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A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

  • Meng Li
  • Chengming HuangEmail author
  • Wanyuan Ming
Original Paper
  • 17 Downloads

Abstract

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis.

Keywords

Nonlinear fractional Schrödinger equation Finite element method Relaxation-type method Conservation Unique solvability Convergence 

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Notes

Funding information

This work was supported by NSF of China (No. 11801527, 11771163) and China Postdoctoral Science Foundation Funded Project (No. 2018M632791).

References

  1. 1.
    Akrivis, G.D.: Finite difference discretization of the cubic Schrödinger equation. IMA J. Numer. Anal. 13(1), 115–124 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrodinger̈ equation. Numer. Math. 59(1), 31–53 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Besse, C.: A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42(3), 934–952 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bhrawy, A., Abdelkawy, M.: A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations. J. Comput. Phys. 294, 462–483 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bhrawy, A., Zaky, M.: An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bhrawy, A.H., Alzaidy, J.F., Abdelkawy, M.A., Biswas, A.: Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations. Nonlinear Dynam. 84(3), 1553–1567 (2016)MathSciNetGoogle Scholar
  7. 7.
    Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293, 264–279 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bu, W., Tang, Y., Yang, J.: Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J. Comput. Phys. 276, 26–38 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chang, Q., Jia, E., Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation. J. Comput. Phys. 148(2), 397–415 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cheng, X., Duan, J., Li, D.: A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equations. Appl. Math. Comput. 346, 452–464 (2019)MathSciNetGoogle Scholar
  11. 11.
    Delfour, M., Fortin, M., Payr, G.: Finite-difference solutions of a non-linear Schrödinger equation. J. Comput. Phys. 44(2), 277–288 (1981)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Deng, W.: Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Duo, S., Zhang, Y.: Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation. Comput. Math. Appl. 71(11), 2257–2271 (2016)MathSciNetGoogle Scholar
  14. 14.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Fei, Z., Perez-Garcia, V.M., Vazquez, L.: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme. Appl. Math. Comput. 71(2), 165–177 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gu, X.M., Huang, T.Z., Li, H.B., Li, L., Luo, W.H.: On k-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations. Appl. Math. Lett. 42, 53–58 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 204(1), 468–477 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Guo, B., Huo, Z.: Global well-posedness for the fractional nonlinear Schrödinger equation. Comm. Partial Diff. Equ. 36(2), 247–255 (2010)zbMATHGoogle Scholar
  19. 19.
    Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082–104 (2006)Google Scholar
  20. 20.
    Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353–384 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51 (1), 445–466 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kirkpatrick, K., Lenzmann, E., Staffilani, G.: On the continuum limit for discrete NLS with long-range lattice interactions. Comm. Math. Phys. 317(3), 563–591 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62(3), 3135 (2000)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4), 298–305 (2000)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66(5), 056–108 (2002)MathSciNetGoogle Scholar
  26. 26.
    Li, C., Zhao, Z., Chen, Y.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62 (3), 855–875 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Li, M., Huang, C., Wang, N.: Galerkin finite element method for nonlinear fractional Ginzburg-Landau equation. Appl. Numer. Math. 118, 131–149 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Li, M., Huang, C., Wang, P.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms 74, 499–525 (2016)zbMATHGoogle Scholar
  29. 29.
    Li, M., Huang, C., Zhang, Z.: Unconditional error analysis of Galerkin FEMs for nonlinear fractional Schrödinger equation. Appl. Anal. 97(2), 295–315 (2018)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Li, M., Zhao, Y.L.: A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator. Appl. Math. Comput. 338, 758–773 (2018)MathSciNetGoogle Scholar
  31. 31.
    Li, S., Vu-Quoc, L.: Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 32(6), 1839–1875 (1995)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time–fractional fourth–order reaction–diffusion problem. Comput. Math. Appl. 70(4), 573–591 (2015)MathSciNetGoogle Scholar
  33. 33.
    Liu, Y., Du, Y., Li, H., Li, J., He, S.: A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative. Comput. Math. Appl. 70(10), 2474–2492 (2015)MathSciNetGoogle Scholar
  34. 34.
    Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40(6), 1117–1120 (2015)Google Scholar
  35. 35.
    Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Roop, J.P.: Variational Solution of the Fractional Advection Dispersion Equation. Ph.D. thesis, Clemson University, South Carolina (2004)Google Scholar
  37. 37.
    Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications, 1st edn. Gordon and Breach Science Publishers, London (1993)zbMATHGoogle Scholar
  38. 38.
    Secchi, S.: Ground state solutions for nonlinear fractional Schrodinger̈ equations in R N. arXiv:1208.2545.  https://doi.org/10.1063/1.4793990 (2012)
  39. 39.
    Secchi, S., Squassina, M.: Soliton dynamics for fractional Schrödinger equations. Appl. Anal. 93(8), 1702–1729 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Stickler, B.: Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Levý crystal. Phys. Rev. E 88(1), 012120 (2013)Google Scholar
  41. 41.
    Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)zbMATHGoogle Scholar
  42. 42.
    Wang, D., Xiao, A., Yang, W.: Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242, 670–681 (2013)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wang, D., Xiao, A., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J. Comput. Phys. 272, 644–655 (2014)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Wang, D., Xiao, A., Yang, W.: Maximum-norm error analysis of a difference scheme for the space fractional CNLS. Appl. Math. Comput. 257, 241–251 (2015)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wang, J.: A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Wang, P., Huang, C.: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer. Algorithms 69(3), 625–641 (2015)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wang, P., Huang, C., Zhao, L.: Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J. Comput. Appl. Math. 306(C), 231–247 (2016)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wei, L., He, Y., Zhang, X., Wang, S.: Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation. Finite Elem. Anal. Des. 59, 28–34 (2012)MathSciNetGoogle Scholar
  50. 50.
    Wei, L., Zhang, X., Kumar, S., Yildirim, A.: A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system. Comput. Math. Appl. 64(8), 2603–2615 (2012)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Weideman, J., Herbst, B.: Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23(3), 485–507 (1986)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34 (1), 200–218 (2010)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Yang, Z.: A class of linearized energy-conserved finite difference schemes for nonlinear space-fractional Schrödinger equations. Int. J. Comput. Math. 93(3), 609–626 (2016)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Zhang, H., Liu, F., Anh, V.: Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217 (6), 2534–2545 (2010)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Zhao, X., Sun, Z.z., Hao, Z.p.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36(6), A2865–A2886 (2014)zbMATHGoogle Scholar
  56. 56.
    Zhao, Z., Li, C.: Fractional difference/finite element approximations for the time–space fractional telegraph equation. Appl. Math. Comput. 219(6), 2975–2988 (2012)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Bu, W., Shu, S., Yue, X., Xiao, A., Zeng, W.: Space-time finite element method for the multi-term time-space fractional diffusion equation on a two-dimensional domain. Comput. Math. Appl.  https://doi.org/10.1016/j.camwa.2018.11.033 (2018)

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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouChina
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina
  3. 3.School of Mathematics and Information SciencesNanchang Hangkong UniversityNanchangChina

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