Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations
Year: 2024
Author: Dongyang Shi, Houchao Zhang
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 4 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2210-m2021-0337
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 4 : pp. 979–998
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Schrödinger-Helmholtz equations Nonconforming FEMs Linearized Crank-Nicolson scheme Optimal error estimates.