Geometric aspects of the Simplicial Discretization of Maxwell's equations

Abstract

Aspects of the geometric discretization of electromagnetic fields on simplicial lattices are considered. First, the convenience of the use of exterior differential forms to represent the field quantities through their natural role as duals (cochains) of the geometric constituents of the lattice (chains = nodes, edges, faces, volumes) is briefly reviewed. Then, the use of the barycentric subdivision to decompose the (ordinary) simplicial primal lattice together with the (twisted) non-simplicial barycentric dual lattice into simplicial elements is considered. Finally, the construction of lattice Hodge operators by using Whitney maps on the first barycentric subdivision is described. The objective is to arrive at a discrete formulation of electromagnetic fields on general lattices which better adheres to the underlying physics.