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