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　efficiency　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.