Uniform discrete Poincar\'e inequalities for Hybrid High-Order differential forms on polyhedral meshes

Abstract

We introduce a Hybrid High-Order framework for differential forms on general polyhedral meshes. For each form degree $k\in\{0,\ldots,n-1\}$, the discrete space combines polynomial differential forms on mesh cells with polynomial trace unknowns on mesh faces, and the exterior derivative is reconstructed through a local Stokes formula. In three dimensions, the corresponding vector proxies recover the usual hybrid gradient, curl, and divergence reconstructions, while the differential-form setting provides a unified exterior-calculus generalisation in arbitrary space dimension. Within this framework, we prove uniform discrete Poincar\'e inequalities at every form degree. The canonical broken--stabilised seminorm controls the distance, in the hybrid $L^2$ norm, to the full kernel of the prescribed reconstructed exterior derivative, uniformly with respect to the mesh size and to every stability-admissible choice of the face spaces. The result holds on domains with arbitrary topology.