  We present numerical upscaling techniques for a class of linear second-order
self-adjoint elliptic partial differential operators (or their high-resolution
finite element discretization). As prototypes for the application of our theory
we consider benchmark multi-scale eigenvalue problems in reservoir modeling and
material science. We compute a low-dimensional generalized (possibly mesh free)
finite element space that preserves the lowermost eigenvalues in a
superconvergent way. The approximate eigenpairs are then obtained by solving
the corresponding low-dimensional algebraic eigenvalue problem. The rigorous
error bounds are based on two-scale decompositions of $H^1_0(\Omega)$ by means
of a certain Cl\'ement-type quasi-interpolation operator.
