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

TL;DR

This work introduces blow-up Whitney forms, a rational generalization of classical Whitney forms with boundary singularities, and shows they form a finite element exterior calculus (FEEC) complex whose DOFs are given by integration over $k$-faces of the blow-up simplex $ ilde T$. A novel Poisson-process interpretation yields explicit bases and a streamlined proof that ordinary Whitney forms embed into the blow-up space, and it connects the exterior derivative with a simple combinatorial rule on flags. The authors prove that the blow-up complex is isomorphic to the cellular cochain complex of $ ilde T$, yielding cohomology that vanishes except in degree zero, and they articulate a framework for extending these ideas to global triangulations and to higher-order spaces. The probabilistic perspective opens avenues for efficient basis construction and suggests a unified approach to discretizing differential forms on manifolds with singular metrics or intricate boundary behavior, with potential intrinsic discretizations for surface Laplacians and higher-order FEEC spaces.

Abstract

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 $\tilde T$ of the simplex $T$. Consequently, we obtain an isomorphism between the cohomology of the complex of shadow forms and the cellular cohomology of $\tilde T$, 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.

Figures (7)

• Figure 1: The 6 blow-up Whitney 0-forms in dimension $n=2$. Top row: $\psi_{012},\psi_{021},\psi_{102}$. Bottom row: $\psi_{120},\psi_{201},\psi_{210}$.
• Figure 3: Degrees of freedom for the function $\psi_{120} = \frac{\lambda_1 \lambda_2}{\lambda_2+\lambda_0} \in b\mathcal{P}_1^-\Lambda^0(T)$. Each labelled point represents the limiting value of the function as one approaches the indicated vertex along the indicated edge.
• Figure 4: Four ways of gluing together the local spaces $b\mathcal{P}_1^-\Lambda^0(T)$. Top left: Degrees of freedom on shared edges are equated, leading to a space of functions whose members are single-valued along edges but multi-valued at vertices. Top right: In addition to equating degrees of freedom on shared edges, degrees of freedom on opposite endpoints of each edge are equated, leading to a space of functions whose members are single-valued and constant along edges but multi-valued at vertices. Bottom left: Degrees of freedom at each vertex are equated, leading to the piecewise affine Lagrange finite element space. Bottom right: Degrees of freedom at each vertex are equated only within individual triangles, leading to the space of discontinuous piecewise affine functions.
• Figure 5: The blow-up $\widetilde{T}$ of a triangle $T$ has 6 faces of dimension 1 that one can loosely think of as the 3 edges of $T$ and 3 infinitesimal arcs at the vertices of $T$. Here, the arcs are depicted as straight lines parallel to the opposite side. This reflects the fact that we can parametrize a point on the arc by specifying a ray from the vertex through the point and seeing where it meets the opposite side. The faces of $\widetilde{T}$ are labelled using shorthand notation for flags as described in the text.
• Figure 6: If $R$ is a triangle and ${\boldsymbol\rho}$ is a point in $R$, then $R_{\boldsymbol\rho}$ is the shaded region depicted above.

Theorems & Definitions (82)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Quasi-cylindrical coordinates
• Proposition 2.4
• proof
• Example 2.5
• Example 2.6
• Example 2.7
• Example 2.8
• Definition 2.11