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Unconditionally optimal error estimates of a new mixed FEM for nonlinear Schrödinger equations

  • Dongyang ShiEmail author
  • Huaijun Yang
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Abstract

In this paper, a new mixed finite element scheme in space and a linearized backward Euler scheme in time are presented and investigated for the nonlinear Schrödinger equations. By introducing a suitable time-discrete system, both the errors in L2- and H1-norms for the original variable and L2-norm for the flux variable are derived without any time-step restriction, while previous works always required certain conditions between time step and space size. Finally, some numerical results are provided to verify the theoretical analysis.

Keywords

Nonlinear Schrödinger equtions New mixed finite element scheme Unconditionally optimal error estimates 

Mathematics Subject Classification (2010)

65M60 65N30 65N15 
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Notes

Funding information

This work is supported by the National Natural Science Foundation of China (Nos. 11671369; 11271340).

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Ablowitz, M.J., Segue, H.: Solitons and the Inverse Scattering Transformation. SIAM, Philadelphia (1981)CrossRefGoogle Scholar
  3. 3.
    Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear schrödinger equations. SIAM J. Sci. Comput. 33, 1008–1033 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear schrödinger equation. Nmer. Math. 59, 31–53 (1911)zbMATHCrossRefGoogle Scholar
  5. 5.
    Akrivis, G.D.: Finite difference discretization of the cubic schrödinger equation. IMA J. Numer. Analysis. 13(1), 115–124 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bratsos, A.G.: A Modified numerical scheme for the cubic schrödinger equation. Numer. Methods Partial Differential Equations 27, 608–620 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Borzi, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the schrödinger equation. J. Comput. Appl. Math. 193, 65–88 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ciarlet, P.G.: The finite element method for elliptic problem. North Holland, Amsterdam (1978)Google 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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cai, W.T., Li, J., Chen, Z.X.: Unconditional convergence and optimal error estimates of the Euler semi-implicit scheme for a generalized nonlinear schrödinger equation. Adv. Comput. Math. 42(6), 1311–1330 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chen, S.C., Chen, H.R.: New mixed element schemes for second order elliptic problem. Math. Numer. Sin. 32, 213–218 (2010). (in Chinese)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, Z.X.: Finite Element Methods and Their Applications. Springer, Berlin (2005)zbMATHGoogle Scholar
  13. 13.
    Dehghan, M., Taleei, A.: Numerical solution of nonlinear schrödinger equation by using time-space pseudo-spectral method. Numer. Methods Partial Differential Equations 26, 979–990 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ebaid, A., Khaled, S.M.: New types of exact solutions for nonlinear schrödinger equation with cubic nonlinearity. J. Comput. Appl. Math. 235, 1984–1992 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gao, Y.L., Mei, L.Q.: Implicit-explicit multistep methods for general two-dimensional nonlinear schrödinger equations. Appl. Nmer. Math. 109, 41–60 (2016)zbMATHCrossRefGoogle Scholar
  16. 16.
    Gao, H.D.: Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations. J. Sci. Comput. 58, 627–647 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Jin, J., Wu, X.: Analysis of finite element method for one-dimensional time-dependent schrödinger equation on unbounded domain. J. Comput. Appl. Math. 220, 240–256 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Hasegawa, A., Kodama, Y.: Solitons in optical communications. Rev. Mod. Phys. 68(2), 423–444 (1996)zbMATHCrossRefGoogle Scholar
  19. 19.
    Li, L.X., Wang, M.: The (G’/G)-expansion method and travelling wave solutions for a high-order nonlinear schrödinger equation. Appl. Math. Comput. 208(2), 440–445 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lin, Q., Liu, X.Q.: Global superconvergence estimates of finite element method for schrödinger equation. J. Comput. Math. 16(6), 521–526 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Liao, H., Sun, Z., Shi, H.: Error estimate of fourth-order compact scheme for linear schrödinger equations. SIAM J. Numer. Anal. 47, 4381–4401 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Li, B.K., Fairweather, G., Bialecki, B.: Discrete-time orthogonal spline collocation methods for schrödinger equations in two space variables. SIAM J. Numer. Anal. 35, 453–477 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Newell, A.C.: Solitons in Mathematical and Physics. SIAM, Philadelphia (1985)zbMATHCrossRefGoogle Scholar
  26. 26.
    Reichel, B., Leble, S.: On convergence and stability of a numerical scheme of coupled nonlinear schrödinger equations. Comput. Math. Appl. 55, 745–759 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sun, Z., Zhao, D.: On the \(l_{\infty }\) convergence of a difference scheme for coupled nonlinear schrödinger equations. Comput. Math. Appl. 59, 3286–3300 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sun, W.W., Wang, J.L.: Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear schrödinger system in 3D. J. Comput. Appl. Math. 317, 685–699 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Sanz-Serna, J.M.: Methods for the numerical solution of nonlinear schrödinger equation. Math. Comput. 43, 21–27 (1984)zbMATHCrossRefGoogle Scholar
  30. 30.
    Shi, D.Y., Wang, J.J.: Unconditional superconvergence analysis of a crank-nicolson galerkin FEM for nonlinear schrödinger equation. J. Sci. Comput. 72(3), 1093–1118 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Shi, D.Y., Wang, P.L., Zhao, Y.M.: Superconvergence analysis of anisotropic linear triangular finite element for nonlinear schrödinger equation. Appl. Math. Lett. 38, 129–134 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Shi, D.Y., Liao, X., Wang, L.L.: A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation. Acta. Math. Sci. 37(3), 584–592 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Shi, D.Y., Liao, X., Wang, L.L.: Superconvergence analysis of conforming finite element method for nonlinear schrödinger equation. Appl. Math. Comput. 289(20), 298–310 (2016)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shi, F., Yu, J.P., Li, K.T.: A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair. Int. J. Comput. Math. 88, 2293–2305 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Tourigny, Y.: Optimal h 1 estimates for two time-discrete Galerkin approximations of a nonlinear schrödinger equation. IMA J. Numer. Anal. 11, 509–523 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Thomee, V.: Galerkin Finite Element Methods for Paraboloc Problems. Springer, Berlin (1977)Google Scholar
  37. 37.
    Wang, J.L.: A new error analysis of crank-nicolson galerkin FEMs for a generalized nonlinear schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Weng, Z.F., Feng, X.L., Huang, P.Z.: A new mixed finite element method based on the Crank-Nicolson scheme for the parabolic problems. Appl. Math. Model. 36, 5068–5079 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Wu, L.: Two-grid mixed finite-element methods for nonlinear schrödinger equations. Numer. Methods for Partial Differential Equations 28, 63–73 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear schrödinger equations. J. Comput. Phys. 205, 72–77 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Zhang, H.: Extended Jacobi elliptic function expansion method and its applications. Commun. Nonlinear Sci. Numer. Simul. 12(5), 627–635 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Zouraris, G.E.: On the convergence of a linear two-step finite element method for the nonlinear schrödinger equation. M2AN Math. Model. Numer. Anal. 35, 389–405 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Zlamal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal. 10, 229–240 (1973)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

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