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Volume 10, Issue 3
Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation

Jianyun Wang & Yunqing Huang

DOI: 10.4208/nmtma.2017.y16008

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 671-688.

Published online: 2017-10

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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 $L^2$ error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

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@Article{NMTMA-10-671, author = {Jianyun Wang and Yunqing Huang}, title = {Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {3}, pages = {671--688}, 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 $L^2$ error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.y16008}, url = {http://global-sci.org/intro/article_detail/nmtma/12364.html} }
TY - JOUR T1 - Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation AU - Jianyun Wang & Yunqing Huang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 671 EP - 688 PY - 2017 DA - 2017/10 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.y16008 UR - https://global-sci.org/intro/article_detail/nmtma/12364.html KW - AB -

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 $L^2$ error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

Jianyun Wang and Yunqing Huang. (2017). Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation. Numerical Mathematics: Theory, Methods and Applications. 10 (3). 671-688. doi:10.4208/nmtma.2017.y16008
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