Convergence and optimality of higher-order adaptive finite element methods for eigenvalue clusters

TL;DR

This paper proves that higher-order adaptive finite element methods reliably converge at optimal rates for clustered eigenvalues of elliptic problems, extending previous results to more complex finite element spaces.

Contribution

It provides the missing link showing equivalence of practical and theoretical estimators for higher-order elements and eigenvalue clusters, ensuring convergence and optimality.

Findings

Higher-order AFEM converges with optimal rate for eigenvalue clusters

The bulk marking parameter can be chosen independently of eigenvalue properties

Convergence results hold under a mesh fineness condition

Abstract

Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai et. al. (2014), who proved convergence and optimality of AFEM for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (non-computable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. Gallistl (2015) used a similar tool in order to prove that a standard adaptive FEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or non-constant diffusion coefficients, however, the arguments of Dai et. al. and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this note we provide this missing key step, thus showing that standard adaptive FEM for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.