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

Abstract

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.

22 pages and 4 figures

References in corpus (1)

• On the degrees of freedom of lattice electrodynamics

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