3 research outputs found
Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters
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.Comment: 10 page