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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 <inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula>, 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.

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