  Based on a geometric discretization scheme for Maxwell equations, we unveil a
mathematical\textit{\}transformation between the electric field intensity $E$
and the magnetic field intensity $H$, denoted as Galerkin duality. Using
Galerkin duality and discrete Hodge operators, we construct two system
matrices, $[ X_{E}] $ (primal formulation) and $[ X_{H} % ] $ (dual
formulation) respectively, that discretize the second-order vector wave
equations. We show that the primal formulation recovers the conventional
(edge-element) finite element method (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 dynamical physical solutions,
the dimensions of the null spaces are different.
