The Karlsruhe Tritium Neutrino (KATRIN) experiment is a tritium beta decay experiment designed to make a direct, model independent measurement of the electron neutrino mass. The experimental apparatus employs strong (O[T]) magnetostatic and (O[10<super>5</super> V/m]) electrostatic fields in regions of ultra high (O[10<super>-11</super> mbar]) vacuum in order to obtain precise measurements of the electron energy spectrum near the endpoint of tritium beta-decay. The electrostatic fields in KATRIN are formed by multiscale electrode geometries, necessitating the development of high performance field simulation software. To this end, we present a Boundary Element Method (BEM) with analytic boundary integral terms in conjunction with the Robin Hood linear algebraic solver, a nonstationary successive subspace correction (SSC) method. We describe an implementation of these techniques for high performance computing environments in the software KEMField, along with the geometry modeling and discretization software KGeoBag. We detail the application of KEMField and KGeoBag to KATRIN's spectrometer and detector sections, and demonstrate its use in furthering several of KATRIN's scientific goals. Finally, we present the results of a measurement designed to probe the electrostatic profile of KATRIN's main spectrometer in comparison to simulated results.