Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations (Journal of Computational Mathematics, 2024)
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 errorand 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 ofthe existence of the consistency errors. This leads to derive optimal spatial error estimatesof orders $\mathcal{O}(h^2)$ in $L^2$-norm and $\mathcal{O}(h)$ in the broken $H^1$-norm under the $H^2$ regularity ofthe solutions for the time-discrete system. At last, two numerical examples are providedto confirm the theoretical analysis. Here, $h$ is the subdivision parameter, and $τ$ is the timestep.
Keywords: Schrödinger-Helmholtz equations; Nonconforming FEMs; Linearized Crank-Nicolson scheme; Optimal error estimates.