Preserving Hamiltonian structure for discretized Maxwell equations with sources*

In the past few decades (e.g. Bossavit) differential forms have been used in computational electromagnetism. We build on the mimetic discretization framework of Bochev and Hyman to construct a mimetic Petrov-Galerkin method for Maxwell equations with general nonlinear polarization and magnetization. Two de Rham complexes are used: one complex represents orientation independent straight forms, the other orientation dependent twisted forms. These complexes are related to each other through the Hodge star operator, which plays a crucial role in modeling the constitutive relations. We found that the Petrov-Galerkin mass matrix is a discrete approximation to the Hodge star operator closely related to existing discrete Hodge star operators in the literature. The two complexes are discretized on staggered cell complexes, yielding a method reminiscent of the Yee scheme but with significant flexibility to prescribe general geometries. The use of mimetic discretization facilitates a strategy for projecting the Hamiltonian structure of the continuous model to that of the discretization,  allowing for exact enforcement of the electromagnetism constaint relations.