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Discrete Poincare inequality for homogeneous Crouzeix–Raviart functions

Statement

Assume Countable Choice. Let $Ω$ be a bounded connected Lipschitz polyhedral domain and $T_{h}$ any finite matching shape-regular simplicial mesh. There is a constant $C$ depending only on $Ω$ , $d$ , and the shape-regularity bound such that every $v_{h} \in V_{h}^{C R, 0}$ satisfies $∥ v_{h} ∥_{L^{2} (Ω)} \leq C ∥ \nabla_{h} v_{h} ∥_{L^{2} (Ω)} .$

Facts & Assumptions

Given: Countable Choice, the mesh and homogeneous CR space of The global homogeneous Crouzeix–Raviart space, $v_{h} \in V_{h}^{C R, 0}$ , and a slab $a < x \cdot e < b$ containing $Ω$ .

[A1]

Countable Choice is required by the conforming-companion interface used below. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), A conforming averaging companion for Crouzeix–Raviart functions)

[F1]

The conforming companion satisfies $E_{h} v_{h} \in H_{0}^{1} (Ω)$ , $∥ v_{h} - E_{h} v_{h} ∥_{2} \leq C h ∥ \nabla_{h} v_{h} ∥_{2}$ , and $∥ \nabla E_{h} v_{h} ∥_{2} \leq C ∥ \nabla_{h} v_{h} ∥_{2}$ . (A conforming averaging companion for Crouzeix–Raviart functions)

[F2]

Every $H_{0}^{1} (Ω)$ function satisfies $∥ w ∥_{2} \leq (b - a) ∥ \nabla w ∥_{2}$ . (A Poincare bound on a slab-bounded domain)

Proof

1.1A1F1F2given

Apply [F2] to $E_{h} v_{h} \in H_{0}^{1} (Ω)$ and use [F1] to obtain $∥ E_{h} v_{h} ∥_{2} \leq (b - a) ∥ \nabla E_{h} v_{h} ∥_{2} \leq C (b - a) ∥ \nabla_{h} v_{h} ∥_{2}$ .

2.1A1F1step 1.1givenalgebra∎

By the triangle inequality and [F1], $∥ v_{h} ∥_{2} \leq ∥ v_{h} - E_{h} v_{h} ∥_{2} + ∥ E_{h} v_{h} ∥_{2} \leq C h ∥ \nabla_{h} v_{h} ∥_{2} + C (b - a) ∥ \nabla_{h} v_{h} ∥_{2} .$ The mesh cells lie in the bounded domain, so $h \leq diam (Ω)$ ; absorb this fixed bound into $C$ .

Depends on

Used by

Energy error estimate for the Crouzeix–Raviart Poisson method Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VIII, Chapter 36, Crouzeix–Raviart approximation (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 25, Non-conforming Finite Element Methods (standard reference, not scraped)