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Convergence Analysis of Nonconforming Quadrilateral Finite Element Methods for Nonlinear Coupled Schrödinger-Helmholtz Equations

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 L2 and H1 estimates of orders O(h2+τ2) and O(h+τ2) 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 O(τ2) in L2 and the broken H1-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 O(h2) in L2-norm and O(h) in the broken H1-norm under the H2 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.

Author Details

Dongyang Shi Email

Houchao Zhang Email