A Cascadic Multigrid Method for GPE Problem

TL;DR

This paper introduces a cascadic multigrid method for efficiently solving the ground state eigenvalue problem in the GPE, achieving optimal convergence and computational efficiency through a multilevel correction scheme.

Contribution

It presents a novel cascadic multigrid approach based on multilevel correction for the GPE problem, improving computational efficiency and convergence.

Findings

Achieves optimal convergence rate

Reduces computational work significantly

Validated by numerical experiments

Abstract

A cascadic multigrid method is proposed for the GPE problem based on the multilevel correction scheme. With this new scheme, the ground state eigenvalue problem on the finest space can be solved by smoothing steps on a series of multilevel finite element spaces and some nonlinear eigenvalue problem solving on a very low-dimensional space. Choosing the appropriate sequence of finite element spaces and the number of smoothing steps, the optimal convergence rate with the optimal computational work can be arrived. Some numerical experiments are presented to validate our theoretical analysis.