  In this paper, we construct discrete versions of some
Bernstein-Gelfand-Gelfand (BGG) complexes, i.e., the Hessian and the divdiv
complexes, on triangulations in 2D and 3D. The sequences consist of finite
elements with local polynomial shape functions and various types of Dirac
measure on subsimplices. 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.
