Discrete exterior calculus
Discrete exterior calculus (DEC) is the analogue of exterior calculus for discrete spaces, usually simplicial but possibly cubical or based on other shapes. Some authors regard DEC as a computational approximation to the continuous calculus; others treat it as an independent mathematical system applicable to inherently discrete systems. Both perspectives can be useful.
Because there is not a unique way to discretize the classical exterior calculus, the DEC is not a single formalism but a family of closely related ones. This is reflected in the organization of this page.
DEC in general
- Grady & Polimeni, 2010: Discrete calculus: Applied analysis on graphs for
computational science (doi)
- Chapter 2 is concise introduction to classical and discrete exterior calculus
- Most of book is about applications to data analysis and network science
- Kotiuga, 2008: Theoretical limitations of discrete exterior calculus in the context of computational electromagnetics (doi)
DEC ala Hirani
The approach to DEC introduced by Desbrun, Hirani, et al has the best claim to being called “standard” and has generated a good-sized literature.
Introductions
- Hirani, 2003, PhD thesis: Discrete exterior calculus (doi)
- A good place to start, with clear and careful development of all the major concepts
- Desbrun, Hirani, Leok, Marsden, 2005: Discrete exterior calculus (arxiv)
- Sections 2-10 are just a compressed version of Hirani’s PhD thesis, which should be consulted for the missing details
- Sections 11-14 seem to follow work by Desbrun and others
- Table 1 helpfully explains how primal orientation induces dual orientation
- Desbrun, Kanso, Tong, 2008: Discrete differential forms for computational modeling (doi)
- Gillette, 2009, lecture notes: Notes on discrete exterior calculus (pdf)
- Compressed summary of DEC
- Section 2.13 has useful worked calculations of the Hodge star and coderivative for subdivision via barycenter, circumcenter, and incenter
- Crane, 2013, lecture notes: Discrete differential geometry: An applied introduction (online )
Hodge star
The discrete Hodge star is the essential ingredient for the metric part of the DEC that goes beyond classical algebraic topology. People are still trying to work out the right definition.
- Mohamed, Hirani, Samtaney, 2016: Comparison of discrete Hodge star operators for surfaces (doi, pdf)
- Ayoub, Hamdouni, Razafindralandy, 2021: A new Hodge operator in discrete exterior calculus. Application to fluid mechanics (doi, arxiv)
Applications
- Mohamed, Hirani, Samtaney, 2016: Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes (doi, arxiv)
- Noguez et al, 2020: Discretization of the 2D convection-diffusion equation using discrete exterior calculus (doi)
Software
- Elcott & Schroder, 2005: Building your own DEC at home (doi, pdf)
- Bell & Hirani, 2012: PyDEC: Software and algorithms for discretization of
exterior calculus (doi, arxiv, GitHub )
- About PyDEC, an actively maintained Python package for DEC and FEEC
- Implementation details for simplicial complexes and cubical complexes
DEC ala Wilson
Wilson introduced another approach to the discrete Hodge star operator, which has the advantage of not requiring a dual complex.
Theory
- Wilson, 2005, PhD thesis: On the algebra and geometry of a manifold’s chains
and cochains (pdf)
- Ch. 1 of the thesis: Wilson, 2005: Geometric structures on the cochains of a manifold (arxiv)
- Wilson, 2007: Cochain algebra on manifolds and convergence under refinement
(doi, pdf)
- Main reference for this approach to the discrete Hodge star
- Wilson, 2008: Conformal cochains (doi)
- Arnold, 2012, PhD thesis: The discrete Hodge star operator and Poincare duality (pdf)
- Tanabe, 2015: Several remarks on the combinatorial Hodge star (pdf)
- Improves the convergence results in (Wilson, 2007)
Applications
- Wilson, 2011: Differential forms, fluids, and finite models (doi)