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Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  20 June 2017

Jianyun Wang  and
Yunqing Huang
Jianyun Wang*
Affiliation:
School of Science, Hunan University of Technology, Zhuzhou 412007, China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
*
*Corresponding author. Email addresses:wjy8137@163.com (J. Y. Wang), huangyq@xtu.edu.cn (Y. Q. Huang)
*Corresponding author. Email addresses:wjy8137@163.com (J. Y. Wang), huangyq@xtu.edu.cn (Y. Q. Huang)
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Abstract

This paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimal L2 error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

Keywords

Finite element methodnonlinear Schrödinger equationsbackward Euler schemeCrank-Nicolson scheme

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M50: Mesh generation and refinement 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 671 - 688
Copyright
Copyright © Global-Science Press 2017 

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