TheoremStatement: Literature-sourcedProof: 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.

Conforming finite-element eigenvalues have a quadratic best-approximation error

Statement

Assume Countable Choice and the compact coercive form setting of Discrete spectrum and min–max principle for a compact coercive form. Let $V_{h} \subseteq V$ be finite-dimensional, let $m \geq 1$ with $\dim V_{h} \geq m$ , and let $E_{m} = span {e_{1}, \dots, e_{m}}$ be the first $m$ continuous eigen-directions. Write $δ_{h} = δ_{a} (E_{m}, V_{h})$ as in Best-approximation gap of a finite-dimensional eigenspace. There are $δ_{0} > 0$ and $C_{m} < \infty$ , depending only on $m$ and the first $m$ continuous eigenvalues, such that if $δ_{h} \leq δ_{0}$ , then the $m$ th conforming Ritz value $λ_{h, m}$ satisfies $λ_{m} \leq λ_{h, m} \leq λ_{m} + C_{m} δ_{h}^{2} .$ The lower bound holds for every conforming $V_{h}$ of dimension at least $m$ ; the upper error bound requires the stated small-gap condition. More generally, let $a_{0}$ be a Hermitian lower-bounded form on the same $V$ , and suppose a specified shift $σ$ makes $a_{σ} (u, v) : = a_{0} (u, v) + σ (J u, J v)_{H}$ bounded, Hermitian and coercive, with the same injective compact dense map $J$ . Define $δ_{h}^{σ}$ using the energy norm $∥ v ∥_{a_{σ}} : = \sqrt{a_{σ} (v, v)}$ and the first $m$ eigenspace directions for $a_{σ}$ . The theorem applies to the shifted problem with $δ_{h}^{σ}$ ; its continuous and discrete eigenvalues are the corresponding unshifted eigenvalues plus $σ$ . Subtracting $σ$ preserves their ordering and the difference $λ_{h, m} - λ_{m}$ .

Facts & Assumptions

Given: Countable Choice, the compact coercive form pair $(V, H, a, J)$ , a finite-dimensional conforming subspace $V_{h}$ , an index $m$ with $\dim V_{h} \geq m$ , and $δ_{h}$ below the threshold in The conforming Ritz matrix differs quadratically in the eigenspace gap.

[F1]

The continuous eigenvalues and Rayleigh quotient satisfy the attained continuous min–max principle (Discrete spectrum and min–max principle for a compact coercive form).

[F2]

The finite-dimensional conforming Ritz eigenvalues satisfy their own min–max formula (Finite-dimensional conforming Ritz values satisfy min–max).

[F3]

The energy projection $P_{h}$ is best approximation and is injective on $E_{m}$ when $δ_{h} < 1$ (The conforming energy projection is the best approximation).

[F4]

On $U_{h} = P_{h} (E_{m})$ , every Rayleigh quotient is at most $λ_{m} + C_{m} δ_{h}^{2}$ under the stated small-gap bound (The conforming Ritz matrix differs quadratically in the eigenspace gap).

Proof

technique · direct

Given: The continuous and discrete form problems, $m$ , $E_{m}$ , $V_{h}$ , $P_{h}$ , and $δ_{h}$ .

1.1F1F2given

For every $m$ -dimensional subspace $W_{h} \subseteq V_{h}$ , conformity gives $W_{h} \subseteq V$ . The continuous min–max principle [F1] therefore gives $\max_{v \in W_{h} ∖ {0}} R (v) \geq λ_{m}$ . Taking the minimum over such $W_{h}$ and applying [F2] yields $λ_{h, m} \geq λ_{m}$ .

1.2givenalgebra

For the shift branch, the form eigenrelation for $a_{σ}$ is $a_{0} (e, v) + σ (J e, J v)_{H} = λ^{σ} (J e, J v)_{H},$ which is equivalent to the unshifted relation with eigenvalue $λ^{0} = λ^{σ} - σ$ . The same equivalence holds on $V_{h}$ , because the mass form is unchanged. Thus every continuous and discrete eigenvalue shifts by the same scalar, preserving the ordering and satisfying $(λ_{h, m}^{σ} - σ) - (λ_{m}^{σ} - σ) = λ_{h, m}^{σ} - λ_{m}^{σ}$ .

2.1F2F3F4step 1.1given

Since $δ_{h} \leq δ_{0} \leq 1 / 2 < 1$ , [F3] makes $U_{h} : = P_{h} (E_{m})$ an $m$ -dimensional subspace of $V_{h}$ . The local Ritz-matrix estimate [F4] gives $\max_{w_{h} \in U_{h} ∖ {0}} R (w_{h}) \leq λ_{m} + C_{m} δ_{h}^{2} .$ By the discrete min–max principle [F2], $λ_{h, m}$ is no greater than this maximum. This proves the upper bound.

3.1givenstep 1.1step 2.1step 1.2algebra∎

Apply steps 1.1–2.1 to the bounded, Hermitian, coercive shifted form $a_{σ}$ using its own best-approximation gap $δ_{h}^{σ}$ . It gives $λ_{m}^{σ} \leq λ_{h, m}^{σ} \leq λ_{m}^{σ} + C_{m}^{σ} (δ_{h}^{σ})^{2}$ under that gap's smallness condition. Step 1.2 subtracts $σ$ from both eigenvalues, leaving the same ordering and error.

Depends on

Used by

Dependency tree · two levels

32 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)