Comparison of guaranteed lower eigenvalue bounds with three skeletal schemes

Abstract

Specially tailored skeletal schemes enable cell and face variables linked with a stabilisation and a fine-tuned parameter can provide guaranteed lower eigenvalue bounds for the Laplacian. This paper briefly presents a unified derivation of skeletal higher-order methods from Carstensen, Zhai, and Zhang (2020), Carstensen, Ern, and Puttkammer (2021), and Carstensen, Gräßle, and Tran (2024). It suggests a paradigm shift from conditional to unconditional lower eigenvalue bounds. Adaptive mesh-refining leads to optimal convergence rates in computational benchmark examples and underlines the superiority of higher-order methods.