Compatible discretizations of Maxwell equations

By applying basic tools of algebraic topology and a discrete analog of differential forms, discrete electromagnetic theory can be constructed from first principles on a general (irregular) primal/dual lattice (cell-complex) (Teixeira, F.L. and Chew, W.C., J. Math. Phys., vol.40, p.169-87, 1999). Using a compatible discretization based on differential forms and a discrete Hodge decomposition, we find that Euler´s formula matches the algebraic properties of the discrete Helmholtz decomposition in an exact way. Furthermore, we show that the number of dynamic degrees of freedom (DoFs) for the electric field equals the number of dynamic DoFs for the magnetic field. We submit an identity that reflects an essential property of Maxwell equations on a lattice and can be thought as a design principle for any numerical scheme. Based on Galerkin duality and the discrete Hodge operators, we can construct two system matrices, [X_E] (primal formulation) and [X_H] (dual formulation) respectively. It can be shown that the primal formulation recovers the conventional (edge-element) FEM and suggests a geometric foundation for it. On the other hand, the dual formulation suggests a new (dual) type of FEM. Although both formulations give identical physical solutions, the dimension of the spaces are different.