Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete Exterior Calculus

Mathieu Desbrun, Anil N. Hirani, Melvin Leok, Jerrold E. Marsden

TL;DR

Discrete Exterior Calculus (DEC) provides a unified framework for calculus on discrete spaces by formulating discrete differential forms and vector fields on simplicial complexes and equipping them with operators $\mathbf{d}$, $\boldsymbol{\delta}$, $\ast$, $\wedge$, $\flat$, $\sharp$, and $\mathbf{i}_X$. The key insight is the circumcentric dual, which enables a coherent primal–dual calculus, yielding discrete analogues of curl, divergence, Laplace--Beltrami, and variational structures that align with continuous counterparts in the limit. The paper develops a full algebraic-topological toolkit, including a discrete Poincaré lemma, groupoid-based extensions for dynamics, and remeshing/coarsening strategies, while connecting to Maxwell’s equations and harmonic problems. The results show that DEC can produce structure-preserving, variational discretizations suitable for simulations in elasticity, fluids, electromagnetism, and beyond, with potential extensions to non-flat manifolds and higher-order tensor theories.

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.

Discrete Exterior Calculus

TL;DR

Discrete Exterior Calculus (DEC) provides a unified framework for calculus on discrete spaces by formulating discrete differential forms and vector fields on simplicial complexes and equipping them with operators , , , , , , and . The key insight is the circumcentric dual, which enables a coherent primal–dual calculus, yielding discrete analogues of curl, divergence, Laplace--Beltrami, and variational structures that align with continuous counterparts in the limit. The paper develops a full algebraic-topological toolkit, including a discrete Poincaré lemma, groupoid-based extensions for dynamics, and remeshing/coarsening strategies, while connecting to Maxwell’s equations and harmonic problems. The results show that DEC can produce structure-preserving, variational discretizations suitable for simulations in elasticity, fluids, electromagnetism, and beyond, with potential extensions to non-flat manifolds and higher-order tensor theories.

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.

Paper Structure

This paper contains 57 sections, 17 theorems, 166 equations, 24 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Applications to Variational Problems.
  3. Structured Constraints.
  4. Lattice Theories.
  5. Some of the Key Theoretical Accomplishments.
  6. Context.
  7. Methodology.
  8. The Objects in DEC.
  9. History and Previous Work
  10. Primal Simplicial Complex and Dual Cell Complex
  11. Circumcentric Subdivision.
  12. Circumcentric Dual.
  13. Orientation of the Dual Cell.
  14. Orientation of the Dual of a Dual Cell.
  15. Support Volume of a Primal Simplex and Its Dual Cell.
  16. ...and 42 more sections

Key Result

Lemma 6.1

For a $k$-form $\alpha^k$,

Figures (24)

  • Figure 1: Primal, and dual meshes, as chains in the first circumcentric subdivision.
  • Figure 2: Orienting the dual of a cell.
  • Figure 3: Examples of chains.
  • Figure 4: Discrete vector fields.
  • Figure 5: Terms in the wedge product of two discrete $1$-forms.
  • ...and 19 more figures

Theorems & Definitions (108)

  • Definition 3.1
  • Example 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Example 3.2
  • Example 3.3
  • ...and 98 more