Finite Element Methods for Linear Maxwell's Equations in Bianisotropic Media Permitting Polarization Fields and Magnetic Currents

TL;DR

This paper develops and demonstrates a finite element method for solving Maxwell's equations in complex bianisotropic media, accounting for polarization, magnetization, and magnetic charges, with numerical validation.

Contribution

It introduces a variational formulation and finite element approximation for Maxwell's equations in bianisotropic media, including magnetic charges and currents, with convergence analysis.

Findings

Finite element method converges for complex media

Numerical examples validate the approach

Handles nonzero polarization and magnetization

Abstract

We review Maxwell's equations and constitutive relations for 3D bianisotropic media in a generalized form: we consider all four variables and allow for nonzero polarization or magnetization, and also nonzero nonzero magnetic charge or current. After a discussion of general boundary conditions, we introduce a time-harmonic variational formulation of linear Maxwell's equations within 3D bianisotropic media in terms of the electric and magnetic fields. We showcase a finite element approximation of our variational formulation, using curl-conforming N\'ed\'elec edge elements of the first kind. Numerical examples illustrate the convergence of the method.