\title{An Iterative Finite Element Method with Adaptivity for Multiple Eigenvalues} \tocauthor{S. Giani} \author{} \institute{} \maketitle \begin{center} {\large Stefano Giani}\\ University of Nottingham\\ {\tt Stefano.Giani@nottingham.ac.uk} \\ \vspace{4mm}{\large Pavel Solin}\\ University of Nevada\\ {\tt solin@unr.edu} \end{center} \section*{Abstract} We consider the task of resolving accurately the $n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate $n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. Especially when adaptivity is used to target an eigenpair of a multiple eigenvalue, the change in the linear combination of eigenfunctions may reduce the convergence rate of the method. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls the eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue-eigenfunction pair on a sequence of locally refined meshes. Both Picard's and Newton's variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp-FEM) but the algorithm also works for standard low-order FEM. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{luka} {\sc L. Grubisic And J. S. Ovall}. {On estimators for eigenvalue/eigenvector approximations}. Mathematics of Computation, 78:266 (2008), pp. 739-770. \bibitem{hp} {\sc P. Solin and D. Andrs and J. Cerveny And M. Simko}. {PDE-independent adaptive hp-fem based on hierarchic extension of finite element spaces}. J. Comput. Appl. Math., 233 (2010), pp. 3086-3094.. \bibitem{solin} {\sc P. Solin and K. Segeth and I. Dolezel}. {Higher-order finite element methods}. Chapman \& Hall, CRC Press, London, 2003.. \end{thebibliography}