Co-authors: Ionut Danaila (1), Frédéric Hecht (2) and Sylvain Auliac (2)

(1) Laboratoire de mathématiques Raphaël Salem, Université de Rouen, France.

(2) Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France.

We present GPFEM, a new numerical tool 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++ [1], 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. The first one is a steepest descent method based on Sobolev gradients, developed in [2]. The second one is a minimization algorithm based on the state-of-the-art optimization library Ipopt [3]. For both methods, mesh adaptivity strategies are implemented to reduce the computational time and increase the local spatial accuracy when vortices are present as done in [4]. 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. 

 This work was  supported by the French ANR ANR-12-MONU-0007 BECASIM (Modèles Numérique call). The authors are grateful to  CRIHAN (Centre de Ressources Informatiques de Haute-Normandie, France) for providing computational resources (project 2015001).

 References:

[1] F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuke, FreeFem++ (manual), www.freefem.org.

[2] I.Danaila and P.Kazemi, A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Computing 32, pp. 2447-2467, 2010.

[3]  A. Wächter and L. T. Biegler, On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, Mathematical Programming 106(1), pp. 25-57, 2006.

[4] I. Danaila and F. Hecht, A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates, SIAM J. Sci. Computing 229, pp. 6946-6960, 2010.