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

Abstract

A discrete differential form approach to solving Maxwell's equations numerically on unstructured meshes is presented. A differential form representation of Maxwell's equations provides a natural and coordinate-free means of studying these equations and their solutions in the presence of curved objects. We begin by reviewing basic properties of differential forms and the operators associated with them for their use in describing electromagnetic fields and sources. Because we are interested in numerically solving Maxwell's equations on unstructured meshes, we introduce discrete representations of these differential forms and the underlying manifolds. This allows a discrete representation of Maxwell's equations in terms of chains and cochains on an arbitrary polyhedral cell complex. The discrete boundary operator, coboundary operator and star operator on cochains are constructed and shown to maintain divergence-free regions. The constructions of the dual of a polyhedral cell complex and the star operator, which give a one-to-one correspondence between the primary and the dual cell complexes, are introduced. This star operation gives the relationship between the magnetic field 1-cochain on the dual cell complex and the magnetic flux 2-cochain on the primary cell complex, and the relationship between the electric field 1-cochain on the primary cell and the electric flux 2-cochain on the dual cell complex. With the construction of these operators, the dual cell complex, and the associated cochains, we have determined the corresponding numerical update equations for the electromagnetic fields on unstructured meshes. The numerical update equations provided by the discrete differential form approach are determined explicitly for cubical, parallelepiped, tetrahedral, and trapezoid cell complexes. For the special cases of an orthogonal complex and a parallelepiped complex, these discrete differential form update equations recover those provided by both Yee's algorithm and the discrete surface integral (DSI) algorithm. It is demonstrated that the discrete differential form update equations differ from those obtained with the DSI approach on the more irregular trapezoid cell complex and, hence, may overcome the known late-time instabilities associated with the DSI approach applied to such highly unstructured meshes.