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 - <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>