DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The face-average Crouzeix–Raviart interpolant

Definition

Assume Countable Choice. For $u \in H^{1} (K)$ , use the facet traces of Fractional face traces and cellwise Green identity on a simplex and define its local face-average interpolant by $I_{K}^{C R} u : = \sum_{F \subset \partial K} (\frac{1}{∣ F ∣} \int_{F} u d S) ϕ_{K, F},$ where $ϕ_{K, F}$ is the explicit polynomial in The local P1 Crouzeix–Raviart element. On a finite matching mesh, define $I_{h}^{C R} u$ cellwise by $(I_{h}^{C R} u) ∣_{K} = I_{K}^{C R} (u ∣_{K})$ . For a global $u \in H_{0}^{1} (Ω)$ , the cell traces agree on interior facets and vanish on boundary facets, so this formula has the intended global Crouzeix–Raviart degrees of freedom. The local estimate item proves its fractional approximation bounds.

Depends on

Used by

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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)