Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations

Abstract

We identify primal and dual formulations in the finite element method (FEM) solution of the vector wave equation using a geometric discretization based on differential forms. These two formulations entail a mathematical duality denoted as Galerkin duality. Galerkin-dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to machine precision), but have static (zero eigenvalue) solution spaces of different dimensions. Algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep-rooted connection between the Hodge-Helmholtz decomposition of differential forms and Descartes-Euler polyhedral formula, and verified numerically. In order to tackle the fullness of dual formulation, algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices