Uniform discrete Poincaré 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 and characterise stability-admissible face spaces in terms of the traces of the polynomial kernel of the exterior derivative. The analysis is based on a cellular-to-hybrid transfer mechanism in which the global part of the stability problem is reduced to a single cellular cochain Poincaré problem, and the resulting control is propagated to the hybrid level using stable polynomial skeletons and local polynomial completions. This mechanism yields a uniform stable conforming lifting and a uniform discrete Poincaré inequality for every form degree in arbitrary space dimension. The lifting preserves the prescribed reconstructed exterior derivative after projection, whereas the Poincaré inequality controls the broken cell polynomial form modulo the continuous conforming kernel of the exterior derivative. Both results hold on domains with arbitrary topology and uniformly over stability-admissible choices of the face spaces. In three dimensions, the conforming-kernel estimate recovers the hybrid Poincaré--Wirtinger inequality for the gradient, the second hybrid Weber estimate under the standard gauge, and the corresponding divergence-kernel estimate.