Blow-up Whitney forms, shadow forms, and Poisson processes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913625841598464 |
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| author | Berchenko-Kogan, Yakov Gawlik, Evan S. |
| author_facet | Berchenko-Kogan, Yakov Gawlik, Evan S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blow-up Whitney forms, shadow forms, and Poisson processes Berchenko-Kogan, Yakov Gawlik, Evan S. Numerical Analysis Differential Geometry 65N30, 58A10 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. |
| title | Blow-up Whitney forms, shadow forms, and Poisson processes |
| topic | Numerical Analysis Differential Geometry 65N30, 58A10 |
| url | https://arxiv.org/abs/2402.03198 |