Discrete Poincaré inequalities: a review on proofs, equivalent formulations, and behavior of constants

2024/12/16 by Ern, Alexandre, Guzmán, Johnny, Potu, Pratyush +1 · 3 citations
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2412.11796

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.

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