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 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.