Duality in finite element exterior calculus and Hodge duality on the sphere
| العنوان: | Duality in finite element exterior calculus and Hodge duality on the sphere |
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| المؤلفون: | Yakov Berchenko-Kogan |
| سنة النشر: | 2019 |
| مصطلحات موضوعية: | Pointwise, Mathematics - Differential Geometry, Degree (graph theory), Differential form, Applied Mathematics, Dimension (graph theory), Scalar (mathematics), Duality (optimization), Numerical Analysis (math.NA), Combinatorics, Computational Mathematics, Finite element exterior calculus, Computational Theory and Mathematics, Differential Geometry (math.DG), FOS: Mathematics, 65N30, 58A10, Mathematics::Metric Geometry, Degree of a polynomial, Mathematics - Numerical Analysis, Analysis, Mathematics |
| الوصف: | Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite element exterior calculus began with the $\mathcal P_r\Lambda^k$ and $\mathcal P_r^-\Lambda^k$ families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $\mathcal P_r\Lambda^k$ and $\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal P_r^-\Lambda^k$ and $\mathring{\mathcal P}_{r+k}\Lambda^{n-k}$. In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard $n$-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $\mathcal P_r\Lambda^k$, $\mathcal P_r^-\Lambda^k$, $\mathring{\mathcal P}_r\Lambda^k$ and $\mathring{\mathcal P}_r^-\Lambda^k$ spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $\mathcal P_r\Lambda^k$ and $\mathring{\mathcal P}_{r+k+1}^-\Lambda^{n-k}$ and between $\mathcal P_r^-\Lambda^k$ and $\mathring{\mathcal P}_{r+k}\Lambda^{n-k}$, which we illustrate with examples. Comment: This revision has been restructured for clarity and includes an expanded introduction. Additionally, there are some new examples and clarifying comments, more care with the edge cases, and a DOI reference to the published version. 27 pages |
| اللغة: | English |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::e29865d4f044ad8ee29bc419977ffac7 http://arxiv.org/abs/1906.06354 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....e29865d4f044ad8ee29bc419977ffac7 |
| قاعدة البيانات: | OpenAIRE |
| الوصف غير متاح. |