Weights as degrees of freedom for high order Whitney finite elements

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.