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Volume 16, Issue 2
Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations

Wanqiu Yuan, Dongfang Li & Chengjian Zhang

DOI: 10.4208/nmtma.OA-2022-0087

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 348-369.

Published online: 2023-04

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  • Abstract

A linearized transformed $L1$ Galerkin finite element method (FEM) is presented for numerically solving the multi-dimensional time fractional Schrödinger equations. Unconditionally optimal error estimates of the fully-discrete scheme are proved. Such error estimates are obtained by combining a new discrete fractional Grönwall inequality, the corresponding Sobolev embedding theorems and some inverse inequalities. While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches. Numerical examples are presented to confirm the theoretical results.

  • Keywords

Optimal error estimates, time fractional Schrödinger equations, transformed $L1$ scheme, discrete fractional Grönwall inequality

  • AMS Subject Headings

34A08, 65M12, 65M60, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-348, author = {Yuan , WanqiuLi , Dongfang and Zhang , Chengjian}, title = {Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations }, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {2}, pages = {348--369}, abstract = {

A linearized transformed $L1$ Galerkin finite element method (FEM) is presented for numerically solving the multi-dimensional time fractional Schrödinger equations. Unconditionally optimal error estimates of the fully-discrete scheme are proved. Such error estimates are obtained by combining a new discrete fractional Grönwall inequality, the corresponding Sobolev embedding theorems and some inverse inequalities. While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches. Numerical examples are presented to confirm the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0087}, url = {http://global-sci.org/intro/article_detail/nmtma/21580.html} }
TY - JOUR T1 - Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations AU - Yuan , Wanqiu AU - Li , Dongfang AU - Zhang , Chengjian JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 348 EP - 369 PY - 2023 DA - 2023/04 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0087 UR - https://global-sci.org/intro/article_detail/nmtma/21580.html KW - Optimal error estimates, time fractional Schrödinger equations, transformed $L1$ scheme, discrete fractional Grönwall inequality AB -

A linearized transformed $L1$ Galerkin finite element method (FEM) is presented for numerically solving the multi-dimensional time fractional Schrödinger equations. Unconditionally optimal error estimates of the fully-discrete scheme are proved. Such error estimates are obtained by combining a new discrete fractional Grönwall inequality, the corresponding Sobolev embedding theorems and some inverse inequalities. While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches. Numerical examples are presented to confirm the theoretical results.

Wanqiu Yuan, Dongfang Li & Chengjian Zhang. (2023). Linearized Transformed $L1$ Galerkin FEMs with Unconditional Convergence for Nonlinear Time Fractional Schrödinger Equations . Numerical Mathematics: Theory, Methods and Applications. 16 (2). 348-369. doi:10.4208/nmtma.OA-2022-0087
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