Distributional Hessian and Divdiv Complexes on Triangulation and Cohomology

We construct discrete versions of some Bernstein-Gelfand-Gelfand (BGG) complexes, i.e., the Hessian and the divdiv complexes, on triangulations in two dimensions and three dimensions. The sequences consist of finite elements with local polynomial shape functions and various types of Dirac measures on subsimplices (generalizations of currents). The construction generalizes Whitney forms (canonical conforming finite elements) for the de Rham complex and Regge calculus/finite elements for the elasticity (Riemannian deformation) complex from discrete topological and Discrete Exterior Calculus perspectives. We show that the cohomology of the resulting complexes is isomorphic to the continuous versions, and thus isomorphic to the de Rham cohomology with coefficients.