Unisolvent and minimal physical degrees of freedom for the second family of polynomial differential forms

TL;DR

This paper develops a minimal and unisolvent set of degrees of freedom for the Nédélec second family of finite elements in 2D, using homological algebra techniques, with potential extensions to higher dimensions.

Contribution

It introduces a new family of minimal degrees of freedom for Nédélec second family elements, extending techniques from the first family and employing homological algebra.

Findings

Provides explicit degrees of freedom for 2D Nédélec second family

Uses homological algebra to derive degrees of freedom

Techniques can be extended to higher dimensions

Abstract

The principal aim of this work is to provide a family of unisolvent and minimal physical degrees of freedom, called weights, for N\'ed\'elec second family of finite elements. Such elements are thought of as differential forms $ \mathcal{P}_r \Lambda^k (T)$ whose coefficients are polynomials of degree $ r $. We confine ourselves in the two dimensional case $ \mathbb{R}^2 $ since it is easy to visualise and offers a neat and elegant treatment; however, we present techniques that can be extended to $ n > 2 $ with some adjustments of technical details. In particular, we use techniques of homological algebra to obtain degrees of freedom for the whole diagram $$ \mathcal{P}_r \Lambda^0 (T) \rightarrow \mathcal{P}_r \Lambda^1 (T) \rightarrow \mathcal{P}_r \Lambda^2 (T), $$ being $ T $ a $2$-simplex of $ \mathbb{R}^2 $. This work pairs its companions recently appeared for N\'ed\'elec first family of finite elements.