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arXiv · 2412.11796

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

Alexandre Ern·Johnny Guzmán·Pratyush Potu·Martin Vohralík
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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.

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Alexandre Ern, Johnny Guzmán, Pratyush Potu, Martin Vohralík. 2024-12-16. Discrete Poincar\'e inequalities: a review on proofs, equivalent formulations, and behavior of constants. https://doi.org/10.1093/imanum%2Fdraf089

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