ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

A broken piecewise-constant trial space can give an eigenvalue below the Dirichlet ground state

Example

Let $Ω = (0, 1)^{2}$ and divide it into two closed triangles by the diagonal from $(0, 0)$ to $(1, 1)$ . Let $w$ equal $1$ on one triangle and $0$ on the other. Take degree $k = 0$ and scalar values $q = 1$ , so $w$ is a nonzero member of the broken piecewise-constant space. For the broken stiffness form $a_{b r} (u, v) : = \sum_{K \in T} \int_{K} \nabla u \cdot \overline{\nabla v} d x$ and mass form $b (u, v) : = \int_{Ω} u \overline{v} d x$ , $w$ is a Ritz eigenvector with eigenvalue $0$ . The continuous Dirichlet ground-state value $λ_{1}$ is positive, so this nonconforming value lies strictly below it.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; the square $Ω$ ; its two-triangle matching mesh; the scalar, degree-zero broken space; and $w$ as defined above.

[F1]

The broken polynomial space consists of cellwise polynomials with no continuity constraint across interior facets; for degree zero it includes the cellwise constants (Matching affine simplicial meshes and shape regularity, Broken and conforming polynomial spaces on a matching mesh).

[F2]

The Dirichlet Laplacian on a nonempty bounded open set has a first eigenvalue $λ_{1} > 0$ under the Axiom of Choice and Countable Choice (The Poincare constant is the reciprocal square root of the first Dirichlet eigenvalue).

Proof

technique · direct

Given: The two cells $K_{1}, K_{2}$ , the broken function $w$ , and the forms $a_{b r}$ and $b$ .

1.1F1givenalgebra

Each triangle has area $1 / 2$ , so $b (w, w) = \int_{K_{1}} 1 d x = 1 / 2 > 0$ . The cellwise gradient of each constant restriction is zero. Hence $a_{b r} (w, v) = 0$ for every broken piecewise-constant $v$ , and in particular $a_{b r} (w, v) = 0 b (w, v)$ for all such $v$ . This is the generalized Ritz equation with eigenvalue zero.

2.1F1F2given∎

The square is nonempty, open and bounded, so [F2] gives $λ_{1} > 0$ . Therefore the broken Ritz eigenvalue $0$ is strictly less than $λ_{1}$ . The usual conforming min–max conclusion does not apply to this trial space: the two traces of $w$ across the diagonal differ, so $w$ is outside the conforming subspace in [F1]. Moreover, the broken stiffness form is only a seminorm on this piecewise-constant space.

Remarks

The example identifies why conformity and coercivity are hypotheses of the upper-bound theorem. It does not say that every nonconforming discretization underestimates the ground state.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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 Finite Element Methods, Fall 2025, Chapter 47 (standard reference, not scraped)
Bertrand Mercier, Topics in Finite Element Solution of Elliptic Problems, Chapter 8 (standard reference, not scraped)