Distributional Finite Element curl\,div Complexes and Application to Quad Curl Problems

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.