Discrete Exterior Calculus

TL;DR

This paper develops a comprehensive theory of discrete exterior calculus on simplicial complexes, incorporating differential forms, vector fields, and their interactions, with a focus on circumcentric duals for improved mathematical structure.

Contribution

It introduces a unified framework for discrete exterior calculus that includes vector fields and operators, utilizing circumcentric duals instead of barycentric ones, advancing the mathematical foundation.

Findings

Includes discrete vector fields and operators in the calculus

Highlights the importance of circumcentric duals over barycentric duals

Provides applications demonstrating the theory's utility

Abstract

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete exterior calculus that admits both vector fields and forms.