Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates (Q6191363)

This paper focus on numerical analysis of guaranteed lower bounds for eigenvalue of \(m\)-th Laplace operator with \(m=1\) or \(m=2\). Guaranteed lower Dirichlet eigenvalue bounds are computed with an extra-stabilized finite element eigensolver, where either the nonconforming Crouzeix-Raviart finite element is used for the standard Laplace operator or the nonconforming Morley finite element is used, respectively. The convergence of the guaranteed lower bounds towards a simple eigenvalue is analyzed and the proof of optimal convergence rates is given with the use of an adaptive mesh-refining algorithm in 3D. The proof is based on a generalization of the axioms of adaptivity. Further, the results of best-approximation is shown in a weaker Sobolev norm with the use of medius analysis.