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We prove the convergence of discontinuous Galerkin approximations for theVlasov-Poisson system written as an hyperbolic system using Hermite polynomialsin velocity. To obtain stability properties, we introduce a suitable weighted L2 space, with a time dependent weight, and first prove global stability for theweighted L 2 norm and propagation of regularity. Then we prove error estimatesbetween the numerical solution and the smooth solution to the Vlasov-Poissonsystem.

On the convergence of discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system