有限元微分形式空间的几何分解与局部基
Geometric decompositions and local bases for spaces of finite element differential forms
AI总结：
针对任意维单纯形网格上的两类有限元微分形式空间，通过一致延拓算子族的概念推导其几何分解，从而得到显式的局部基，推广了拉格朗日有限元的Bernstein基。
摘要：
我们研究了在任意空间维数下，相对于单纯形网格的两类主要有限元微分形式空间。这些空间是经典向量场有限元空间（通常称为Raviart-Thomas、Brezzi-Douglas-Marini和Nedelec空间）的推广。在本文中，我们推导了这些空间的几何分解，从而直接得到显式的局部基，推广了普通拉格朗日有限元的Bernstein基。该方法适用于两类有限元空间，任意多项式次数、任意阶微分形式以及任意空间维数的任意单纯形三角剖分。在构造中，一致延拓算子族的概念扮演了重要角色，它在抽象框架中表达了导出有限元空间几何分解并得到局部基的充分条件。
英文摘要：
We study the two primary families of spaces of finite element differential forms with respect to a simplicial 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
