The paper addresses the challenge of constructing conforming finite element spaces for high-order differential operators in high dimensions, with a focus on the curl div operator in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element 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：本文针对高维高阶微分算子协调有限元空间的构造难题展开研究，重点聚焦于三维空间中的旋度散度算子。为构建分布型有限元旋度散度复形，引入了切向-法向连续性。所构造的空间被用于离散四阶旋度问题，并展现出最优收敛阶。此外，本文提出一种杂交技术，论证了其与非协调有限元及弱Galerkin方法的等价性。