The authors consider the approximation of the eigenvalues of the Dirichlet problem for the second-order elliptic operator \[ - \nabla \cdot (A \nabla u) + cu = \lambda u \] with \(c \geq 0\) using continuous piecewise polynomial finite elements of order \(k \in \mathbb{N}\). Several authors have studied the convergence and complexity of adaptive finite element approximations including a posteriori error estimates for simple eigenvalues. In the present paper, multiple eigenvalues are admitted.
The authors define an a posteriori error estimator which is used to design an adaptive finite element algorithm. Error estimates for the gap between the continuous and discrete invariant subspaces of eigenfunctions are given. The contraction property and the asymptotic quasi-optimal convergence of the algorithm is proved. The proofs are involved, frequently extending over several pages.
Three numerical examples related to the Laplacian with different potentials are presented to support the theory.