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

• Published in: arXiv.org
• Main Author: Kamenski, Lennard
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.06.2021
• Subjects: Computer Science - Numerical Analysis; Degrees of freedom; Eigenvalues; Finite element method; Lower bounds; Mathematical analysis; Mathematics - Numerical Analysis; Nonuniformity; Regularity
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1908.03460

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.