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This paper addresses the challenge of constructing finite element curl div complexes 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 the 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. 

In the field of solving partial differential equations (PDEs), Hilbert complexes have become highly significant. Recent advances focus on creating new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as shown by Arnold and Hu [Found. Comput. Math. 21 (2021), pp. 1739–1774]. This paper extends their approach to three-dimensional finite element complexes. The finite element Hessian, elasticity, and divdiv complexes are systematically derived by applying techniques such as smooth finite element de Rham complexes, the $t$- $n$decomposition, and trace complexes, along with related two-dimensional finite element analogs. The construction includes two reduction operations and one augmentation operation to address continuity differences in the BGG diagram, ultimately resulting in a comprehensive and effective framework for constructing finite element complexes, which have various applications in PDE solving. 

A unified construction of H(div)-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to (n-1)-dimensional faces. The developed finite element spaces are H(div)-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established. 

A new H(divdiv)-conforming finite element is presented, which avoids the need for supersmoothness by redistributing the degrees of freedom to edges and faces. This leads to a hybridizable mixed method with superconvergence for the biharmonic equation. Moreover, new finite element divdiv complexes are established. Finally, new weak Galerkin and C0 discontinuous Galerkin methods for the biharmonic equation are derived. 

Finite element de Rham complexes and finite element Stokes complexes with varying degrees of smoothness in three dimensions are systematically constructed in this paper. Smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. H(div)-conforming finite elements and H(curl)-conforming finite elements with varying degrees of smoothness are devised based on these smooth scalar finite elements. The finite element de Rham complexes with corresponding smoothness and commutative diagrams are induced by these elements. The div stability of the H(div)-conforming finite elements is established, and the exactness of these finite element complexes is proven. 

A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The H 1 H^1 conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an H ( inc ) H(\operatorname {inc}) -conforming finite element of minimum polynomial degree 6 6 for symmetric tensors is the focus of this paper. Our construction appears to be the first H ( inc ) H(\operatorname {inc}) -conforming finite elements on tetrahedral meshes without further splitting. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the inc \operatorname {inc} operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition. The trace of the inc \operatorname {inc} operator is induced from a Green’s identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Two-dimensional smooth finite element Hessian complex and div ⁡ div \operatorname {div}\operatorname {div} complex are constructed. 

Finite element spaces on a tetrahedron are constructed for div div -conforming symmetric tensors in three dimensions. The key tools of the con- struction are the decomposition of polynomial tensor spaces and the charac- terization of the trace operators. First, the div div Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial com- plexes. Second, traces for the divdiv operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element div div complex, as well as the bubble functions complex, in three dimensions is established.