On hp-Streamline Diffusion and Nitsche schemes for the Relativistic Vlasov-Maxwell System

We study stability and convergence of $hp$-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the $hp$ scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space $H^{s+1}(\Omega)$, we derive global {\sl a priori} error bound of order ${\mathcal O}(h/p)^{s+1/2}$, where $h (= \max_K h_K)$ is the mesh parameter and $p (= \max_K p_K)$ is the spectral order. This estimate is based on the local version with $h_K=\mbox{ diam } K$ being the diameter of the {\sl phase-space-time} element $K$ and $p_K$ is the spectral order (the degree of approximating finite element polynomial) for $K$. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an {\sl elliptic type} equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of ${\mathcal O}(h^2+k^2)$, where $h$ is the spatial mesh size and $k$ is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [20].