Topological and geometrical considerations for maxwell's equations on unstructured meshes

A discrete differential form approach to solving Maxwell's equations on unstructured meshes is presented. Discrete representations of differential forms, the underlying manifolds, and associated operators are developed. The discrete boundary, coboundary, and hodge star operators are shown to maintain divergence-free regions. With the construction of a cell complex, its dual complex, and the associated discrete operators, we have determined the numerical update equations for the electromagnetic fields on unstructured meshes with second-order accuracy. The discrete differential form approach generalizes to the Yee algorithm on an orthogonal complex and to the discrete surface integral algorithm on a parallelepiped complex.