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Geometric decompositions and local bases for spaces of finite element differential forms

机译：有限元微分形式空间的几何分解和局部基

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摘要

We study the two primary families of spaces of finite element differential forms with respect to a simpli-cial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.
机译：我们研究了在任意数量的空间尺寸中相对于简单网格的有限元差分形式的两个主要空间族。这些空间是矢量场经典有限元空间的概括，通常被称为Raviart-Thomas，Brezzi-Douglas-Marini和Nedelec空间。在本文中，我们导出了这些空间的几何分解，从而直接导致了它们的显式局部底基，从而将普通拉格朗日有限元的伯恩斯坦基础进行了推广。对于任意多项式度，任意阶的微分形式以及任意数量的空间维数中的任意单纯三角剖分，该方法适用于两个有限元空间族。一致的扩展算子族的概念在构造中发挥了重要作用，它在抽象框架中表达了用于导出导致局部基础的有限元空间的几何分解的充分条件。

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