Blow-up Whitney forms, shadow forms, and Poisson processes

The Whitney forms on a simplex T admit high-order generalizations that have received a great deal of attention in numerical analysis. Less well-known are the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize the Whitney forms, but have rational coefficients, allowing singularities near the faces of T. Motivated by numerical problems that exhibit these kinds of singularities, we introduce degrees of freedom for the shadow k-forms that are well-suited for finite element implementations. In particular, we show that the degrees of freedom for the shadow forms are given by integration over the k-dimensional faces of the blow-up T̃ of the simplex T. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of T̃, which vanishes except in degree zero. Additionally, we discover a surprising probabilistic interpretation of shadow forms in terms of Poisson processes. This perspective simplifies several proofs and gives a way of computing bases for the shadow forms using a straightforward combinatorial calculation.