Geometric finite element discretization of Maxwell equations in primal and dual spaces

Based on a geometric discretization scheme for Maxwell equations, we unveil a mathematical 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, [XE][XE] (primal formulation) and [XH][XH] (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.