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Volume 42, Issue 4
Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations

Dongyang Shi & Houchao Zhang

J. Comp. Math., 42 (2024), pp. 979-998.

Published online: 2024-04

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  • Abstract

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods (FEMs) for the nonlinear coupled Schrödinger-Helmholtz equations. Optimal $L^2$ and $H^1$ estimates of orders $\mathcal{O}(h^2+τ^2)$ and $\mathcal{O}(h+τ^2)$ are derived respectively without any grid-ratio condition through the following two keys. One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error, which leads to optimal temporal error estimates of order $\mathcal{O}(τ^2)$ in $L^2$ and the broken $H^1$-norms, as well as the uniform boundness of numerical solutions in $L^∞$-norm. The other is that a novel projection is utilized, which can iron out the difficulty of the existence of the consistency errors. This leads to derive optimal spatial error estimates of orders $\mathcal{O}(h^2)$ in $L^2$-norm and $\mathcal{O}(h)$ in the broken $H^1$-norm under the $H^2$ regularity of the solutions for the time-discrete system. At last, two numerical examples are provided to confirm the theoretical analysis. Here, $h$ is the subdivision parameter, and $τ$ is the time step.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-979, author = {Shi , Dongyang and Zhang , Houchao}, title = {Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {42}, number = {4}, pages = {979--998}, abstract = {

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods (FEMs) for the nonlinear coupled Schrödinger-Helmholtz equations. Optimal $L^2$ and $H^1$ estimates of orders $\mathcal{O}(h^2+τ^2)$ and $\mathcal{O}(h+τ^2)$ are derived respectively without any grid-ratio condition through the following two keys. One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error, which leads to optimal temporal error estimates of order $\mathcal{O}(τ^2)$ in $L^2$ and the broken $H^1$-norms, as well as the uniform boundness of numerical solutions in $L^∞$-norm. The other is that a novel projection is utilized, which can iron out the difficulty of the existence of the consistency errors. This leads to derive optimal spatial error estimates of orders $\mathcal{O}(h^2)$ in $L^2$-norm and $\mathcal{O}(h)$ in the broken $H^1$-norm under the $H^2$ regularity of the solutions for the time-discrete system. At last, two numerical examples are provided to confirm the theoretical analysis. Here, $h$ is the subdivision parameter, and $τ$ is the time step.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2210-m2021-0337}, url = {http://global-sci.org/intro/article_detail/jcm/23043.html} }
TY - JOUR T1 - Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations AU - Shi , Dongyang AU - Zhang , Houchao JO - Journal of Computational Mathematics VL - 4 SP - 979 EP - 998 PY - 2024 DA - 2024/04 SN - 42 DO - http://doi.org/10.4208/jcm.2210-m2021-0337 UR - https://global-sci.org/intro/article_detail/jcm/23043.html KW - Schrödinger-Helmholtz equations, Nonconforming FEMs, Linearized Crank-Nicolson scheme, Optimal error estimates. AB -

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods (FEMs) for the nonlinear coupled Schrödinger-Helmholtz equations. Optimal $L^2$ and $H^1$ estimates of orders $\mathcal{O}(h^2+τ^2)$ and $\mathcal{O}(h+τ^2)$ are derived respectively without any grid-ratio condition through the following two keys. One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error, which leads to optimal temporal error estimates of order $\mathcal{O}(τ^2)$ in $L^2$ and the broken $H^1$-norms, as well as the uniform boundness of numerical solutions in $L^∞$-norm. The other is that a novel projection is utilized, which can iron out the difficulty of the existence of the consistency errors. This leads to derive optimal spatial error estimates of orders $\mathcal{O}(h^2)$ in $L^2$-norm and $\mathcal{O}(h)$ in the broken $H^1$-norm under the $H^2$ regularity of the solutions for the time-discrete system. At last, two numerical examples are provided to confirm the theoretical analysis. Here, $h$ is the subdivision parameter, and $τ$ is the time step.

Dongyang Shi & Houchao Zhang. (2024). Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations. Journal of Computational Mathematics. 42 (4). 979-998. doi:10.4208/jcm.2210-m2021-0337
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