EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 58 / No 6 (November-December 2024) ESAIM: M2AN, 58 6 (2024) 2317-2349 References
Open Access
Issue
ESAIM: M2AN
Volume 58, Number 6, November-December 2024
Special issue - To commemorate Assyr Abdulle
Page(s) 2317 - 2349
DOI https://doi.org/10.1051/m2an/2024040
Published online 04 December 2024
  1. A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM. Multiscale Model. Simul. 4 (2005) 447–459. [CrossRef] [MathSciNet] [Google Scholar]
  2. A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, in Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics. Vol. 31 of GAKUTO Internat. Ser. Math. Sci. Appl. Gakkotosho, Tokyo (2009) 133–181. [Google Scholar]
  3. A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems. J. Comput. Phys. 231 (2012) 7014–7036. [CrossRef] [MathSciNet] [Google Scholar]
  4. A. Abdulle and P. Henning, A reduced basis localized orthogonal decomposition. J. Comput. Phys. 295 (2015) 379–401. [CrossRef] [MathSciNet] [Google Scholar]
  5. A. Abdulle and P. Henning, Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comput. 86 (2017) 549–587. [Google Scholar]
  6. A. Abdulle and P. Henning, Multiscale methods for wave problems in heterogeneous media, in Handbook of Numerical Methods for Hyperbolic Problems. Vol. 18 of Handb. Numer. Anal. Elsevier/North-Holland, Amsterdam (2017) 545–576. [Google Scholar]
  7. A. Abdulle and T. Pouchon, A priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time. SIAM J. Numer. Anal. 54 (2016) 1507–1534. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Model. Simul. 3 (2004/2005) 195–220. [CrossRef] [Google Scholar]
  9. A. Abdulle and E. Weinan, Finite difference heterogeneous multi-scale method for homogenization problems. J. Comput. Phys. 191 (2003) 18–39. [CrossRef] [MathSciNet] [Google Scholar]
  10. A. Abdulle, E. Weinan, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21 (2012) 1–87. [CrossRef] [MathSciNet] [Google Scholar]
  11. A. Abdulle, D. Arjmand and E. Paganoni, Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems. C. R. Math. Acad. Sci. Paris 357 (2019) 545–551. [Google Scholar]
  12. G.D. Akrivis, V.A. Dougalis and O.A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59 (1991) 31–53. [CrossRef] [MathSciNet] [Google Scholar]
  13. T. Aktosun, T. Busse, F. Demontis and C. van der Mee, Exact solutions to the nonlinear Schrödinger equation, in Topics in Operator Theory. Volume 2. Systems and Mathematical Physics. Vol. 203 of Oper. Theory Adv. Appl. Birkh¨auser Verlag, Basel (2010) 1–12. [Google Scholar]
  14. 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]
  15. R. Altmann, P. Henning and D. Peterseim, Numerical homogenization beyond scale separation. Acta Numer. 30 (2021) 1–86. [CrossRef] [MathSciNet] [Google Scholar]
  16. 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]
  17. X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: computation of stationary solutions. Comput. Phys. Commun. 185 (2014) 2969–2991. [CrossRef] [Google Scholar]
  18. X. Antoine and R. Duboscq, Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose–Einstein condensates. J. Comput. Phys. 258 (2014) 509–523. [CrossRef] [MathSciNet] [Google Scholar]
  19. X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184 (2013) 2621–2633. [CrossRef] [MathSciNet] [Google Scholar]
  20. 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]
  21. W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50 (2012) 492–521. [CrossRef] [MathSciNet] [Google Scholar]
  22. W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6 (2013) 1–135. [CrossRef] [MathSciNet] [Google Scholar]
  23. 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]
  24. W. Bao, S. Jin and P.A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25 (2003) 27–64. [CrossRef] [MathSciNet] [Google Scholar]
  25. C. Besse, A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42 (2004) 934–952. [CrossRef] [MathSciNet] [Google Scholar]
  26. C. Besse, S. Descombes, G. Dujardin and I. Lacroix-Violet, Energy-preserving methods for nonlinear Schrödinger equations. IMA J. Numer. Anal. 41 (2021) 618–653. [CrossRef] [MathSciNet] [Google Scholar]
  27. J. Bunder and A. Roberts, High-order homogenisation by learning spatial discretisations of PDEs that provably preserve self-adjointnessr. Part. Differ. Equ. Appl. Math. 6 (2022) 1–16. [Google Scholar]
  28. E. Cancès, R. Chakir and Y. Maday, Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45 (2010) 90–117. [CrossRef] [MathSciNet] [Google Scholar]
  29. T. Cazenave, Semilinear Schrödinger Equations. Vol. 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003). [CrossRef] [Google Scholar]
  30. E.A. Cornell and C.E. Wieman, Nobel lecture: Bose–Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Rev. Mod. Phys. 74 (2002) 875–893. [CrossRef] [Google Scholar]
  31. J. Cui, W. Cai and Y. Wang, A linearly-implicit and conservative Fourier pseudo-spectral method for the 3D Gross–Pitaevskii equation with angular momentum rotation. Comput. Phys. Commun. 253 (2020) 107160, 26. [CrossRef] [MathSciNet] [Google Scholar]
  32. 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]
  33. 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]
  34. K. Dekker and J.G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations. Vol. 2 of CWI Monographs. North-Holland Publishing Co., Amsterdam (1984). [Google Scholar]
  35. C.M. Dion and E. Cancès, Ground state of the time-independent Gross–Pitaevskii equation. Comput. Phys. Comm. 177 (2007) 787–798. [CrossRef] [MathSciNet] [Google Scholar]
  36. C. Döding and P. Henning, Uniform L∞-bounds for energy-conserving higher-order time integrators for the Gross–Pitaevskii equation with rotation. IMA J. Numer. Anal. (2023). [Google Scholar]
  37. D.A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21 (1985) 1129–1148. [CrossRef] [Google Scholar]
  38. C. Engwer, P. Henning, A. Målqvist and D. Peterseim, Efficient implementation of the localized orthogonal decomposition method. Comput. Methods Appl. Mech. Eng. 350 (2019) 123–153. [CrossRef] [Google Scholar]
  39. L. Erd˝os, B. Schlein and H.-T. Yau, Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. of Math. 172 (2010) 291–370. [CrossRef] [MathSciNet] [Google Scholar]
  40. M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415 (2002) 39–41. [NASA ADS] [CrossRef] [Google Scholar]
  41. E.P. Gross, Structure of a quantized vortex in boson systems. Nuovo Cimento 20 (1961) 454–477. [CrossRef] [Google Scholar]
  42. M. Hauck and D. Peterseim, Super-localization of elliptic multiscale problems. Math. Comput. 92 (2023) 981–1003. [Google Scholar]
  43. 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) 110165. [CrossRef] [Google Scholar]
  44. F. Hellman, T. Keil and A. Målqvist, Numerical upscaling of perturbed diffusion problems. SIAM J. Sci. Comput. 42 (2020) A2014–A2036. [CrossRef] [Google Scholar]
  45. F. Hellman, A. Målqvist and S. Wang, Numerical upscaling for heterogeneous materials in fractured domains. ESAIM Math. Model. Numer. Anal. 55 (2021) S761–S784. [CrossRef] [EDP Sciences] [Google Scholar]
  46. P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36 (2014) A1609–A1634. [CrossRef] [MathSciNet] [Google Scholar]
  47. P. Henning and A. Målqvist, The finite element method for the time-dependent Gross–Pitaevskii equation with angular momentum rotation. SIAM J. Numer. Anal. 55 (2017) 923–952. [CrossRef] [MathSciNet] [Google Scholar]
  48. P. Henning and A. Persson, On optimal convergence rates for discrete minimizers of the Gross–Pitaevskii energy in localized orthogonal decomposition spaces. Multiscale Model. Simul. 21 (2023) 993–1011. [CrossRef] [MathSciNet] [Google Scholar]
  49. P. Henning and D. Peterseim, Oversampling for the multiscale finite element method. SIAM Multiscale Model. Simul. 11 (2013) 1149–1175. [CrossRef] [MathSciNet] [Google Scholar]
  50. P. Henning and D. Peterseim, Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials. Math. Models Methods Appl. Sci. 27 (2017) 2147–2184. [CrossRef] [MathSciNet] [Google Scholar]
  51. 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]
  52. P. Henning and J. W¨arnegård, Numerical comparison of mass-conservative schemes for the Gross–Pitaevskii equation. Kinet. Relat. Models 12 (2019) 1247–1271. [CrossRef] [MathSciNet] [Google Scholar]
  53. P. Henning and J. W¨arnegård, A note on optimal H1-error estimates for Crank–Nicolson approximations to the nonlinear Schrödinger equation. BIT Numer. Math. 61 (2021) 37–59. [CrossRef] [Google Scholar]
  54. P. Henning and J. W¨arnegård, Superconvergence of time invariants for the Gross–Pitaevskii equation. Math. Comput. 91 (2022) 509–555. [Google Scholar]
  55. P. Henning, A. Målqvist and D. Peterseim, Two-level discretization techniques for ground state computations of Bose–Einstein condensates. SIAM J. Numer. Anal. 52 (2014) 1525–1550. [CrossRef] [MathSciNet] [Google Scholar]
  56. 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]
  57. O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method. SIAM J. Numer. Anal. 36 (1999) 1779–1807. [CrossRef] [MathSciNet] [Google Scholar]
  58. P. Keast, Moderate-degree tetrahedral quadrature formulas. Comput. Methods Appl. Mech. Eng. 55 (1986) 339–348. [CrossRef] [Google Scholar]
  59. W. Ketterle, Nobel lecture: when atoms behave as waves: Bose–Einstein condensation and the atom laser. Rev. Mod. Phys. 74 (2002) 1131–1151. [CrossRef] [Google Scholar]
  60. F. Kröpfl, R. Maier and D. Peterseim, Operator compression with deep neural networks. Adv. Contin. Discrete Models 2022 (2022) 29. [CrossRef] [Google Scholar]
  61. E. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61 (2000). [CrossRef] [Google Scholar]
  62. E.H. Lieb, R. Seiringer and J. Yngvason, A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional Bose gas. Comm. Math. Phys. 224 (2001) 17–31. Dedicated to Joel L. Lebowitz. [CrossRef] [MathSciNet] [Google Scholar]
  63. P. Ljung, A. Målqvist and A. Persson, A generalized finite element method for the strongly damped wave equation with rapidly varying data. ESAIM Math. Model. Numer. Anal. 55 (2021) 1375–1403. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  64. C. Lubich, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–2153. [CrossRef] [Google Scholar]
  65. R. Maier, A high-order approach to elliptic multiscale problems with general unstructured coefficients. SIAM J. Numer. Anal. 59 (2021) 1067–1089. [CrossRef] [MathSciNet] [Google Scholar]
  66. A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems. Math. Comput. 83 (2014) 2583–2603. [CrossRef] [Google Scholar]
  67. A. Målqvist and D. Peterseim, Numerical Homogenization by Localized Orthogonal Decomposition. Vol. 5 of SIAM Spotlights. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2021). [Google Scholar]
  68. A. Målqvist and B. Verfürth, An offline-online strategy for multiscale problems with random defects. ESAIM Math. Model. Numer. Anal. 56 (2022) 237–260. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  69. M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman and E.A. Cornell, Vortices in a Bose–Einstein condensate. Phys. Rev. Lett. 83 (1999) 2498–2501. [CrossRef] [Google Scholar]
  70. J.M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1996) 289–314. [CrossRef] [Google Scholar]
  71. L.P. Pitaevskii, Vortex lines in an imperfect Bose gas. Number 13. Soviet Physics JETP-USSR (1961). [Google Scholar]
  72. L.P. Pitaevskii and S. Stringari, Bose–Einstein Condensation. Oxford University Press, Oxford (2003). [Google Scholar]
  73. C. Raman, J.R. Abo-Shaeer, J.M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a stirred Bose–Einstein condensate. Phys. Rev. Lett. 87 (2001) 210402. [CrossRef] [PubMed] [Google Scholar]
  74. J.M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schrödinger equation. Math. Comput. 43 (1984) 21–27. [CrossRef] [Google Scholar]
  75. J.M. Sanz-Serna and J.G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation. IMA J. Numer. Anal. 6 (1986) 25–42. [CrossRef] [MathSciNet] [Google Scholar]
  76. P. Solin, K. Segeth and I. Dolezel, Higher-Order Finite Element Methods, 1st edition. Chapman and Hall/CRC (2003). [CrossRef] [Google Scholar]
  77. M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258. [CrossRef] [MathSciNet] [Google Scholar]
  78. Y. Tourigny, Optimal H1 estimates for two time-discrete Galerkin approximations of a nonlinear Schrödinger equation. IMA J. Numer. Anal. 11 (1991) 509–523. [CrossRef] [MathSciNet] [Google Scholar]
  79. P. Upadhyaya, E. Jarlebring and E.H. Rubensson, A density matrix approach to the convergence of the self-consistent field iteration. Numer. Algebra Control Optim. 11 (2021) 99–115. [CrossRef] [MathSciNet] [Google Scholar]
  80. J.G. Verwer and J.M. Sanz-Serna, Convergence of method of lines approximations to partial differential equations. Computing 33 (1984) 297–313. [CrossRef] [MathSciNet] [Google Scholar]
  81. J. Wang, A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60 (2014) 390–407. [CrossRef] [MathSciNet] [Google Scholar]
  82. Z. Wu and Z. Zhang, Convergence analysis of the localized orthogonal decomposition method for the semiclassical Schrödinger equations with multiscale potentials. J. Sci. Comput. 93 (2022) 73. [CrossRef] [Google Scholar]
  83. G.E. Zouraris, On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation. M2AN Math. Model. Numer. Anal. 35 (2001) 389–405. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you