Discrete Poincar\'e inequalities: a review on proofs, equivalent formulations, and behavior of constants

TL;DR

This paper reviews discrete Poincaré inequalities in 3D, analyzing their proofs, equivalent formulations, and how the constants depend on mesh properties, polynomial degree, and shape regularity.

Contribution

It provides a comprehensive review of proofs, equivalent formulations, and the behavior of constants in discrete Poincaré inequalities for piecewise polynomial spaces in 3D.

Findings

Characterization of constant dependence on mesh and polynomial degree

Review of equivalent formulations including stability and inf-sup conditions

Analysis of constants on local tetrahedral mesh patches

Abstract

We investigate discrete Poincar\'e 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 Poincar\'e inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincar\'e maps), and existence of graph-stable commuting projections.