In this paper, we establish unconditionally optimal error estimates for linearized backward Euler Galerkin finite element methods (FEMs) applied to nonlinear Schrödinger-Helmholtz equations. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and derive error estimates in temporal direction. Second, by the above achievements, we obtain the boundedness of the numerical solution in \(L^{\infty }\)-norm. Then, the optimal L² error estimates for r-order FEMs are derived without any restriction on the time step size. Numerical results in both two- and three-dimensional spaces are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods.

在本文中，我们建立了应用于非线性Schrödinger-Helmholtz方程的线性反向Euler Galerkin有限元方法（FEM）的无条件最优误差估计。通过使用时空误差分裂技术，我们将精确解和数值解之间的误差分为两个部分，分别称为时间误差和空间误差。首先，通过引入时离散系统，我们证明了该时离散系统在一些强范数上的一致有界性，并在时间方向上导出了误差估计。其次，通过以上成就，我们获得了\（L ^ {\ infty} \）-范数中数值解的有界性。然后，针对r的最优L ²误差估计阶FEM不受时间步长的限制。提供了二维和三维空间中的数值结果，以说明理论预测并证明方法的有效性。