A Multigrid Method for the Ground State Solution of Bose-Einstein Condensates

TL;DR

This paper introduces a multigrid method combining finite element and multilevel correction techniques to efficiently compute the ground state of Bose-Einstein condensates, significantly reducing computational effort.

Contribution

It presents a novel multigrid scheme that integrates linear boundary problem solutions and nonlinear eigenvalue problems for efficient Bose-Einstein condensate simulations.

Findings

Achieves near-optimal computational complexity

Validates efficiency through numerical experiments

Improves simulation speed for Bose-Einstein condensates

Abstract

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.