Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. Besides the normal-normal component, another trace involving combination of first order derivatives of stress should be continuous across the face. Due to the rigid of polynomials, the symmetric stress tensor element is  continuous at vertices, and on the plane orthogonal to each edge. Hilbert complex and polynomial complexes are presented and several decomposition of polynomial vector and tensors spaces are revealed from the complexes . The constructed divdiv conforming elements are exploited to discretize the mixed formulation of the biharmonic equation. Optimal order and superconvergence error analysis is provided. Hybridization is given for the ease of implementation .
Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor should be 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. Hilbert complex and polynomial complexes are presented and several decomposition of polynomial vector and tensors spaces are revealed from the complexes .
R
stress
the tensor
R
rigid
smoothness
D
stress
A
also
D
. The constructed divdiv conforming elements are exploited to discretize the mixed formulation of the biharmonic equation. Optimal order and superconvergence error analysis is provided. Hybridization is given for the ease of implementation