SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 983-997, January 2021.
A proof for the lower bound is provided for the smallest eigenvalue of finite element equations with arbitrary conforming simplicial meshes. The bound has a similar form to the one by Graham and McLean [SIAM J. Numer. Anal., 44 (2006), pp. 1487--1513] but doesn't require any mesh regularity assumptions, neither global nor local. In particular, it is valid for highly adaptive, anisotropic, or nonregular meshes without any restrictions. In three and more dimensions, the bound depends only on the number of degrees of freedom $N$ and the Hölder mean $M_{1-d/2} (\lvert \tilde{\omega} \rvert / \lvert \omega_i \lvert)$ taken to the power $1-2/d$, $\lvert \tilde{\omega} \rvert$ and $\lvert \omega_i \rvert$ denoting the average mesh patch volume and the volume of the patch corresponding to the $i$th mesh node, respectively. In two dimensions, the bound depends on the number of degrees of freedom $N$ and the logarithmic term $(1 + \lvert \ln (N \lvert \omega_{\min} \rvert) \rvert)$, $\lvert \omega_{\min} \rvert$ denoting the volume of the smallest patch. Provided numerical examples demonstrate that the bound is more accurate and less dependent on the mesh nonuniformity than the previously available bounds.

中文翻译：

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无规则假设的任意网格有限元方程的最小特征值的尖锐界

SIAM数值分析学报，第59卷，第2期，第983-997页，2021年1月。
对于具有任意一致的简单网格的有限元方程的最小特征值，提供了下界的证明。装订线的格式与Graham和McLean [SIAM J. Numer。Anal。，44（2006），pp。1487--1513]，但不需要任何网格规则性假设，无论是全局的还是局部的。尤其适用于没有任何限制的高度自适应，各向异性或非规则网格。在三个或更多个维度中，边界仅取决于自由度$ N $的数量，而霍尔德均值$ M_ {1-d / 2}（\ lvert \ tilde {\ omega} \ rvert / \ lvert \ omega_i \ lvert）$乘以幂1-2 / d $，$ \ lvert \ tilde {\ omega} \ rvert $和$ \ lvert \ omega_i \ rvert $表示平均网格补丁量以及与该补丁对应的补丁量第i个网格节点。在两个维度上 边界取决于自由度$ N $的数量和对数项$（1 + \ lvert \ ln（N \ lvert \ omega _ {\ min} \ rvert）\ rvert）$，$ \ lvert \ omega _ {\ min} \ rvert $表示最小补丁的数量。提供的数值示例表明，与以前可用的边界相比，边界更准确，并且对网格非均匀性的依赖性较小。