  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].
