The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker-Planck equation with Lorentz force. This model is studied both, from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practicable discretization scheme is proposed based on implicit time-stepping through the energy levels and a spherical-harmonics finite-element discretization with respect to the remaining variables. A full error analysis of the resulting scheme is given, and numerical results are presented to illustrate the theoretical results and the performance of the proposed method.

翻译：在磁场存在下，带电粒子通过散射介质的传播可由带洛伦兹力的福克-普朗克方程描述。本研究从理论和数值两个角度对该模型进行探讨。首先推导了相关函数空间上的特定迹估计，以明确边界值的含义；随后，利用Rothe方法证明了弱解的存在性。在研究的第二步中，我们提出了一种完全可实现的离散化方案，该方案基于通过能级的隐式时间步进以及剩余变量的球谐函数-有限元离散化。给出了该方案的完整误差分析，并展示了数值结果，以阐释理论成果和所提方法的性能。