(untitled) (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 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.