  Two types of finite element spaces on a tetrahedron are constructed for
divdiv conforming symmetric tensors in three dimensions. The key tools of the
construction are the decomposition of polynomial tensor spaces and the
characterization of the trace operators. First, the divdiv Hilbert complex and
its corresponding polynomial complexes are presented. Several decompositions of
polynomial vector and tensors spaces are derived from the polynomial complexes.
Then, traces for div-div 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. Third, a finite element for
sym curl-conforming trace-free tensors is constructed following the same
approach. Finally, a finite element divdiv complex, as well as the bubble
functions complex, in three dimensions are established.
