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

TL;DR

This paper introduces a novel multigrid method based on Newton iteration for efficiently computing the ground state of Bose-Einstein condensates by solving a nonlinear eigenproblem as a single entity.

Contribution

It proposes a new multigrid approach that treats the eigenpair as a combined element, simplifying the solution process and enhancing computational efficiency.

Findings

Reduces the complexity of solving nonlinear eigenproblems.

Improves efficiency in simulating Bose-Einstein condensates.

Uses linear boundary value problems at each refinement level.

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 method. Instead of treating eigenvalue $\lambda$ and eigenvector $u$ respectively, we regard the eigenpair $(\lambda, u)$ as one element in the composite space $\R \times H_0^1(\Omega)$ 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.