analysis. By now, a series of satisfactory progress has been made in the two-dimensional problems of the Poisson Equation and the plane elasticity. The present paper intends to give a brief report on the recent progress and some numerical results. The paper briefly describes the idea of the super-convergent EEP method and corresponding self-adaptive strategy for two-dimensional FEMOL analysis, which forms a clean, simple, effective and reliable algorithm that can adaptively produce FEMOL results on arbitrary geometric domains with the displacement accuracy point-wisely satisfying the user specified error tolerance in max-norm. Sufficient and representative numerical examples are given to demonstrate the effectiveness and reliability of the proposed algorithm.