arXiv:2611.05698v1 Announce Type: new Abstract: We present discrete Poincar\'e and Bogovskii operators for the common finite element discretizations of the de Rham complex of arbitrary polynomial order. We first discuss how such operators arise from a decomposition of the finite element spaces into a lowest order component and a remainder. This results in a fast algorithm for computing grad, curl, and div potentials. We then introduce a nonlocal Poincar\'e operator on contractible domains, which generalizes existing constructions based on cone operators to arbitrary order by integrating over cones against extended test forms. This requires developing commuting extension operators, which turn out to be related to rational forms. We also show how the nonlocal construction is modified to obtain a discrete Bogovskii operator preserving homogeneous boundary conditions. We further discuss how the ideas of the construction fit into the theory of finite element systems.