We construct finite element de~Rham complexes of higher and possibly non-uniform polynomial order in finite element exterior calculus (FEEC). Starting from the finite element differential complex of lowest-order, known as the complex of Whitney forms, we incrementally construct the higher-order complexes by adjoining exact local complexes associated to simplices. We define a commuting canonical interpolant. On the one hand, this research provides a base for studying $hp$-adaptive methods in finite element exterior calculus. On the other hand, our construction of higher-order spaces enables a new tool in numerical analysis which we call "partially localized flux reconstruction". One major application of this concept is in the area of equilibrated a~posteriori error estimators: we generalize the Braess-Sch\"oberl error estimator to edge elements of higher and possibly non-uniform order.

翻译：我们构造了有限元外微积分（FEEC）中高阶且可能非均匀多项式阶的有限元de Rham复形。从最低阶有限元微分复形（即Whitney形式复形）出发，通过附加与单纯形相关的精确局部复形，我们逐步构建高阶复形。我们定义了一个满足交换性的典范插值算子。一方面，本研究为有限元外微积分中的$hp$自适应方法提供了理论基础。另一方面，我们的高阶空间构造为数值分析提供了一种新工具，称为"部分局部化通量重建"。该概念的一个主要应用领域是平衡后验误差估计器：我们将Braess-Schöberl误差估计器推广到高阶且可能非均匀阶的棱边元。