Analysis and Systematic Discretization of a Fokker–Planck Equation with Lorentz Force

Abstract

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 practical discretization scheme is proposed based on an implicit Euler method for the energy variable and a spherical-harmonics finite-element discretization with respect to the remaining variables. A complete error analysis of the resulting scheme is given and numerical tests are presented to illustrate the theoretical results and the performance of the proposed method.