Discrete Poincaré inequalities: a review on proofs, equivalent formulations, and behavior of constants
Alexandre Ern, Johnny Guzmán, Pratyush Potu, Martin Vohralík
TL;DR
The article analyzes discrete Poincaré inequalities on $H({\mathop{\bf curl}})$ and $H({\mathrm{div}})$-conforming finite element spaces in 3D, focusing on how the discrete constants depend on the continuous constants, mesh shape-regularity, patch size, and polynomial degree $p$. It presents three equivalent formulations—stability of constrained minimization, discrete inf-sup conditions, and bounds for vector-potential operators—connecting the inequalities to operator norms and commuting projections. The main contribution is Theorem on discrete Poincaré inequalities, plus three proof routes: via equivalence to continuous minimizers, via stable commuting projections, and via direct piecewise Piola transformations, yielding insights into $p$-robustness and independence from the number of elements under suitable geometric assumptions. The results are framed within the finite element exterior calculus context and extend to arbitrary space dimensions, highlighting implications for vector Laplacians, eigenvalue bounds, and robust projections in local star patches.
Abstract
We investigate discrete Poincaré inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincaré inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincaré maps), and existence of graph-stable commuting projections.