Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces

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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] (primal formulation) and [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.