@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 = {<p style="text-align: justify;">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.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2210-m2021-0337},
url = {http://global-sci.org/intro/article_detail/jcm/23043.html}
}