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A unified construction of H(div)-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to (n-1)-dimensional faces. The developed finite element spaces are H(div)-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.
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This content will become publicly available on February 1, 2026
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.
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- PAR ID:
- 10574898
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Results in Applied Mathematics
- Volume:
- 25
- Issue:
- C
- ISSN:
- 2590-0374
- Page Range / eLocation ID:
- 100529
- Subject(s) / Keyword(s):
- Finite element exterior calculus, Shadow forms, Whitney forms, Blowup, Finite elements, Cohomology
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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