A multigrid method for the ground state solution of Bose–Einstein condensates based on Newton iteration

Abstract

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of treating eigenvalue \(\lambda \) and eigenvector u separately, we regard the eigenpair \((\lambda , u)\) as one element in the composite space \({\mathbb {R}} \times H_0^1(\varOmega )\) and then Newton iteration step is adopted for the nonlinear problem. Thus in this multigrid scheme, the main computation is 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.