This work presents a new methodology for computing ground states of Bose-Einstein
condensates based on finite element discretizations on two different scales of numerical
resolution. In a pre-processing step, a low-dimensional (coarse) generalized finite
element space is constructed. It is based on a local orthogonal decomposition and
exhibits high approximation properties. The non-linear eigenvalue problem that characterizes
the ground state is solved by some suitable iterative solver exclusively in this low-dimensional
space, without loss of accuracy when compared with the solution of the full fine scale
problem. The pre-processing step is independent of the types and numbers of bosons.
A post-processing step further improves the accuracy of the method. We present rigorous
a priori error estimates that predict convergence rates H^3 for the ground state eigenfunction
and H^4 for the corresponding eigenvalue without pre-asymptotic effects; H being the
coarse scale discretization parameter. Numerical experiments indicate that these high
rates may still be pessimistic.