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的协调有限元空间构造问题展开研究，重点探讨三维情形下的旋度散度算子。通过引入切向-法向连续性，建立了分布有限元旋度散度复形。所构造的空间被应用于离散化四重旋度问题，并证明了最优收敛阶。此外，提出了一种杂交技术，论证了其与非协调有限元及弱伽辽金方法的等价性。