Convergence of hp-Streamline Diffusion Method for Vlasov–Maxwell System. Issue 7 (10th November 2019) - Asadzadeh, M.; Kowalczyk, P.; Standar, C.
Abstract: In this paper we study stability and convergence for hp -streamline diffusion (SD) finite element method for the, relativistic, time-dependent Vlasov–Maxwell (VM) system. We consider spatial domain Ω x ⊂ R 3 and velocities v ∈ Ω v ⊂ R 3 . The objective is to show globally optimal a priori error bound of order O ( h / p ) s + 1 / 2, for the SD approximation of the VM system; where h   ( = max K h K ) is the mesh size and p   ( = max K p K ) is the spectral order. Our estimates rely on the local version with hK being the diameter of the phase-space-time element K and pK the spectral order for K . The optimal hp estimates require an exact solution in the Sobolev space H s + 1 ( Ω ) . Numerical implementations, performed for examples in one space- and two velocity dimensions, are justifying the robustness of the theoretical results.