Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters

Abstract

Proofs of convergence of adaptive finite element methods (AFEMs) for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai, He, and Zhou in [IMA J. Numer. Anal., 35 (2015), pp. 1934--1977], who proved convergence and optimality of AFEMs for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (noncomputable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. In [Numer. Math., 130 (2015), pp. 467--496], Gallistl used a similar tool to prove that a standard AFEM 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 nonconstant diffusion coefficients, however, the arguments of Dai, He, and Zhou and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this article we provide this missing key step, thus showing that standard adaptive FEMs 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.