  In this paper, we propose a new numerical method to compute the ground state
solution of trapped interacting Bose-Einstein condensation (BEC) at zero or
very low temperature by directly minimizing the energy functional via finite
element approximation. As preparatory steps we begin with the 3d
Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and
show how to reduce it to 2d and 1d GPEs. The ground state solution is
formulated by minimizing the energy functional under a constraint, which is
discretized by the finite element method. The finite element approximation for
1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical
symmetry are presented in detail and approximate ground state solutions, which
are used as initial guess in our practical numerical computation of the
minimization problem, of the GPE in two extreme regimes: very weak interactions
and strong repulsive interactions are provided. Numerical results in 1d, 2d
with radial symmetry and 3d with spherical symmetry and cylindrical symmetry
for atoms ranging up to millions in the condensation are reported to
demonstrate the novel numerical method. Furthermore, comparisons between the
ground state solutions and their Thomas-Fermi approximations are also reported.
Extension of the numerical method to compute the excited states of GPE is also
presented.
