Adaptive hp -FEM for eigenvalue computations

We design an adaptive procedure for approximating a selected eigenvalue and its eigen-space for a second-order elliptic boundary-value problem, using an hp finite element method. Such iterative procedure judiciously alternates between a stage in which a near-optimal hp-mesh for the current level of accuracy is generated, and a stage in which such mesh is sufficiently refined to produce a new, enhanced approximation of the eigenfunctions. We identify conditions on the initial mesh and the operator coefficients under which the procedure yields approximations that converge at a geometric rate independent of any discretization parameter, using a number of degrees of freedom comparable to the smallest number needed to get the achieved accuracy. We detail the second stage for a single eigenvalue, relying on a p-robust saturation property.