| Abstract: |
| We present several numerical tools using classical finite elements with mesh adaptivity for solving
different models used for the study of Bose-Einstein condensates. The programs are written as a toolbox for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms [1]. \
For solving the stationary (imaginary-time) Gross-Pitaevskii equation, we use two robust and optimised numerical methods:
a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization
library IPOPT. A very recent conjugate-gradient method using concepts of Riemannian optimization is also presented [2]. \
For the Bogoliubov-de Gennes system, representing a linearisation of the Gross-Pitaevskii equation, a Newton method
and a fast algorithm based on ARPACK for the calculation of eigenvalues are available. For the real-time Gross-Pitaevskii
equation, classical splitting and relaxation methods were implemented and intensively tested. \
Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments:
single-line vortex, Abrikosov lattice, giant vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates [3].\
[1] G. Vergez, I. Danaila, S. Auliac, F.Hecht, A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation,
Computer Physics Communications, 209, p. 144-162, 2016. \
[2] I. Danaila, B. Protas, Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization, SIAM J. Sci. Comput.
Vol. 39, No. 6, pp. B1102-B1129, 2017. \
[3] I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent
Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016. |
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