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 PrΛk(T) whose coefficients are polynomials of degree r. In this paper we confine ourselves in the two dimensional case ℝ2, as in this framework the Five Lemma offers a neat and elegant treatment avoiding computations on the middle space. The majority of definitions and constructions are meaningful for n > 2 as well and, when possible, they are thus given in such a generality, although more complicated techniques shall be invoked to replace the graceful role of the Five Lemma. In particular, we use techniques of homological algebra to obtain degrees of freedom for the whole diagram $$ {\mathcal{P}}_r{\mathrm{\Lambda }}^0(T)\to {\mathcal{P}}_{r-1}{\mathrm{\Lambda }}^1(T)\to {\mathcal{P}}_{r-2}{\mathrm{\Lambda }}^2(T), $$ being T a 2-simplex of ℝ2. This work pairs its companions recently appeared for Nédélec first family of finite elements.