A mimetic discretization of the macroscopic Maxwell equations in Hamiltonian form

A mimetic discretization of the Hamiltonian structure of the macroscopic Maxwell equations with periodic boundary conditions is presented. The model accommodates general (and possibly nonlinear) polarizations and magnetizations. The fields are modeled as either straight or twisted differential forms, so each variable associated with one of the vector spaces in the double de Rham complex. The discretization strategy is an adaptation of the mimetic discretization framework of Bochev and Hyman with special attention given to the Poincar{\'e} duality structure inherent in the double de Rham complex. The $L^2$ and Poincar{\'e} duality pairings induce the Hodge star operator at both the continuous and discrete levels which act as maps between the two de Rham complexes. Additionally, the discrete duality pairings provide a convenient framework for the discretization of variational derivatives. These discretized variational derivatives may then be used as a tool for discretizing the Poisson bracket of the macroscopic Maxwell equations. The discretized macroscopic Maxwell equations possess Hamiltonian structure; the use of mimetic spaces and and a natural discretization of the variational derivatives ensure the existence of discrete Casimir invariants of the Maxwell bracket. As a simple test case, a one-dimensional version of Maxwell's equations are discretized in the same manner and its computed solutions are given.