The adaptive finite element approximation of multiple eigenvalues
leads to the situation of eigenvalue clusters because the
eigenvalues of interest and their multiplicities may not
be resolved by the initial mesh.
In practice, also little perturbations in coefficients or in
the geometry immediately lead to an eigenvalue cluster.
The first part of this talk presents an adaptive algorithm for eigenvalue
clusters and shows optimal decay rates of the error in terms
of degrees of freedom. Numerical tests suggest that the adaptive cluster
approximation is superior compared to the use of an adaptive scheme for
each eigenvalue separately, even if the multiplicities are known.

The optimality in terms of the concept of nonlinear approximation
classes is concerned with the approximation of invariant subspaces
spanned by eigenfunctions of an eigenvalue cluster.
In order to obtain eigenvalue error estimates in the case of
nonconforming finite element approximations, the second part of
the talk presents new error estimates for nonconforming finite elements
which relate the error of the eigenvalue approximation to the error of
the approximation of the invariant subspace.