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Discrete exterior calculus

## Summary

In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes[1] (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated (e.g., near an obstruction to a fluid flow), while using less computational power than if a uniformly fine mesh were used.

## The discrete exterior derivative

Stokes' theorem relates the integral of a differential (n − 1)-form ω over the boundaryM of an n-dimensional manifold M to the integral of dω (the exterior derivative of ω, and a differential n-form on M) over M itself:

${\displaystyle \int _{M}\mathrm {d} \omega =\int _{\partial M}\omega .}$

One could think of differential k-forms as linear operators that act on k-dimensional "bits" of space, in which case one might prefer to use the bracket notation for a dual pairing. In this notation, Stokes' theorem reads as

${\displaystyle \langle \mathrm {d} \omega \mid M\rangle =\langle \omega \mid \partial M\rangle .}$

In finite element analysis, the first stage is often the approximation of the domain of interest by a triangulation, T. For example, a curve would be approximated as a union of straight line segments; a surface would be approximated by a union of triangles, whose edges are straight line segments, which themselves terminate in points. Topologists would refer to such a construction as a simplicial complex. The boundary operator on this triangulation/simplicial complex T is defined in the usual way: for example, if L is a directed line segment from one point, a, to another, b, then the boundary ∂L of L is the formal difference b − a.

A k-form on T is a linear operator acting on k-dimensional subcomplexes of T; e.g., a 0-form assigns values to points, and extends linearly to linear combinations of points; a 1-form assigns values to line segments in a similarly linear way. If ω is a k-form on T, then the discrete exterior derivative dω of ω is the unique (k + 1)-form defined so that Stokes' theorem holds:

${\displaystyle \langle \mathrm {d} \omega \mid S\rangle =\langle \omega \mid \partial S\rangle .}$

For every (k + 1)-dimensional subcomplex of T, S.

Other operators and operations such as the discrete wedge product,[2] Hodge star, or Lie derivative can also be defined.

## Notes

1. ^ Ptáčková, Lenka; Velho, Luiz (June 2021). "A simple and complete discrete exterior calculus on general polygonal meshes". Computer Aided Geometric Design. 88: 102002. arXiv:2401.15436. doi:10.1016/j.cagd.2021.102002. S2CID 235613614.
2. ^ Ptackova, Lenka; Velho, Luiz (2017). "A Primal-to-Primal Discretization of Exterior Calculus on Polygonal Meshes". Symposium on Geometry Processing 2017- Posters: 2 pages. doi:10.2312/SGP.20171204. ISBN 9783038680475. ISSN 1727-8384.

## References

• A simple and complete discrete exterior calculus on general polygonal meshes, Ptackova, Lenka and Velho, Luiz, Computer Aided Geometric Design, 2021, DOI: 10.1016/j.cagd.2021.102002
• Discrete Calculus, Grady, Leo J., Polimeni, Jonathan R., 2010
• Hirani Thesis on Discrete Exterior Calculus
• A Primal-to-Primal Discretization of Exterior Calculus on Polygonal Meshes, Ptackova, L. and Velho, L., Symposium on Geometry Processing 2017, DOI: 10.2312/SGP.20171204
• Convergence of discrete exterior calculus approximations for Poisson problems, E. Schulz & G. Tsogtgerel, Disc. Comp. Geo. 63(2), 346 - 376, 2020
• On geometric discretization of elasticity, Arash Yavari, J. Math. Phys. 49, 022901 (2008), DOI:10.1063/1.2830977
• Discrete Differential Geometry: An Applied Introduction, Keenan Crane, 2018