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），提出了一种隐式Crank-Nicolson有限元格式。在数值格式中，时间离散采用隐式Crank-Nicolson方法，空间离散采用协调有限元方法。我们证明了所提方法的适定性，并确保其在离散层面上保持质量和能量守恒。进一步地，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率与守恒性质。