TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

Exact RT0/P0 element mass balance and normal continuity

Statement

Assume Countable Choice, the mesh conventions of Affine simplicial meshes and RT/BDM spaces, and let $(u_{h}, p_{h}) \in V_{h} \times Q_{h}$ solve the discrete Darcy equations with source $f \in L_{0}^{2} (D)$ . Then on every cell $K$ , $div u_{h} ∣_{K} = ∣ K ∣^{- 1} \int_{K} f, \int_{K} div u_{h} = \int_{K} f .$ The normal flux is single-valued on every interior facet and zero on every boundary facet.

Facts & Assumptions

Given: A source $f \in Q$ , a conforming mesh, and a discrete solution $(u_{h}, p_{h})$ .

[A1]

Countable Choice is inherited through the weak normal-trace and divergence-mean suppliers. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The discrete second equation is $- \int_{D} q_{h} div u_{h} = - \int_{D} f q_{h}$ for every $q_{h} \in Q_{h}$ . (No-flow mixed Darcy problem and coefficient norms)

[F2]

$div u_{h}$ is cellwise constant and has zero mean. (Affine simplicial meshes and RT/BDM spaces, Zero mean of the divergence under zero normal trace)

[F3]

Cell averaging maps $f$ to a mean-zero piecewise constant and is the $L^{2}$ projection onto cell constants. (Cell averaging is the $L^{2}$ projection onto P0)

[F4]

Global RT0 conformity means matching interior normal flux and zero boundary normal flux. (RT0 facet-flux unisolvence and global H(div) conformity)

Proof

1.1A1F1F2F3constructalgebra

Let $P_{0} f$ be the cellwise average of $f$ . By [F2] and [F3], both $div u_{h}$ and $P_{0} f$ are in $Q_{h}$ . Put $g_{h} = div u_{h} - P_{0} f \in Q_{h}$ . For every $q_{h} \in Q_{h}$ , [F1] and the orthogonality property in [F3] give $\int_{D} g_{h} q_{h} = 0$ . Taking $q_{h} = g_{h}$ yields $∥ g_{h} ∥_{2}^{2} = 0$ .

2.1F2F4step 1.1algebra∎

On each cell the equality $div u_{h} = P_{0} f$ says the constant divergence equals the cell average of $f$ ; multiplying by $∣ K ∣$ gives the exact element balance. The field $u_{h}$ is globally RT0 and in H0(div), so [F4] supplies matching interior normal values and zero boundary normal values. The H(div) conformity result equates the flux with the weak normal trace, hence the asserted normal conditions hold.

Depends on

Used by

Two-triangle hybrid RT0 local elimination Example
Hybridization, static condensation, and pressure gauge Theorem

Dependency tree · two levels

30 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 III Chapter 14: H(div) finite elements (standard reference, not scraped)
Francisco-Javier Sayas, A gentle introduction to the Finite Element Method (2015) (standard reference, not scraped)