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Volume 42, Issue 2
Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term

Meng Li, Jikun Zhao, Zhongchi Wang & Shaochun Chen

DOI: 10.4208/jcm.2209-m2021-0038

J. Comp. Math., 42 (2024), pp. 454-499.

Published online: 2024-01

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  • Abstract

This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrödinger equations with trapped term. We mainly consider three types of virtual elements, including $H^2$ conforming virtual element, $C^0$ nonconforming virtual element and Morley-type nonconforming virtual element. The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time. We prove the mass and energy conservation, the boundedness and the unique solvability of the fully discrete schemes. After introducing a new type of the Ritz projection, the optimal and unconditional error estimates for the fully discrete schemes are presented and proved. Finally, two numerical examples are investigated to confirm our theoretical analysis.

  • Keywords

$H^2$ conforming virtual element, $C^0$ nonconforming virtual element, Morley-type nonconforming virtual element, Nonlinear Schrödinger equation, Conservation, Convergence.

  • AMS Subject Headings

65N35, 65N12, 76D07, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-454, author = {Li , MengZhao , JikunWang , Zhongchi and Chen , Shaochun}, title = {Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {2}, pages = {454--499}, abstract = {

This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrödinger equations with trapped term. We mainly consider three types of virtual elements, including $H^2$ conforming virtual element, $C^0$ nonconforming virtual element and Morley-type nonconforming virtual element. The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time. We prove the mass and energy conservation, the boundedness and the unique solvability of the fully discrete schemes. After introducing a new type of the Ritz projection, the optimal and unconditional error estimates for the fully discrete schemes are presented and proved. Finally, two numerical examples are investigated to confirm our theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2209-m2021-0038}, url = {http://global-sci.org/intro/article_detail/jcm/22889.html} }
TY - JOUR T1 - Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term AU - Li , Meng AU - Zhao , Jikun AU - Wang , Zhongchi AU - Chen , Shaochun JO - Journal of Computational Mathematics VL - 2 SP - 454 EP - 499 PY - 2024 DA - 2024/01 SN - 42 DO - http://doi.org/10.4208/jcm.2209-m2021-0038 UR - https://global-sci.org/intro/article_detail/jcm/22889.html KW - $H^2$ conforming virtual element, $C^0$ nonconforming virtual element, Morley-type nonconforming virtual element, Nonlinear Schrödinger equation, Conservation, Convergence. AB -

This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrödinger equations with trapped term. We mainly consider three types of virtual elements, including $H^2$ conforming virtual element, $C^0$ nonconforming virtual element and Morley-type nonconforming virtual element. The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time. We prove the mass and energy conservation, the boundedness and the unique solvability of the fully discrete schemes. After introducing a new type of the Ritz projection, the optimal and unconditional error estimates for the fully discrete schemes are presented and proved. Finally, two numerical examples are investigated to confirm our theoretical analysis.

Meng Li, Jikun Zhao, Zhongchi Wang & Shaochun Chen. (2024). Conservative Conforming and Nonconforming VEMs for Fourth Order Nonlinear Schrödinger Equations with Trapped Term. Journal of Computational Mathematics. 42 (2). 454-499. doi:10.4208/jcm.2209-m2021-0038
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