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.