High order Whitney forms, also known as trimmed polynomial differential forms, are a celebrated family of differential forms. They find their roots in Hassler Whitney's book Geometric integration theory, published in 1957, where a low degree counterpart was used to prove the famous de Rham's Theorem. It was only in the '80s that they were recognised as a powerful tool in numerical analysis, when they were proved to parametrise Nédélec first family of finite elements. In Whitney's spirit, one may choose weights, namely integrals of k-forms on k-simplices, as degrees of freedom for those spaces. To do this, the concept of small simplices shall be introduced. A small simplex is a piace of a (virtual) partitioning of a simplex. We show that weights associated with appropriate small simplices ensure unisolvence for Nédélec's first family and we offer a strategy to shape the geometry of small simplices. This allows to generalise classical concepts peculiar to Lagrangian finite elements and Lagrangian interpolation to high dimensional frameworks. We first develop the corresponding theory and then offer numerical examples, relating results with the geometry of the problem.

Weights as degrees of freedom for high order Whitney finite elements / Bruni Bruno, Ludovico. - (2022 Oct 28).

### Weights as degrees of freedom for high order Whitney finite elements

#### Abstract

High order Whitney forms, also known as trimmed polynomial differential forms, are a celebrated family of differential forms. They find their roots in Hassler Whitney's book Geometric integration theory, published in 1957, where a low degree counterpart was used to prove the famous de Rham's Theorem. It was only in the '80s that they were recognised as a powerful tool in numerical analysis, when they were proved to parametrise Nédélec first family of finite elements. In Whitney's spirit, one may choose weights, namely integrals of k-forms on k-simplices, as degrees of freedom for those spaces. To do this, the concept of small simplices shall be introduced. A small simplex is a piace of a (virtual) partitioning of a simplex. We show that weights associated with appropriate small simplices ensure unisolvence for Nédélec's first family and we offer a strategy to shape the geometry of small simplices. This allows to generalise classical concepts peculiar to Lagrangian finite elements and Lagrangian interpolation to high dimensional frameworks. We first develop the corresponding theory and then offer numerical examples, relating results with the geometry of the problem.
##### Scheda breve Scheda completa Scheda completa (DC)
28-ott-2022
XXXIV
2020-2021
Matematica (29/10/12-)
Mathematics
Alonso Rodriguez, Ana Maria
Rapetti, Francesca
FRANCIA
Inglese
Settore MAT/08 - Analisi Numerica
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/365011`
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