  Operator splitting methods combined with finite element spatial
discretizations are studied for time-dependent nonlinear Schr\"odinger
equations. In particular, the Schr\"odinger-Poisson equation under homogeneous
Dirichlet boundary conditions on a finite domain is considered. A rigorous
stability and error analysis is carried out for the second-order Strang
splitting method and conforming polynomial finite element discretizations. For
sufficiently regular solutions the classical orders of convergence are
retained, that is, second-order convergence in time and polynomial convergence
in space is proven. The established convergence result is confirmed and
complemented by numerical illustrations.
