In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal $L^2$ error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

翻译：本文针对一类非线性薛定谔型系统（包括薛定谔-亥姆霍兹系统和薛定谔-泊松系统）提出了隐式Crank-Nicolson有限元格式。在数值方案中，采用隐式Crank-Nicolson方法进行时间离散，并使用协调有限元方法进行空间离散。该方法的适定性得以证明，并在离散层面确保质量和能量守恒。此外，我们证明了全离散解的最优$L^2$误差估计。最后，通过数值算例验证了收敛速度与守恒性质。