A posteriori error analysis for nonconforming approximation of multiple eigenvalues

TL;DR

This paper develops and validates an a posteriori error estimator for nonconforming finite element approximations of Laplace eigenvalues, effectively handling multiple eigenvalues and supporting adaptive algorithms.

Contribution

It extends existing error estimators to be robust for multiple eigenvalues in nonconforming finite element methods, with theoretical and numerical validation.

Findings

Estimator is robust for multiple eigenvalues

Numerical examples confirm theoretical predictions

Adaptive algorithm converges effectively with multiple eigenvalues

Abstract

In this paper we study an a posteriori error indicator introduced in E. Dari, R.G. Duran, C. Padra, Appl. Numer. Math., 2012, for the approximation of the Laplace eigenvalue problem with Crouzeix-Raviart non-conforming finite elements. In particular, we show that the estimator is robust also in presence of eigenvalues of multiplicity greater than one. Some numerical examples confirm the theory and illustrate the convergence of an adaptive algorithm when dealing with multiple eigenvalues.