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

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 O(h 2 +τ 2 ) and 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 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 O(h 2 ) in L 2 -norm and 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.