We discuss the adaptive numerical solution of the Schrödinger-Poisson
equation on a truncated finite domain with an underlying space discretization by conforming
piecewise polynomial finite elements, where we truncate to a
sufficiently large finite domain and impose homogeneous Dirichlet boundary conditions. The
motivation for this approach is the possibility to treat the Poisson equation separately
by dedicated solvers for the arising linear equations. The classical convergence orders
in both the time and space discretization are established theoretically under
natural assumptions on the regularity of the exact solution and illustrated
by numerical experiments. Adaptive time-stepping relying on a defect-based
error estimator is shown to correctly reflect the solution behaviour.