discrete exterior calculus
/ DEC /
Smooth calculus on manifolds is built from differential forms, the exterior derivative d, and the Hodge star — the machinery behind grad, curl, div, and Stokes' theorem. Discrete exterior calculus rebuilds that same machinery on a mesh, with forms replaced by numbers stored on the mesh elements. The slogan is: a k-form is something you integrate over k-dimensional pieces, so on a mesh just store, for each k-cell, the value of that integral. Forms become cochains, and the whole apparatus becomes exact linear algebra.
Here is how it works. On a simplicial complex, a discrete k-form is a cochain: an assignment of one real number to every oriented k-simplex (a value per vertex for 0-forms, per edge for 1-forms, per triangle for 2-forms). The discrete exterior derivative d is the coboundary operator, the transpose of the boundary operator: (d omega)(sigma) = sum of omega over the faces of sigma with signs from orientation. Because the boundary of a boundary is zero, automatically d composed with d is zero — Stokes' theorem holds exactly and by construction, not approximately: the integral of d omega over a chain equals the integral of omega over its boundary, which here is just the pairing of a cochain with a chain. To turn this metric-free top half into real vector calculus you add the discrete Hodge star, which maps a k-form on the primal mesh to an (n-k)-form on the dual mesh and is the only place the geometry (edge lengths, areas, the cotangent weights) enters.
DEC matters because it gives structure-preserving numerics: schemes that, by construction, satisfy discrete versions of d-d = 0, Stokes, and the Hodge decomposition, so conserved quantities stay conserved and spurious modes are avoided — invaluable in electromagnetism, fluid simulation, and geometry processing. It is closely related to finite-element exterior calculus and to Whitney forms, which provide the interpolating basis that makes the two viewpoints agree. The honest caveat: the topological half (d as coboundary) is canonical and exact, but the metric half (the Hodge star) is a modeling choice — the diagonal 'circumcentric' Hodge star needs well-centered or Delaunay meshes to stay positive, and a bad star can wreck accuracy even though the d-d = 0 identity still holds perfectly.
Encode a vector field on a mesh as a discrete 1-form: store on each edge the field's integral (circulation) along that edge. Then the discrete d of this 1-form is a 2-form whose value on each triangle is the sum of the three edge circulations around its boundary — the discrete curl. And d-d = 0 says taking d again (to a 3-form) vanishes, the exact discrete identity curl of grad = 0 and div of curl = 0.
Storing forms on cells turns d into the signed boundary sum, and d-d=0 becomes an exact combinatorial fact.
DEC splits cleanly into a canonical topological part and a chosen metric part: the coboundary d is forced and gives d-d = 0 exactly, but the Hodge star is a modeling decision. The diagonal circumcentric star requires Delaunay/well-centered meshes to stay positive-definite, so accuracy is a property of the star, not of the exact structural identities.