International Journal of Nonlinear Analysis and Applications

Weak Galerkin finite element method for the nonlinear Schrodinger equation

Document Type : Research Paper

Authors

College of Education for Pure Sciences, University of Thi-Qar, Iraq

Abstract

The numerical technique for a two-dimensional time-dependent nonlinear Schrodinger equation is the subject of this work. The approximations are produced using the weak Galerkin finite element technique with semi-discrete and fully discrete finite element methods, respectively, using the backward Euler method and the crank-Nicolson method in time. Using the elliptic projection operator, we provide optimum L2 error estimates for semi and fully discrete weak Galerkin finite elements. Finally, we present numerical examples provided to verify our theoretical results.

Keywords

[1] G.D. Akrivis, Finite difference discretization of the cubic Schrodinger equation, IMAJ. Numer. Anal. 13 (1993),
115–124.
[2] W.Z. Bao, S. Jin and P.A. Markowich Numerical study of time -splitting spectral discretization of nonlinear
Schrodinger equations in the semi classical regimes, SIAMJ. Sci. Comput. 25 (2003), 27–64.
[3] Q.S. Chang, E.Jia and W. Sun: Difference schemes for solving the generalized nonlinear Schrodinger equation, J.
Comput .Phys., 148(1999), 397-415.
[4] M. Dehghan and A. Taleei, Numerical solution of nonlinear Schrodinger equation by using time-space pseudo
-spectral method, Numer. Meth. Partial Differ. Equ. 26 (2010), 979–992.
[5] X.L. HU and L.M. ZHANG, Conservative compact difference schemes for the coupled nonlinear Schrodinger
system, Numer. Meth. Partial Differ. Eq. 30 (2014), 749-772 .
[6] J.C. Jin, N. Wei and H.M. Zhang, A two-grid finite -element method for the nonlinear Schrodinger equation, J.
Comput. Math. 33 (2015), 146–157.
[7] H. A. Kashkool, and A.H. Aneed, Full-discrete weak Galerkin finite element method for solving diffusion -
convection problem, J. Adv. Math. 13 (2017), no. 4, 7333–7344.
[8] Y. LIU and H. LI, H1
-Galerkin mixed finite element method for the linear Schrodinger equation, ADV. Math.
39 (2010), 429–442.
[9] V. Thomee, Galerkin finite element method for parabolic problems, (Springer Series in Computational Mathematics
), NJ, 1984.[10] J. Wang ,Y. Huang. : Fully discrete Galerkin finite element method for cubic nonlinear Schrodinger equation,
Numer. Math. Teor. Meth. Appl. 10 (2017), no. 3, 671–688.
[11] J. P. Wang, and X. Ye, :A weak Galerkin finite element method for second-order elliptic problems, J. Comput.
Appl. Math. 241 (2013), no. 1, 103–115.
[12] H.M. Zhang, J.C. Jin and J.Y. Wang, Two-grid finite -element method for the two -dimensional time -dependent
Schrodinger equation, ADV. APPI. Math. Mech. 5 (2013), 180–193.
[13] Y.M. Zhao, D.Y. Shi and F.Wang, High accuracy analysis of a new mixed finite element method for nonlinear
Schrodinger equation, Math. Numer. Sin. 37 (2015), 162–177.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2453-2468
  • Receive Date: 03 January 2022
  • Revise Date: 19 February 2022
  • Accept Date: 12 March 2022