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In mathematics, the discrete exterior calculus (DEC) is the extension of the
exterior calculus In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
to
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
spaces including graphs and finite element meshes. 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 In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
of a differential (''n'' − 1)-form ''ω'' over the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
∂''M'' of an ''n''-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al manifold ''M'' to the integral of d''ω'' (the exterior derivative of ''ω'', and a differential ''n''-form on ''M'') over ''M'' itself: :\int_ \mathrm \omega = \int_ \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 bra–ket notation for a dual pairing. In this notation, Stokes' theorem reads as :\langle \mathrm \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 d''ω'' of ''ω'' is the unique (''k'' + 1)-form defined so that Stokes' theorem holds: :\langle \mathrm{d} \omega \mid S \rangle = \langle \omega \mid \partial S \rangle. For every (''k'' + 1)-dimensional subcomplex of ''T'', ''S''. Other concepts such as the discrete
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
and the discrete
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
can also be defined.


See also

*
Discrete differential geometry Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, g ...
*
Discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology com ...
*
Topological combinatorics The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in top ...
*
Discrete calculus Discrete calculus or the calculus of discrete functions, is the mathematical study of ''incremental'' change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word ''ca ...


References


Discrete Calculus
Grady, Leo J., Polimeni, Jonathan R., 2010
Hirani Thesis on Discrete Exterior CalculusConvergence 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 Finite element method Multilinear algebra