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 based on classical field theory. Borrowing from QED, we modify the Lagrangian by adding an implicit gauge-fixing term. Our formulation, in the language of differential geometry, shows that conventional edge elements should be replaced by the simpler nodal 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.