Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

TL;DR

This paper proves optimal convergence rates for adaptive algorithms computing guaranteed lower bounds of eigenvalues of the Laplace operator, using stabilized nonconforming finite element methods and advanced analytical techniques.

Contribution

It introduces a new convergence analysis for adaptive eigenvalue solvers with proven optimal rates, extending known methods and including a medius analysis for best-approximation.

Findings

Proven optimal convergence rates for adaptive eigenvalue bounds.

Development of a new discrete reliability estimate in 3D.

Enhanced analysis techniques for locally refined triangulations.

Abstract

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the $m$-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix-Raviart ($m=1$) or Morley ($m=2$) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated $L^2$ error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in $3$D and highlights a new version of discrete reliability.