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} }