  In this work, following the Discrete de Rham (DDR) paradigm, we develop an
arbitrary-order discrete divdiv complex on general polyhedral meshes. The
construction rests 1) on discrete spaces that are spanned by vectors of
polynomials whose components are attached to mesh entities and 2) on discrete
operators obtained mimicking integration by parts formulas. We provide an
in-depth study of the algebraic properties of the local complex, showing that
it is exact on mesh elements with trivial topology. The new DDR complex is used
to design a numerical scheme for the approximation of biharmonic problems, for
which we provide detailed stability and convergence analyses.
