A Classical Field Theory Formulation for the Numerical Solution of Time Harmonic Electromagnetic Fields

Abstract

Finite element representations of Maxwell's equations pose unusual challenges inherent to the variational representation of the “curl-curl” equation for the fields. We present a variational formulation which solves for the four-potential instead, based on classical field theory. Borrowing from quantum electrodynamics, we modify the Lagrangian by adding an implicit gauge-fixing term. This reformulation explicitly accounts for Gauss' law through the coupling between φ and ρ, and enables the use of nodal basis functions instead of edge elements for time-harmonic problems. We demonstrate how this formulation, adhering to the deeper underlying symmetries of the four-dimensional covariant field description, provides a highly general, robust numerical framework.