  We propose a positivity preserving finite element discretization for the
nonlinear Gross-Pitaevskii eigenvalue problem. The method employs mass lumping
techniques, which allow to transfer the uniqueness up to sign and positivity
properties of the continuous ground state to the discrete setting. We further
prove that every non-negative discrete excited state up to sign coincides with
the discrete ground state. This allows one to identify the limit of fully
discretized gradient flows, which are typically used to compute the discrete
ground state, and thereby establish their global convergence. Furthermore, we
perform a rigorous a priori error analysis of the proposed non-standard finite
element discretization, showing optimal orders of convergence for all unknowns.
Numerical experiments illustrate the theoretical results of this paper.
