A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation

TL;DR

This paper introduces a finite-element toolbox for computing stationary solutions of the Gross-Pitaevskii equation with rotation, featuring mesh adaptivity, multiple numerical methods, and user-friendly interfaces for simulating Bose-Einstein condensates.

Contribution

It provides a versatile, easy-to-use finite-element toolbox integrated with FreeFem++ for efficient simulation of rotating Bose-Einstein condensates, including mesh adaptivity and post-processing tools.

Findings

Efficient computation of stationary solutions with mesh adaptivity.

Implementation of robust numerical methods for energy minimization.

Versatile tools for physical feature extraction from simulations.

Abstract

We present a new numerical system using classical finite elements with mesh adaptivity for computing stationary solutions of the Gross-Pitaevskii equation. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software available for all existing operating systems. This offers the advantage to hide all technical issues related to the implementation of the finite element method, allowing to easily implement various numerical algorithms.Two robust and optimised numerical methods were implemented to minimize the Gross-Pitaevskii energy: a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization library Ipopt. For both methods, mesh adaptivity strategies are implemented to reduce the computational time and increase the local spatial accuracy when vortices are present. Different run cases are made available for 2D and 3D configurations of Bose-Einstein condensates in rotation. An optional graphical user interface is also provided, allowing to easily run predefined cases or with user-defined parameter files. We also provide several post-processing tools (like the identification of quantized vortices) that could help in extracting physical features from the simulations. The toolbox is extremely versatile and can be easily adapted to deal with different physical models.