Sharp bounds on the smallest eigenvalue of finite element equations with arbitrary meshes without regularity assumptions

TL;DR

This paper establishes mesh-regularity-independent lower bounds for the smallest eigenvalue of finite element matrices on arbitrary meshes, including highly adaptive and anisotropic ones, improving accuracy over previous bounds.

Contribution

It provides the first mesh-regularity-free lower bound for finite element eigenvalues applicable to arbitrary conforming simplicial meshes.

Findings

The bounds are valid for highly non-uniform meshes.

Numerical examples show improved accuracy over previous bounds.

Bounds depend only on degrees of freedom and local mesh properties.

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.