Parallel Spectral Numerical Methods/Maxwell's Equations
Maxwell’s equations are a set of hyperbolic partial differential equations that describe the propagation of electric and magnetic fields. They can be numerically approximated by several different methods, common methods include: finite element, finite differences, and spectral methods. In this section, we wish to explain how to develop a spectral method scheme to solve Maxwell's equations. To do so, we first introduce Maxwell's equations.
Maxwell’s equations for an observer in the laboratory frame, where no free charges and currents are present, can be written as:

(

(
with the relation between the electromagnetic components given by the constitutive relations:

(

(
where and are the electromagnetic material properties, analytically they are symmetrical second rank tensors with non zero offdiagonal entries; and are the vacuum’s permittivity and permeability, respectively; the functions and are the polarization and magnetization, respectively, material’s response to the incident field; and is known as the dichroic response of the material .^{[1]}.