An optimal adaptive FEM for eigenvalue clusters

The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.