Finite Elements for div- and divdiv-Conforming Symmetric Tensors in Arbitrary Dimension

Complexes from Complexes: Finite Element Complexes in Three Dimensions

“…Finite element Hessian complexes, elasticity complexes, and divdiv complexes have been constructed recently in [11,17,15,18,21,22,23,24] but in a case by case manner. A natural question arises: can we apply the BGG framework to unify these constructions?…”

“…As b e ≥ 0, the test function space in (16b) can be changed to q ∈ P k−2(r v +1)+j (e). For ease of notation, we use DoF k (r) to denote the set of DoFs (16) and DoF k (s; r) for the part on the sub-simplex s. The unisolvence can be written as…”

“…Compared with de Rham complex (1), those derived complexes involve Sobolev spaces for tensor functions which are much harder to discretize. Finite element Hessian complexes, elasticity complexes, and divdiv complexes have been constructed recently case by case in [11,17,15,18,21,22,23,24]. Our goal is to extend the BGG construction to finite element complexes and thus unify these scattered results and produce more in a systematical way.…”

“…Examples include the finite element divdiv complex in [17]: r 0 = (1, 0, 0) , r 1 = r 0 − 1, r 2 = (0, −1, −1) and r 3 = r 2 2, and the finite element divdiv complex in [22]: r 0 = (2, 0, 0) , r 1 = r 0 − 1, r 2 = r 1 1, and r 3 = r 2 2.…”

“…(2) the t − n decomposition approach for constructing div-conforming elements developed in [12]; (3) the trace complexes found in [11,17,15] and 2D finite element complexes developed in [13]. We have mentioned smooth finite element de Rham complexes before and give a brief summary of the t − n decomposition next.…”