×

A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation. (English) Zbl 1380.65389

Summary: 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++ (http://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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y15 Packaged methods for numerical algorithms
65Z05 Applications to the sciences
65-04 Software, source code, etc. for problems pertaining to numerical analysis
82B10 Quantum equilibrium statistical mechanics (general)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Pitaevskii, L. P.; Stringari, S., Bose-Einstein condensation, (2003), Clarendon Press Oxford · Zbl 1110.82002
[2] Minguzzi, A.; Succi, S.; Toschi, F.; Tosi, M. P.; Vignolo, P., Phys. Rep., 395, 223-355, (2004)
[3] Bao, W., (Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-González, R., Transport Phenomena and Kinetic Theory: Applications to Gases, Semiconductors, Photos, and Biological Systems, Series Modeling and Simulation in Science, (2006), Engineering and Technology, Birkhauser), 215-255
[4] Bao, W.; Cai, Y., Kinet. Relat. Models, 6, 1-135, (2013)
[5] Antoine, X.; Besse, C.; Bao, W., Comput. Phys. Comm., 184, 12, 2621-2633, (2013)
[6] W. Bao, Proceedings of the International Congress of Mathematicians (Seoul 2014) IV, 2014, pp. 971-996.
[7] (Barenghi, C. F.; Donnelly, R. J.; Vinen, W. F., Quantized Vortex Dynamics and Superfluid Turbulence, Lecture Notes in Physics, vol. 571, (2001), Springer) · Zbl 0972.00023
[8] (Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-González, R., Emergent Nonlinear Phenomena in Bose-Einstein Condensates, Atomic, Optical, and Plasma Physics, vol. 45, (2008), Springer) · Zbl 1137.82003
[9] (Barenghi, C. F.; Sergeev, Y. A., Vortices and Turbulence at Very Low Temperatures, CISM International Centre for Mechanical Sciences, vol. 501, (2008), Springer)
[10] (Halperin, B.; Tsubota, M., Quantum Turbulence, Progress in Low Temperature Physics, vol. 16, (2009), Springer)
[11] Kasamatsu, K.; Machida, M.; Sasa, N.; Tsubota, M., Phys. Rev. A, 71, (2005)
[12] Berloff, N. G., Phys. Rev. A, 69, (2004)
[13] Aftalion, A.; Danaila, I., Phys. Rev. A, 68, (2003), 023603(1-6)
[14] Aftalion, A.; Danaila, I., Phys. Rev. A, 69, (2004), 033608(1-6)
[15] Danaila, I., Phys. Rev. A, 72, (2005), 013605(1-6)
[16] Kasamatsu, K.; Tsubota, M., Prog. Low Temp. Phys., 16, 351-403, (2008)
[17] García-Ripoll, J. J.; Pérez-García, V. M., Phys. Rev. A, 64, (2001)
[18] García-Ripoll, J. J.; Pérez-García, V. M., SIAM J. Sci. Comput., 23, 1315-1333, (2001)
[19] Zeng, R.; Zhang, Y., Comput. Phys. Comm., 180, 854-860, (2009)
[20] Bao, W.; Du, Q., SIAM J. Sci. Comput., 25, 1674, (2004)
[21] Bao, W.; Chern, I.-L.; Lim, F. Y., J. Comput. Phys., 219, 836-854, (2006)
[22] Bao, W.; Shen, J., J. Comput. Phys., 227, 9778-9793, (2008)
[23] Farhat, C.; Toivanen, J., J. Comput. Phys., 231, 4709-4722, (2012)
[24] Tiwari, R. P.; Shukla, A., Comput. Phys. Comm., 174, 12, 966-982, (2006)
[25] Dion, C. M.; Cancès, E., Comput. Phys. Comm., 177, 787-798, (2007)
[26] Hohenester, U., Comput. Phys. Comm., 185, 1, 194-216, (2014)
[27] Muruganandam, P.; Adhikari, S., Comput. Phys. Comm., 180, 10, 1888-1912, (2009)
[28] Caliari, M.; Rainer, S., Comput. Phys. Comm., 184, 3, 812-823, (2013)
[29] Vudragović, D.; Vidanović, I.; Balaz, A.; Muruganandam, P.; Adhikari, S. K., Comput. Phys. Comm., 183, 9, 2021-2025, (2012)
[30] Caplan, R., Comput. Phys. Comm., 184, 4, 1250-1271, (2013)
[31] Antoine, X.; Duboscq, R., Comput. Phys. Comm., 185, 11, 2969-2991, (2014)
[32] Aftalion, A.; Du, Q., Phys. Rev. A, 64, (2001)
[33] Bao, W.; Tang, W., J. Comput. Phys., 187, 230-254, (2003)
[34] Baksmaty, L. O.; Liub, Y.; Landmanc, U.; Bigelowd, N. P.; Pu, H., Math. Comput. Simulation, 80, 131-138, (2009)
[35] Danaila, I.; Hecht, F., J. Comput. Phys., 229, 6946-6960, (2010)
[36] Hecht, F., J. Numer. Math., 20, 251-266, (2012)
[37] F. Hecht, O. Pironneau, A.L. Hyaric, K. Ohtsuke, FreeFem++ (manual), 2007, www.freefem.org.
[38] Danaila, I.; Kazemi, P., SIAM J. Sci. Comput., 32, 2447-2467, (2010)
[39] A. Wächter, An interior point algorithm for large-scale nonlinear optimization with applications in process engineering (Ph.D. thesis), Carnegie Mellon University, Pittsburgh, PA, USA.
[40] Dapogny, C. D.C.; Frey, P., J. Comput. Phys., 262, 358-378, (2014)
[41] C. Dobrzynski, P. Frey, MMG3D: User Guide. [Technical Report] RT-0422, INRIA hal-00681813, 2012.
[42] Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Rev. Modern Phys., 71, 463-512, (1999)
[43] Bagnato, V. S.; Frantzeskakis, D. J.; Kevrekidis, P. G.; Malomed, B. A.; Mihalache, D., Romanian Rep. Phys., 67, 5-50, (2015)
[44] Fetter, A. L.; Jackson, B.; Stringari, S., Phys. Rev. A, 71, (2005)
[45] Tsubota, M.; Kasamatsu, K.; Ueda, M., Phys. Rev. A, 65, (2002)
[46] Aftalion, A.; Rivière, T., Phys. Rev. A, 64, (2001)
[47] Bretin, V.; Stock, S.; Seurin, Y.; Dalibard, J., Phys. Rev. Lett., 92, (2004)
[48] Aftalion, A., Vortices in Bose-Einstein condensates, (2006), Birkhauser · Zbl 1129.82004
[49] Madison, K. W.; Chevy, F.; Wohlleben, W.; Dalibard, J., J. Modern Opt., 47, 2715, (2000)
[50] Madison, K. W.; Chevy, F.; Bretin, V.; Dalibard, J., Phys. Rev. Lett., 86, 4443, (2001)
[51] Rosenbusch, P.; Bretin, V.; Dalibard, J., Phys. Rev. Lett., 89, (2002)
[52] M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi, R. Ulerich, GNU Scientific Library Reference Manual third ed., ISBN 0954612078, 2015, www.gnu.org.
[53] Borouchaki, H.; Castro-Diaz, M. J.; George, P. L.; Hecht, F.; Mohammadi, B., (5th Inter. Conf. on Numerical Grid Generation in Computational Field Simulations, (1996), Mississipi State Univ.)
[54] Castro-Diaz, M.; Hecht, F.; Mohammadi, B., Int. J. Comput. Fluid Dyn., 25, 475-491, (2000)
[55] Hecht, F.; Mohammadi, B., AIAA Pap., 97, 0859, (1997)
[56] George, P. L.; Borouchaki, H., Delaunay triangulation and meshing, (1998), Hermès Paris
[57] P.J. Frey, Medit: An Interactive Mesh Visualisation Software, RT-0253, INRIA, 2001.
[58] Nocedal, A. W.J.; Waltz, R. A., SIAM J. Optim., 19, 4, 1674-1693, (2008)
[59] Wächter, A.; Biegler, L. T., Math. Program., 106, 1, 25-57, (2006)
[60] S. Auliac, Développement d’outils d’optimisation pour Freefem++, Thèse, Université Pierre et Marie Curie, Paris, France, 2014.
[61] Madison, K. W.; Chevy, F.; Wohlleben, W.; Dalibard, J., Phys. Rev. Lett., 84, 806, (2000)
[62] Bretin, V.; Rosenbusch, P.; Chevy, F.; Shlyapnikov, G.; Dalibard, J., Phys. Rev. Lett., 90, (2003)
[63] Stringari, S., Phys. Rev. Lett., 82, 4371, (1999)
[64] T. Williams, C. Kelley, Gnuplot 5.0 : An Interactive Plotting Programm, 2015, http://www.Gnuplot.info/.
[65] M. Kilgard, GLUT 3.7, 2000, https://www.opengl.org/resources/libraries/glut/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.