An Efficient Multigrid Method for Ground State Solution of Bose-Einstein Condensates

TL;DR

This paper introduces a multigrid method that efficiently computes the ground state of Bose-Einstein condensates using finite element discretization, achieving optimal convergence and computational efficiency regardless of nonlinearity.

Contribution

It presents a novel multigrid approach that combines nonlinear eigenvalue problem solving with efficient nonlinear iteration, reducing computational work to near that of linear problems.

Findings

Achieves optimal convergence rate.

Computational work is asymptotically independent of nonlinearity.

Numerical experiments validate efficiency and accuracy.

Abstract

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 work 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.