Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes forthe approximation of the one-dimensional periodic Vlasov-Poissonsystem. The methods are based on the coupling of discontinuousGalerkin approximation to the Vlasov equation and several finiteelement (conforming, non-conforming and mixed) approximations forthe Poisson problem. We show optimal error estimates for all theproposed methods in the case of smooth compactly supported initialdata. The issue of energy conservation is also analyzed for someof the methods.