Finite elements for divdiv conforming symmetric tensors in three dimensions

Finite element spaces on a tetrahedron are constructed for div ? \operatorname {div}\operatorname {div} -conforming symmetric tensors in three dimensions. The key tools of the construction decomposition polynomial tensor and characterization trace operators. First, Hilbert complex its corresponding complexes presented. Several decompositions vector derived from complexes. Second, traces operator characterized through Green’s identity. Besides normal-normal component, another involving combination first order derivatives is continuous across face. Due to smoothness polynomials, also at vertices, plane orthogonal each edge. Besides, finite alttext="s y m c u r l"> s y m c u r l encoding="application/x-tex">symcurl -conforming trace-free following same approach. Putting all together, complex, as well bubble functions dimensions established.