On convergence of a h-p Streamline Diffusion and Discontinuous Galerkin Methods for the Vlasov-Poisson-Fokker-Planck System

Paper in proceeding, 2009

In this paper we investigate the basic ingredients for global superconvergence strategy of streamline diffusion (SD) and discontinuous Galerkin (DG) finite element approximations in $H^{1}$ and $W^{1,\infty}$-norms (see \cite{Adams:75}) for the solution of the Vlasov--Poisson--Fokker--Planck system. This study is an extension of the results in \cite{Asadzadeh:90}-\cite{Asadzadeh.Sopasakis:2007}, to finite element schemes including discretizations of the Poisson term, where we also introduce results of an extension of the $h$-versions of SD and DG to the corresponding $hp$-versions. Optimal convergence results presented in the paper relay on the estimates for the regularized Green's functions with memory terms where some interpolation postprocessing techniques play important roles, see \cite{Baouendi.Grisvard:86}.