A Multigrid Method for the Ground State Solution of Bose-Einstein Condensates Based on Newton Iteration

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose-Einstein condensates based on Newton iteration method. Instead of treating eigenvalue $\lambda$ and eigenvector $u$ respectively, we regard the eigenpair $(\lambda ,u)$ as one element in the composite space $\mathbb{R}\times H_0^1(\mathrm{\Omega })$ and then Newton iteration method is adopted for the nonlinear problem. Thus in this multigrid scheme, we only need to solve a linear discrete boundary value problem in every refined space, which can improve the overall efficiency for the simulation of Bose-Einstein condensations.