In this paper we study stability and convergence for hp-streamline diffusion (SD) finite element method for the, relativistic, timedependent Vlasov-Maxwell (VM) system. We consider spatial domain X x & R 3 and velocities v 2 X v & R 3 : The objective is to show globally optimal a priori error bound of order Oðh=pÞ sþ1=2 , for the SD approximation of the VM system; where h ð¼ max K h K Þ is the mesh size and p ð¼ max K p K Þ is the spectral order. Our estimates rely on the local version with h K

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... being the diameter of the phase-space-time element K and p K the spectral order for K. The optimal hp estimates require an exact solution in the Sobolev space H sþ1 ðXÞ: Numerical implementations, performed for examples in one space-and two velocity dimensions, are justifying the robustness of the theoretical results. KEYWORDS Vlasov-Maxwell; stability; convergence; hp method Hence, the name of the method (the streamline diffusion). Such an extra diffusion would improve both the stability and convergence properties of the underlying Galerkin scheme. It is well known that the standard Galerkin method, used for hyperbolic problems, has a suboptimal behavior: Converges as Oðh sÀ1 Þ (versus Oðh s Þ for the elliptic and parabolic problems with exact solution in the same space: H s ðXÞ). The SD method improves this drawback by Oðh 1=2 Þ, also, having an upwinding character, enhances the stability. The properties that are achieved by the discontinuous Galerkin as well. The hp-approach is to capture local behavior in the sense CONTACT M. Asadzadeh