  The ground states of Bose-Einstein condensates in a rotating frame can be
described as constrained minimizers of the Gross-Pitaevskii energy functional
with an angular momentum term. In this paper we consider the corresponding
discrete minimization problem in Lagrange finite element spaces of arbitrary
polynomial order and we investigate the approximation properties of discrete
ground states. In particular, we prove a priori error estimates of optimal
order in the $L^2$- and $H^1$-norm, as well as for the ground state energy and
the corresponding chemical potential. A central issue in the analysis of the
problem is the missing uniqueness of ground states, which is mainly caused by
the invariance of the energy functional under complex phase shifts. Our error
analysis is therefore based on an Euler-Lagrange functional that we restrict to
certain tangent spaces in which we have local uniqueness of ground states. This
gives rise to an error decomposition that is ultimately used to derive the
desired a priori error estimates. We also present numerical experiments to
illustrate various aspects of the problem structure.
