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Two-Grid Finite-Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

Published online by Cambridge University Press:  03 June 2015

Hongmei Zhang ,
Jicheng Jin  and
Jianyun Wang
Hongmei Zhang*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China School of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, China
Jicheng Jin*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, China
Jianyun Wang
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
*
Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

In this paper, we construct semi-discrete two-grid finite element schemes and full-discrete two-grid finite element schemes for the two-dimensional time-dependent Schrödinger equation. The semi-discrete schemes are proved to be convergent with an optimal convergence order and the full-discrete schemes, verified by a numerical example, work well and are more efficient than the standard finite element method.

Keywords

65N3065N55Schrödinger equationtwo-grid methodfinite element method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 180 - 193
Copyright
Copyright © Global-Science Press 2013

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References

[1]Hannabuss, K., An Introduction to Quantum Theory, Oxford University Press Inc., USA, 1997.Google Scholar
[2]Schmidt, F. and Deuflhard, P., Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation, Comput. Math. Appl., 29 (1995), pp. 53–76.CrossRefGoogle Scholar
[3]Collino, F., Perfectly matched absorbing layers for the paraxial equations, J. Comput. Phys., 131 (1997), pp. 164–180.Google Scholar
[4]Hall, D. S., Matthews, M. R., Ensher, J. R., Wieman, C. E. and Cornell, E. A., Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), pp. 1539–1542.Google Scholar
[5]Wang, T., Chen, J. and Zhang, L., Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator, Appl. Math. Comput., 182 (2006), pp. 1780–1794.Google Scholar
[6]Jin, J. and Wu, X., Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip, J. Comput. Appl. Math., 234 (2010), pp. 777–793.Google Scholar
[7]Alonso-Mallo, I. and Reguera, N., A high order finite element discretization with local absorbing boundary conditions of the linear Schrödinger equation, J. Comput. Phys., 220 (2006), pp. 409–421.Google Scholar
[8]Xu, J., A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29 (1992), pp. 303–319.CrossRefGoogle Scholar
[9]Xu, J., Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal., 33 (1996), pp. 1759–1777.Google Scholar
[10]Xu, J., A novel two-grid method for semilinear equations, SIAM J. Sci. Comput., 15 (1994), pp. 231–237.Google Scholar
[11]Marion, M. and Xu, J., Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal., 32 (1995), pp. 1170–1184.Google Scholar
[12]Xu, J. and Zhou, A., Local and parallel finite element algorithms for eigenvalue problems, Acta Math. Appl. Sin. Eng. Ser., 18 (2002), pp. 185–200.Google Scholar
[13]Xu, J. and Zhou, A., Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems, Adv. Comput. Math., 14 (2001), pp. 293–327.Google Scholar
[14]Xu, J. and Zhou, A., Local and parallel finite element algorithms based on two-grid discretizations, Math. Comput., 69 (2000), pp. 881–909.Google Scholar
[15]Jin, J., Shu, S. and Xu, J., A two-grid discretization method for decoupling systems of partial differential equations, Math. Comput., 75 (2006), pp. 1617–1626.CrossRefGoogle Scholar
[16]Axelsson, O. and Padiy, A., On a two level Newton type procedure applied for solving nonlinear elasticity problems, Int. J. Numer. Meth. Eng., 49 (2000), pp. 1479–1493.Google Scholar
[17]Layton, W. and Lenferink, W., Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comput., 69 (1995), pp. 263–274.Google Scholar
[18]Layton, W., Meir, A. and Schmidt, P., A two-level discretization method for the stationary MHD equations, Electron. Trans. Numer. Anal., 6 (1997), pp. 198–210.Google Scholar
[19]Wu, L. and Allen, M. B., A two-grid method for mixed finite-element solution of reaction-diffusion equations, Numer. Methods Partial Differential Equations, 15 (1999), pp. 317–332.Google Scholar
[20]Chien, C. S., Huang, H. T., Jeng, B. W. and Lid, Z. C., Two-grid discretization schemes for nonlinear Schröinger equations, J. Comput. Appl. Math., 214 (2008), pp. 549–571.CrossRefGoogle Scholar
[21]Wu, L., Two-grid strategy for unsteady state nonlinear Schrödinger equations, Int. J. Pure Appl. Math., 68 (2011), pp. 465–475.Google Scholar
[22]Wu, L., Two-grid mixed finite-element methods for nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations, 28 (2012), pp. 63–73.Google Scholar
[23]Lin, Q. and Liu, X., Global superconvergence estimates of finite element method for Schrödinger equation, J. Comput. Math., 6 (1998), pp. 521–526.Google Scholar