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Operator splitting methods combined with finite element spatialdiscretizations are studied for time-dependent nonlinear Schr\"odingerequations. In particular, the Schr\"odinger-Poisson equation under homogeneousDirichlet boundary conditions on a finite domain is considered. A rigorousstability and error analysis is carried out for the second-order Strangsplitting method and conforming polynomial finite element discretizations. Forsufficiently regular solutions the classical orders of convergence areretained, that is, second-order convergence in time and polynomial convergencein space is proven. The established convergence result is confirmed andcomplemented by numerical illustrations.

Convergence of a Strang splitting finite element discretization for the Schr\"odinger-Poisson equation