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All issues Volume 60 / No 3 (May-June 2026) ESAIM: M2AN, 60 3 (2026) 1297-1326 References
Open Access
Issue
ESAIM: M2AN
Volume 60, Number 3, May-June 2026
Page(s) 1297 - 1326
DOI https://doi.org/10.1051/m2an/2026034
Published online 01 June 2026
  1. A. Aftalion and Q. Du, Vortices in a rotating Bose-Einstein condensate: critical angular velocities and energy diagrams in the Thomas-Fermi regime. Phys. Rev. A 64 (2001). [Google Scholar]
  2. R. Altmann, P. Henning and D. Peterseim, The J-method for the Gross Pitaevskii eigenvalue problem. Numer. Math. 148 (2021) 575–610. [CrossRef] [MathSciNet] [Google Scholar]
  3. R. Altmann, D. Peterseim and T. Stykel, Energy-adaptive Riemannian optimization on the Stiefel manifold. ESAIM Math. Model. Numer. Anal. 56 (2022) 1629–1653. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  4. R. Altmann, D. Peterseim and T. Stykel, Riemannian Newton methods for energy minimization problems of Kohn-Sham type. J. Sci. Comput. 101 (2024). [Google Scholar]
  5. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269 (1995) 198–201. [Google Scholar]
  6. X. Antoine, A. Levitt and Q. Tang, Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods. J. Comput. Phys. 343 (2017) 92–109. [CrossRef] [MathSciNet] [Google Scholar]
  7. P. Antonelli, P. Cannarsa and B. Shakarov, Existence and asymptotic behavior for L2-norm preserving nonlinear heat equations. Calc. Var. Partial Differ. Equ. 63 (2024) 1–31. [Google Scholar]
  8. I. Bailleul and F. Bernicot, Heat semigroup and singular PDEs. J. Funct. Anal. 270 (2016) 3344–3452. [Google Scholar]
  9. W. Bao and W. Tang, Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional. J. Comput. Phys. 187 (2003) 230–254. [Google Scholar]
  10. W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674–1697. [CrossRef] [MathSciNet] [Google Scholar]
  11. W. Bao and H. Wang, A mass and magnetization conservative and energy-diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates. SIAM J. Numer. Anal. 45 (2007) 2177–2200. [Google Scholar]
  12. W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Mod. 6 (2013) 1–135. [Google Scholar]
  13. W. Bao, I.-L. Chern and F. Y. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates. J. Comput. Phys. 219 (2006) 836–854. [Google Scholar]
  14. S. Bose, Plancks Gesetz und Lichtquantenhypothese. Z. Phys. 26 (1924) 178–181. [NASA ADS] [CrossRef] [Google Scholar]
  15. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, in Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York (2008). [Google Scholar]
  16. E. Cancès, SCF algorithms for HF electronic calculations, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, Vol. 74 of Lecture Notes in Chemistry. Springer, Berlin (2000) 17–43. [Google Scholar]
  17. E. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree Fock equations. ESAIM Math. Model. Numer. Anal. 34 (2000) 749–774. [Google Scholar]
  18. E. Cancès, R. Chakir and Y. Maday, Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45 (2010) 90–117. [CrossRef] [MathSciNet] [Google Scholar]
  19. E. Cancès, G. Kemlin and A. Levitt, Convergence analysis of direct minimization and self-consistent iterations. SIAM J. Matrix Anal. Appl. 42 (2021) 243–274. [CrossRef] [MathSciNet] [Google Scholar]
  20. M.M. Cerimele, M.L. Chiofalo, F. Pistella, S. Succi and M.P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates. Phys. Rev. E 62 (2000) 1382–1389. [Google Scholar]
  21. H. Chen, L. He and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics. Comput. Methods Appl. Mech. Eng. 200 (2011) 1846–1865. [Google Scholar]
  22. Z. Chen, J. Lu, Y. Lu and X. Zhang, On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem. SIAM J. Numer. Anal. 62 (2024) 667–691. [CrossRef] [MathSciNet] [Google Scholar]
  23. Z. Chen, J. Lu, Y. Lu and X. Zhang, Fully discretized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem. Math. Comput. 94 (2025) 2723–2760. [Google Scholar]
  24. M.L. Chiofalo, S. Succi and M. Tosi, Ground state of trapped interacting Bose Einstein condensates by an explicit imaginary-time algorithm. Phys. Rev. E 62 (2000) 7438–7444. [Google Scholar]
  25. I. Danaila and P. Kazemi, A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation. SIAM J. Sci. Comput. 32 (2010) 2447–2467. [CrossRef] [MathSciNet] [Google Scholar]
  26. I. Danaila and B. Protas, Computation of ground states of the Gross-Pitaevskii functional via Riemannian optimization. SIAM J. Sci. Comput. 39 (2017) B1102–B1129. [CrossRef] [Google Scholar]
  27. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75 (1995) 3969–3973. [Google Scholar]
  28. C.M. Dion and E. Cancès, Ground state of the time-independent Gross-Pitaevskii equation. Comput. Phys. Commun. 177 (2007) 787–788. [Google Scholar]
  29. G. Dusson and Y. Maday, A posteriori analysis of a nonlinear Gross-Pitaevskii-type eigenvalue problem. IMA J. Numer. Anal. 37 (2017) 94–137. [Google Scholar]
  30. A. Einstein, Quantentheorie des einatomigen idealen Gases. Sitzungsberichte der Preussischen Akademie der Wissenschaften (1924) 261–267. [Google Scholar]
  31. L.C. Evans, Partial Differential Equations, in Vol. 19 of Graduate Studies in Mathematics, 2nd edition. American Mathematical Society (2010). [Google Scholar]
  32. E. Faou and T. Jézéquel, Convergence of a normalized gradient algorithm for computing ground states. IMA J. Numer. Anal. 38 (2018) 360–376. [CrossRef] [MathSciNet] [Google Scholar]
  33. A. Friedman, Partial Differential Equations of Parabolic Type. Dover Books on Mathematics. Dover Publications (2013). [Google Scholar]
  34. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. [Google Scholar]
  35. P. Grisvard, Elliptic Problems in Nonsmooth Domains, in Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). [Google Scholar]
  36. E.P. Gross, Structure of a quantized vortex in boson systems. Il Nuovo Cimento (1955–1965) 20 (1961) 454–477. [Google Scholar]
  37. P. Heid, B. Stamm and T.P. Wihler, Gradient flow finite element discretizations with energy-based adaptivity for the Gross–Pitaevskii equation. J. Comput. Phys. 436 (2021). [Google Scholar]
  38. P. Henning, The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem. Math. Models Methods Appl. Sci. 33 (2023) 1517–1544. [CrossRef] [MathSciNet] [Google Scholar]
  39. P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: global convergence and computational efficiency. SIAM J. Numer. Anal. 58 (2020) 1744–1772. [CrossRef] [MathSciNet] [Google Scholar]
  40. P. Henning and E. Jarlebring, The Gross–Pitaevskii equation and eigenvector nonlinearities: numerical methods and algorithms. SIAM Rev. 67 (2025) 256–317. [Google Scholar]
  41. P. Henning and M. Yadav, On discrete ground states of rotating Bose–Einstein condensates. Math. Comput. 94 (2025) 1–32. [Google Scholar]
  42. J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. [CrossRef] [MathSciNet] [Google Scholar]
  43. E. Jarlebring, S. Kvaal and W. Michiels, An inverse iteration method for eigenvalue problems with eigenvector nonlinearities. SIAM J. Sci. Comput. 36 (2014) A1978–A2001. [CrossRef] [Google Scholar]
  44. J. Jost, The Heat Equation, Semigroups, and Brownian Motion. Springer, New York (2013) 173–206. [Google Scholar]
  45. P. Kazemi and M. Eckart, Minimizing the Gross–Pitaevskii energy functional with the Sobolev gradient–Analytical and numerical results. Int. J. Comput. Methods 7 (2010) 453–475. [CrossRef] [Google Scholar]
  46. W. Liu and Y. Cai, Normalized gradient flow with Lagrange multiplier for computing ground states of Bose–Einstein condensates. SIAM J. Sci. Comput. 43 (2021) B219–B242. [Google Scholar]
  47. A. Maugeri, D.K. Palagachev and L.G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, in Vol. 109 of Mathematical Research. Wiley-VCH Verlag, Berlin (2000). [Google Scholar]
  48. PHG. http://lsec.cc.ac.cn. [Google Scholar]
  49. L.P. Pitaevskii, Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13 (1911) 451–454. [Google Scholar]
  50. P. Upadhyaya, E. Jarlebring and E.H. Rubensson, A density matrix approach to the convergence of the self-consistent field iteration. NACO 11 (2021) 99–115. [Google Scholar]
  51. H. Wang, A projection gradient method for computing ground state of spin-2 Bose–Einstein condensates. J. Comput. Phys. 274 (2014) 473–488. [Google Scholar]
  52. X. Wu, Z. Wen and W. Bao, A regularized Newton method for computing ground states of Bose–Einstein condensates. J. Sci. Comput. 73 (2017) 303–329. [CrossRef] [MathSciNet] [Google Scholar]
  53. E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. Springer-Verlag, New York (1985). [Google Scholar]
  54. Z. Zhang, Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenvalue problems. Commun. Math. Sci. 20 (2022) 377–403. [CrossRef] [MathSciNet] [Google Scholar]
  55. A. Zhou, An analysis of finite-dimensional approximations for the ground state solution of Bose–Einstein condensates. Nonlinearity 17 (2003) 541–550. [Google Scholar]
  56. Q. Zhuang and J. Shen, Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose–Einstein condensates. J. Comput. Phys. 396 (2019) 72–88. [Google Scholar]

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