  An efficient multigrid method is proposed to compute the ground state
solution of Bose-Einstein condensations by the finite element method based on
the combination of the multigrid method for nonlinear eigenvalue problem and an
efficient implementation for the nonlinear iteration. The proposed numerical
method not only has the optimal convergence rate, but also has the
asymptotically optimal computational work which is independent from the
nonlinearity of the problem. The independence from the nonlinearity means that
the asymptotic estimate of the computational work can reach almost the same as
that of solving the corresponding linear boundary value problem by the
multigrid method. Some numerical experiments are provided to validate the
efficiency of the proposed method.
