This dissertation consists of two parts. Part I constitutes an introduction to some of the elementary concepts of the classical theory of differential geometry that are of relevance to computational electromagnetics. The motivation here is to aid researchers in overcoming the significant investment in mathematics that is typically required in order to learn and apply the theory to problems of practical interest in engineering. Another motivation for introducing the mathematical background in such detail is to provide a springboard to the algebraic model that takes up the second part of the dissertation.

The algebraic model of Part II represents a somewhat radical approach to computation. In this approach, rather than take the classical theory based on the continuum as a given and constructing approximate numerical techniques from there, work is done toward constructing an alternative to the continuum theory that is built up from scratch in a discrete setting. The goal is to take each mathematical objects that is necessary in order to write down the classical electromagnetic theory and develop a corresponding discrete analog. The theory builds upon standard concepts in algebraic topology and is motivated by recent progress in noncommutative differential geometry.