Boundary element method is used for the purpose of ray tracing of charged particles. The results presented here are based on a commercially available package; Lorentz, by Integrated Engineering Software; a.k.a. Enginia Research Inc. First, a brief description of different numerical methods is presented, followed by some known theoretical examples. In all cases, excellent agreement between the theoretical results and the numerical ones is observed. Several of the examples deal with the space charge issue. Child's law and Langmuir-Blodgett's are used to verify these results. Also an example, 743 Test, of launching a particle close to a field discontinuity is presented to show the power of BEM method when it comes to dealing with extreme ratios in the dimensionality (>106) within a device.
"Boundary element method (BEM) for charged particle optics", Proc. SPIE 4510, Charged Particle Detection, Diagnostics, and Imaging, (21 December 2001); doi: 10.1117/12.451274; https://doi.org/10.1117/12.451274