Arnold, Falk, & Winther, in "Finite element exterior calculus, homological techniques, and applications" (2006), show how to geometrically decompose the full and trimmed polynomial spaces on simplicial elements into direct sums of trace-free subspaces and in "Geometric decompositions and local bases for finite element differential forms" (2009) the same authors give direct constructions of extension operators for the same spaces. The two families -- full and trimmed -- are treated separately, using differently defined isomorphisms between each and the other's trace-free subspaces and mutually incompatible extension operators. This work describes a single operator $\mathring{\star}_T$ that unifies the two isomorphisms and also defines a weighted-$L^2$ norm appropriate for defining well-conditioned basis functions and dual-basis functionals for geometric decomposition. This work also describes a single extension operator $\dot{E}_{\sigma,T}$ that implements geometric decompositions of all differential forms as well as for the full and trimmed polynomial spaces separately.
翻译:Arnold, Falk, & Winther, in“Finite 元素外部微积分、同理技术和应用”(2006年),展示了如何从几何上分解成无痕子空间的直接总和,以及“有限元素差异表的地球分解和地方基数”(2009年),同样的作者为同一空间直接建造了扩展操作员。两个家庭 -- -- 完整和细化 -- -- 分别处理,使用不同定义的无痕量子空间和相互不兼容的扩展操作员之间的异形。这项工作描述了一个单一操作员$\mathring\star_T$,该操作员统一了两个无色子空间,并定义了用于界定良好基础功能和大地分解功能的加权-2美元规范。这项工作还描述了一个单一的扩展操作员 $\dot{E ⁇ sigma,T$,用于所有差异表的几何分解位置,以及用于完整和三面聚体空间。