  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.
