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 αFEM parameter. The αFEM 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.

中文翻译：

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平滑有限元方法优于传统有限元法的理论证明


大量仿真表明，平滑有限元法 （S-FEM） 的性能优于标准 FEM。然而，这种说法缺乏严格的数学证明。这项任务具有挑战性，因为 S-FEM 有很多变体，而且由于平滑梯度，Sobolev 空间中的标准 FEM 理论不适用于 S-FEM。另一个长期悬而未决的问题是建立 αFEM 参数的理论。αFEM 可能是最灵活、最快速的 S-FEM 变体。如果对参数进行微调，它的能量甚至会很精确。所以这个问题很实用，也很有趣。借助非线性基本边界（几何）、外尔不等式（代数）和矩阵微分（分析），该参数问题引导我们估计 S-FEM 和 FEM 之间的特征值间隙和能量间隙。因此，我们在统一的框架中为长期存在的 S-FEM 优势问题提供了明确的答案。基本边界、特征值和能量通过四个新的必要和充分条件联系在一起，这些条件简单、实用且超出我们的预期。标准 S-FEM 源代码可以重复使用，因此便于数值实现。最后，对悬臂和圆孔无限板进行仿真，验证了验证。