The pairing matrix in discrete electromagnetism

¹ CERN AT/MEL, 1211 Geneva 23, Switzerland
² ETAS GmbH, Stuttgart, Germany

Corresponding author: bernhard.auchmann@cern.ch

Received: 18 January 2007
Revised: 19 March 2007
Accepted: 3 April 2007
Published online: 28 June 2007

Abstract

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.