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

The principal aim of this work is to provide a family of unisolvent and minimal physical degrees of freedom, called weights, for Nédélec second family of finite elements. Such elements are thought of as differential forms $ \mathcal{P}_r Λ^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 $$ \mathcal{P}_r Λ^0 (T) \rightarrow \mathcal{P}_r Λ^1 (T) \rightarrow \mathcal{P}_r Λ^2 (T), $$ being $ T $ a $2$-simplex of $ \mathbb{R}^2 $. This work pairs its companions recently appeared for Nédélec first family of finite elements.