A multi-mesh adaptive finite element method for solving the Gross-Pitaevskii equation

TL;DR

This paper introduces a multi-mesh adaptive finite element method for efficiently solving the Gross-Pitaevskii equation, reducing computational costs while maintaining accuracy by using multiple meshes tailored to different solution regions.

Contribution

The paper presents a novel multi-mesh adaptive finite element approach for the GPE, improving efficiency over traditional single-mesh methods by adaptively refining multiple meshes based on solution variation.

Findings

Multi-mesh method achieves same accuracy as single-mesh with less computation

Numerical experiments validate efficiency and accuracy improvements

Method effectively handles regions with sharp and smooth variations

Abstract

It is found that the wave functions of the Gross-Pitaevskii equation (GPE) often vary significantly in different spatial regions, with some components exhibiting sharp variations while others remain smooth. Solving the GPE on a single mesh, even with adaptive refinement, can lead to excessive computational costs due to the need to accommodate the most oscillatory solution. To address this issue, we present a multi-mesh adaptive finite element method for solving the GPE. To this end, we first convert it into a time-dependent equation through the imaginary time propagation method. Then the equation is discretized by the backward Euler method temporally and the multi-mesh adaptive finite element method spatially. The proposed method is compared with the single-mesh adaptive method through a series of numerical experiments, which demonstrate that the multi-mesh adaptive method can achieve the same numerical accuracy with less computational consumption.