The Galactic synchrotron emission contains abundant physics related to not only the Galactic magnetized interstellar medium but also has prominent effect on understanding the Cosmic microwave background especially the B-mode polarization. To catch up with the growing precision in astrophysical observations, we need to build a consistent numerical framework where simulating the cosmic ray (electron) propagation is a major task. In the master project, we propose to use the finite element method for solving cosmic ray (electron) transport equation within the phase-space of dimension varying from two to six. The numeric package BIFET is developed on top of the deal.II library with support in the adaptive mesh refinement. We mainly introduce the design and methods in solving advection-diffusion problems and demonstrate its capability and precision with physically simplified tests and examples.
