A Hamiltonian structure preserving discretization of Maxwell's equations with general constitutive laws

It is frequently advantageous to characterize the microscopic response of a medium to an electromagnetic field with constitutive relations for the polarization and magnetization of that medium which self-consistently depend on the fields. This basic idea accounts for the electromagnetic component of many reduced models in plasma physics and nonlinear optics. It is possible to describe such models in a general Hamiltonian framework. We consider a Hamiltonian structure preserving spatial discretization of Maxwell's equations in general media using a spectral element finite element exterior calculus method. This method is capable of supporting arbitrary (and even nonlinear) constitutive laws. When the spatially discretized model is time evolved using a Hamiltonian splitting method, the energy is approximately conserved, and Gauss's laws for the electric and magnetic fields are exactly conserved. Moreover, because of its Hamiltonian structure preservation, it is well suited to be used as the field solver in a PIC scheme.