# A classical field theory formulation for the numerical solution of time harmonic electromagnetic fields

## TL;DR

This paper introduces a classical field theory-based variational formulation for solving time-harmonic electromagnetic fields, simplifying finite element methods by replacing edge elements with nodal elements, and leveraging gauge fixing and differential geometry.

## Contribution

It presents a novel variational approach inspired by classical field theory and QED, enabling the use of simpler nodal elements in finite element analysis of Maxwell's equations.

## Key findings

- Replaces edge elements with nodal elements for simplicity.
- Provides a robust, symmetry-preserving numerical framework.
- Demonstrates effectiveness in time-harmonic electromagnetic problems.

## 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.