The Pairing Matrix in Discrete Electromagnetism On the Geometry of Discrete de Rham Currents

We introduce pairing matrices on simplicial cell complexes in discrete electromagnetism as a means to avoid the explicit construction of a topologically dual complex. Interestingly, the Finite Element Method with first-order Whitney elements – when it is looked upon from a cell-method perspective – features pairing matrices and thus an implicitly defined dual mesh. We show that the pairing matrix can be used to construct discrete energy products. In this exercise we find that different formalisms lead to equivalent matrix representations. Discrete de Rham currents are an elegant way to subsume these geometrically equivalent but formally distinct ways of defining energy-products.