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

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 code various numerical algorithms. Two robust and optimized 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 used 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. Program summaryProgram title: GPFEMCatalogue identifier: AFBD_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AFBD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Apache 2.0No. of lines in distributed program, including test data, etc.: 49149No. of bytes in distributed program, including test data, etc.: 407572Distribution format: tar.gzProgramming language: FreeFem++ (free software, www.freefem.org).Computer: PC, Mac, Super-computer.Operating system: Windows, Mac OS, Linux.Classification: 2.7, 4.9, 7.7.Nature of problem:The software computes 2D or 3D stationary solutions of the Gross–Pitaevskii equation with rotation. The main application is the computation of different types of vortex states (Abrikosov vortex lattice, giant vortex) in rotating Bose Einstein condensates. The software can be easily modified to take into account different related physical models.Solution method:The user has the choice between two robust and optimized numerical methods for the direct minimization of 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.Running time:From minutes for 2D configurations to hours for 3D cases (on a personal laptop). Complex 3D cases (with hundreds of vortices) may require several days of computational time.