The paper addresses the challenge of constructing conforming finite element spaces for high-order differential operators in high dimensions, with a focus on the $\textrm{curl\,div}$ operator in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element $\textrm{curl\,div}$ complexes. The spaces constructed are applied to discretize a quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods.

翻译：本文解决了高维高阶微分算子协同有限元空间构造的挑战，重点聚焦三维空间中的$\textrm{curl\,div}$算子。通过引入切向-法向连续性，建立了分布型有限元$\textrm{curl\,div}$复形。所构造的空间被应用于四旋度问题的离散化，展示了最优收敛阶。此外，提出了一种混合化技术，并证明了其与非协调有限元及弱伽辽金方法的等价性。