Sharp Bounds on the Smallest Eigenvalue of Finite Element Equations with Arbitrary Meshes without Regularity Assumptions

Abstract

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 as 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 non-regular meshes without any restrictions. In three and more dimensions, the bound depends only on the number of degrees of freedom $N$ and the H\"older 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 $i^{\text{th}}$ 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 non-uniformity than the previously available bounds.