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Geometric finite element discretization of Maxwell equations in primal and dual spaces. (English) Zbl 1195.78058
Summary: 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, $\left[XE\right]$ (primal formulation) and $\left[XH\right]$ (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.
##### MSC:
 78M10 Finite element methods (optics) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)