A theoretical proof of superiority of Smoothed Finite Element Method over the conventional FEM

Numerous simulations have shown that Smoothed Finite Element Method (S-FEM) performs better than the standard FEM. However, there is lack of rigorous mathematical proof on such a claim. This task is challenging since there are so many variants of S-FEM and the standard FEM theory in Sobolev space does not work for S-FEM because of the Smoothed Gradient. Another long-standing open problem is to establish the theory of $\alpha$FEM parameter. The $\alpha$FEM could be the most flexible and fastest S-FEM variant. Its energy is even exact if the parameter is fine-tuned. So this problem is practical and interesting. By the help of nonlinear essential boundary (geometry), Weyl inequalities (algebra) and matrix differentiation (analysis), this parameter problem leads us to estimate the eigenvalue-gap and energy-gap between S-FEM and FEM. Consequently, we provide a definite answer to the long-standing S-FEM superiority problem in a unified framework. The essential boundary, eigenvalue and energy are linked together by four new necessary and sufficient conditions which are simple, practical and beyond our expectations. The standard S-FEM source code can be reused so it is convenient to numerically implement. Finally, the cantilever and infinite plate with a circular hole are simulated to verify the proof.