KNOWPIA
WELCOME TO KNOWPIA

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.

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

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

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** d*ω* of *ω* is the unique (*k* + 1)-form defined so that Stokes' theorem holds:

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.

**^**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.**^**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.

- 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