Spherical Harmonics and a Semidiscrete Finite Element Approximation for the Transport Equation

Abstract

This work is the first part in a series of two articles, where the objective is to construct, analyze, and implement realistic particle transport models relevant in applications in radiation cancer therapy. Here we use spherical harmonics and derive an energy-dependent model problem for the transport equation. Then we show stability and derive optimal convergence rates for semidiscrete (discretization in energy) finite element approximations of this model problem. The fully discrete problem that also considers the study of finite element discretizations in radial and spatial domains as well is the subject of a forthcoming article.

Keywords:

• spherical harmonics
• transport equation
• finite element method
• charged particle beams

Acknowledgments

This research is partially supported by the Swedish Foundation of Strategic Research (SSF) in Gothenburg Mathematical Modeling Center (GMMC).