  In this paper we study finite element discretizations of a surface
vector-Laplace eigenproblem. We consider two known classes of finite element
methods, namely one based on a vector analogon of the Dziuk-Elliott surface
finite element method and one based on the so-called trace finite element
technique. A key ingredient in both classes of methods is a penalization method
that is used to enforce tangentiality of the vector field in a weak sense. This
penalization and the perturbations that arise from numerical approximation of
the surface lead to essential nonconformities in the discretization of the
variational formulation of the vector-Laplace eigenproblem. We present a
general abstract framework applicable to such nonconforming discretizations of
eigenproblems. Error bounds both for eigenvalue and eigenvector approximations
are derived that depend on certain consistency and approximability parameters.
Sharpness of these bounds is discussed. Results of a numerical experiment
illustrate certain convergence properties of such finite element
discretizations of the surface vector-Laplace eigenproblem.
