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Volume 42, Issue 4
Two-Grid Finite Element Method for Time-Fractional Nonlinear Schrödinger Equation

Hanzhang Hu, Yanping Chen & Jianwei Zhou

DOI: 10.4208/jcm.2302-m2022-0033

J. Comp. Math., 42 (2024), pp. 1124-1144.

Published online: 2024-04

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  • Abstract

A two-grid finite element method with $L1$ scheme is presented for solving two-dimensional time-fractional nonlinear Schrödinger equation. The finite element solution in the $L^∞$-norm are proved bounded without any time-step size conditions (dependent on spatial-step size). The classical $L1$ scheme is considered in the time direction, and the two-grid finite element method is applied in spatial direction. The optimal order error estimations of the two-grid solution in the $L^p$-norm is proved without any time-step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

  • Keywords

Time-fractional nonlinear Schrödinger equation, Two-grid finite element method, The $L1$ scheme.

  • AMS Subject Headings

65M06, 65M12, 65M15, 65M55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1124, author = {Hu , HanzhangChen , Yanping and Zhou , Jianwei}, title = {Two-Grid Finite Element Method for Time-Fractional Nonlinear Schrödinger Equation}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {4}, pages = {1124--1144}, abstract = {

A two-grid finite element method with $L1$ scheme is presented for solving two-dimensional time-fractional nonlinear Schrödinger equation. The finite element solution in the $L^∞$-norm are proved bounded without any time-step size conditions (dependent on spatial-step size). The classical $L1$ scheme is considered in the time direction, and the two-grid finite element method is applied in spatial direction. The optimal order error estimations of the two-grid solution in the $L^p$-norm is proved without any time-step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2302-m2022-0033}, url = {http://global-sci.org/intro/article_detail/jcm/23049.html} }
TY - JOUR T1 - Two-Grid Finite Element Method for Time-Fractional Nonlinear Schrödinger Equation AU - Hu , Hanzhang AU - Chen , Yanping AU - Zhou , Jianwei JO - Journal of Computational Mathematics VL - 4 SP - 1124 EP - 1144 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2302-m2022-0033 UR - https://global-sci.org/intro/article_detail/jcm/23049.html KW - Time-fractional nonlinear Schrödinger equation, Two-grid finite element method, The $L1$ scheme. AB -

A two-grid finite element method with $L1$ scheme is presented for solving two-dimensional time-fractional nonlinear Schrödinger equation. The finite element solution in the $L^∞$-norm are proved bounded without any time-step size conditions (dependent on spatial-step size). The classical $L1$ scheme is considered in the time direction, and the two-grid finite element method is applied in spatial direction. The optimal order error estimations of the two-grid solution in the $L^p$-norm is proved without any time-step size conditions. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse, with no loss in the order of accuracy.

Hanzhang Hu, Yanping Chen & Jianwei Zhou. (2024). Two-Grid Finite Element Method for Time-Fractional Nonlinear Schrödinger Equation. Journal of Computational Mathematics. 42 (4). 1124-1144. doi:10.4208/jcm.2302-m2022-0033
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