Discrete exterior calculus (DEC) is a discretization of exterior calculus on an oriented simplicial complex that preserves identities
such as 
and separates topological operators from metric-dependent ones.
A discrete 
-form
is a cochain that assigns a number to every oriented 
-simplex, with the sign reversed
when the orientation is reversed. The discrete exterior
derivative 
is the coboundary operator.
For a discrete 
-form 
, it is given by
![(d_kalpha)([v_0,...,v_(k+1)])=sum_(i=0)^(k+1)(-1)^ialpha([v_0,...,v_i^^,...,v_(k+1)]).](/images/equations/DiscreteExteriorCalculus/NumberedEquation1.svg) |
(1)
|
Equivalently, the matrix of 
is the signed incidence
matrix between 
- and 
-simplices. The identity 
follows from the fact that the boundary of a boundary
is zero.
Metric information is supplied by a discrete Hodge star, which maps primal 
-forms to dual 
-forms. It determines adjoints 
and hence the discrete Hodge Laplacian
 |
(2)
|
In finite dimensions this gives the discrete Hodge decomposition
 |
(3)
|
whose three summands are the exact, coexact, and harmonic discrete forms, respectively.
DEC is used in numerical differential equations, computational geometry, and geometry processing. For example, Dodik et al. (2025) represent an impossible figure by a locally consistent edge-depth 1-form whose nonzero harmonic component is the obstruction to globally consistent depths.
