Advertisement

Applied Mathematics and Mechanics

, Volume 39, Issue 9, pp 1353–1372 | Cite as

Superconvergence analysis of bi-k-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

  • Jianyun Wang
  • Yanping ChenEmail author
Article

Abstract

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(hk+1) in the H1 norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(hk+1 + τ2) in the H1 norm can be obtained in the Crank-Nicolson fully discrete scheme.

Key words

superconvergence elliptic projection Schrödinger equation interpolation post-processing 

Chinese Library Classification

O241.82 

2010 Mathematics Subject Classification

65M12 65M15 65M60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We would like to thank anonymous referees for their insightful comments that improved this paper.

References

  1. [1]
    BAO, W. Z., JIN, S., and MARKOWICH, P. A. Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM Journal on Scientific Computing, 25(1), 27–64 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    FEIT, M. D., FLECK, J. A., and STEIGER, A. Solution of the Schrödinger equation by a spectral method. Journal of Computational Physics, 47, 412–433 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    AKRIVIS, G. D. Finite difference discretization of the cubic Schrödinger equation. IMA Journal of Numerical Analysis, 13(1), 115–124 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    BAO, W. Z. and CAI, Y. Y. Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM Journal on Numerical Analysis, 50(2), 492–521 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    HAN, H. D., JIN, J. C., and WU, X. N. A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain. Computers and Mathematics with Applications, 50(8), 1345–1362 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    AKRIVIS, G. D., DOUGALIS, V. A., and KARAKASHIAN, O. A. On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numerische Mathematik, 59(1), 31–53 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    ANTONOPOULOU, D. C., KARALI, G. D., PLEXOUSAKIS, M., and ZOURARIS, G. E. Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain. Mathematics of Computation, 84(294), 1571–1598 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    JIN, J. C. and WU, X. N. Convergence of a finite element scheme for the two-dimensional timedependent Schrödinger equation in a long strip. Journal of Computational and Applied Mathematics, 234(3), 777–793 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    KYZA, I. A posteriori error analysis for the Crank-Nicolson method for linear Schrödinger equations. ESAIM Mathematical Modelling and Numerical Analysis, 45(4), 761–778 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    LEE, H. Y. Fully discrete methods for the nonlinear Schrödinger equation. Computers and Math-ematics with Applications, 28(6), 9–24 (1994)CrossRefGoogle Scholar
  11. [11]
    TANG, Q., CHEN, C. M., and LIU, L. H. Space-time finite element method for Schrödinger equation and its conservation. Applied Mathematics and Mechanics (English Edition), 27(3), 335–340 (2006)  https://doi.org/10.1007/s10483-006-0308-z MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    WANG, J. Y. and HUANG Y. Q. Fully discrete Galerkin finite element method for the cubic nonlinear Schrödinger equation. Numerical Mathematics: Theory, Methods and Applications, 10(3), 670–687 (2017)Google Scholar
  13. [13]
    ANTONOPOULOU, D. C. and PLEXOUSAKIS, M. Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains. Numerische Mathematik, 115(4), 585–608 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    KARAKASHIAN, O. A. and MAKRIDAKIS C. A space-time finite element method for the nonlinea. Schrödinger equation: the discontinuous Galerkin method. Mathematics of Computation, 67(222), 479–499 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    LU, W. Y., HUANG, Y. Q., and LIU, H. L. Mass preserving discontinuous Galerkin methods for Schrödinger equations. Journal of Computational Physics, 282, 210–226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    GUO, L. and XU, Y. Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator. Journal of Scientific Computing, 65(2), 622–647 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    WANG, W. and SHU, C. W. The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode. Journal of Scientific Computing, 40(1-3), 360–374 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    XU, Y. and SHU, C. W. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. Journal of Computational Physics, 205, 72–97 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    CHEN, C. M. and HUANG Y. Q. High Accuracy Theory of Finite Element Methods (in Chinese), Hunan Science Press, Changsha, 235–248 (1995)Google Scholar
  20. [20]
    LIN, Q. and YAN, N. N. Construction and Analysis of High Efficient Finite Elements (in Chinese), Hebei University Press, Baoding, 175–185 (1996)Google Scholar
  21. [21]
    WAHLBIN, L. B. Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 48–64 (1995)Google Scholar
  22. [22]
    YAN, N. N. Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Meth-ods, Science Press, Beijing, 35–156 (2008)Google Scholar
  23. [23]
    ARNOLD, D. N., DOUGLAS, J., Jr., and THOMEE, V. Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable. Mathematics of Computation, 36(153), 53–63 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    CHEN, C. M. and HU, S. F. The highest order superconvergence for bi-k degree rectangular elements at nodes: a proof of 2k-conjecture. Mathematics of Computation, 82(283), 1337–1355 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    CHEN, Y. P. Superconvergence of mixed finite element methods for optimal control problems. Mathematics of Computation, 77(263), 1269–1291 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    CHEN, Y. P., HUANG, Y. Q., LIU, W. B., and YAN, N. N. Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. Journal of Scientific Computing, 42(3), 382–403 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    HUANG, Y. Q., LI, J. C., WU, C., and YANG, W. Superconvergence analysis for linear tetrahedral edge elements. Journal of Scientific Computing, 62(1), 122–145 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    HUANG, Y. Q., YANG, W., and YI, N. Y. A posteriori error estimate based on the explicit polynomial recovery. Natural Science Journal of Xiangtan University, 33(3), 1–12 (2011)zbMATHGoogle Scholar
  29. [29]
    LIN, Q. and ZHOU, J. M. Superconvergence in high-order Galerkin finite element methods. Computer Methods in Applied Mechanics and Engineering, 196(37), 3779–3784 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    SHI, D. Y. and PEI, L. F. Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L2 projection method. Applied Mathematics and Mechanics (English Edition), 34(7), 861–874 (2013)  https://doi.org/10.1007/s10483-013-1713-x MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    WHEELER, M. F. and WHITEMAN, J. R. Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems. Numerical Methods for Partial Differential Equations, 10(3), 271–294 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    LIN, Q. and LIU, X. Q. Global superconvergence estimates of finite element method for Schrödinger equation. Journal of Computational Mathematics, 16(6), 521–526 (1998)MathSciNetzbMATHGoogle Scholar
  33. [33]
    SHI, D. Y., WANG, P. L., and ZHAO, Y. M. Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrödinger equation. Applied Mathematics Letters, 38, 129–134 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    TIAN, Z. K., CHEN, Y. P., and WANG J. Y. Superconvergence analysis of bilinear finite element for the nonlinear Schrödinger equation on the rectangular mesh. Advances in Applied Mathematics and Mechanics, 10(2), 468–484 (2018)Google Scholar
  35. [35]
    WANG, J. Y., HUANG, Y. Q., TIAN, Z. K., and ZHOU, J. Superconvergence analysis of finite element method for the time-dependent Schrödinger equation. Computers and Mathematics with Applications, 71(10), 1960–1972 (2016)MathSciNetCrossRefGoogle Scholar
  36. [36]
    ZHOU, L. L., XU, Y., ZHANG, Z. M., and CAO, W. X. Superconvergence of local discontinuous Galerkin method for one-dimensional linear Schrödinger equations. Journal of Scientific Computing, 73(2/3), 1290–1315 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    HU, H. L., CHEN, C. M., and PAN, K. J. Time-extrapolation algorithm (TEA) for linear parabolic problems. Journal of Computational Mathematics, 32(2), 183–194 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceHunan University of TechnologyZhuzhouChina
  2. 2.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina

Personalised recommendations