  A multigrid method is proposed to compute the ground state solution of
Bose-Einstein condensations by the finite element method based on the
multilevel correction for eigenvalue problems and the multigrid method for
linear boundary value problems. In this scheme, obtaining the optimal
approximation for the ground state solution of Bose-Einstein condensates
includes a sequence of solutions of the linear boundary value problems by the
multigrid method on the multilevel meshes and a series of solutions of
nonlinear eigenvalue problems on the coarsest finite element space. The total
computational work of this scheme can reach almost the optimal order as same as
solving the corresponding linear boundary value problem. Therefore, this type
of multigrid scheme can improve the overall efficiency for the simulation of
Bose-Einstein condensations. Some numerical experiments are provided to
validate the efficiency of the proposed method.
