Blow-up Whitney forms, shadow forms, and Poisson processes
<|reference_start|>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 $\tilde T$ 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.<|reference_end|>
@article{berchenko-kogan2024blow-up,
    title={Blow-up Whitney forms, shadow forms, and Poisson processes},
    author={Yakov Berchenko-Kogan and Evan S. Gawlik},
    journal={arXiv preprint arXiv:2402.03198},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03198},
    primaryClass={math.NA cs.NA math.DG}
}