  A rigorous convergence analysis of the Strang splitting algorithm with a
discontinuous Galerkin approximation in space for the Vlasov--Poisson equations
is provided. It is shown that under suitable assumptions the error is of order
$\mathcal{O}(\tau^2+h^q +h^q / \tau)$, where $\tau$ is the size of a time step,
$h$ is the cell size, and $q$ the order of the discontinuous Galerkin
approximation. In order to investigate the recurrence phenomena for
approximations of higher order as well as to compare the algorithm with
numerical results already available in the literature a number of numerical
simulations are performed.
