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

Abstract

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 ${\mathcal{P}}_r{\Lambda }^k(T)$ 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 $${\mathcal{P}}_r{\Lambda }^0(T)\to {\mathcal{P}}_r{\Lambda }^1(T)\to {\mathcal{P}}_r{\Lambda }^2(T),$$ being $T$ a $2$-simplex of ${\mathbb{R}}^2$. This work pairs its companions recently appeared for Nédélec first family of finite elements.

13 pages, 2 figures, 2 tables