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 ${\Omega }_x\subset {\mathbf{R}}^3$ and velocities $\mathbf{v}\in {\Omega }_v\subset {\mathbf{R}}^3.$ The objective is to show globally optimal a priori error bound of order $\mathcal{O}{(h/p)}^{s+1/2},$ 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 h_K 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 $H^{s+1}(\Omega ).$ Numerical implementations, performed for examples in one space- and two velocity dimensions, are justifying the robustness of the theoretical results.

在本文中，我们研究相对论性的，时间相关的Vlasov-Maxwell（VM）系统的hp流线扩散（SD）有限元方法的稳定性和收敛性。我们考虑空间域${\Omega }_X\subset {\mathbf{[R}}^3$ 和速度 $\mathbf{v}\in {\Omega }_v\subset {\mathbf{[R}}^3。$目的是为了显示全局最优顺序的先验误差界$\mathcal{Ø}{（H/p）}^{s+\mathrm{1个}/2}，$用于VM系统的SD近似值；哪里$H （={\mathrm{最高}}_{ķ}{H}_{ķ}）$ 是网眼尺寸和 $p （={\mathrm{最高}}_{ķ}{p}_{ķ}）$是频谱阶数 我们的估计依靠当地版本^ h _ķ是的直径相时空元素ķ和p _ķ光谱顺序ķ。最佳功率估计需要在Sobolev空间中提供精确的解决方案$H^{s+\mathrm{1个}}（\Omega ）。$ 例如在一维和两个速度维度上进行的数值实施证明了理论结果的鲁棒性。