论文标题
第二个多项式差异形式家庭的单回收和最小的物理自由度
Unisolvent and minimal physical degrees of freedom for the second family of polynomial differential forms
论文作者
论文摘要
这项工作的主要目的是为有限元素的第二个家庭提供一个统一和最小的物理自由度（称为重量）的家庭。这样的元素被认为是差异表单$ \ MATHCAL {p}_rλ^k（t）$，其系数是$ r $的多项式。我们将自己局限于两个维度的情况$ \ mathbb {r}^2 $，因为它易于可视化并提供整洁而优雅的治疗；但是，我们提出的技术可以扩展到$ n> 2 $，并进行一些调整的技术细节。特别是，我们使用同源代数的技术来获得整个图的自由度$ \ MATHCAL {p}_rλ^0（t）\ rightarrow \ Mathcal {p}_rλ^1（t） \ mathbb {r}^2 $。这项工作将其同伴最近出现在Nédélec的第一家族有限元素的第一家族中。
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.
