TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck 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.

Finite-dimensional VMS residual closure has an explicit Schur defect bound

Statement

Use the block operators and residual closure of Finite-dimensional resolved–unresolved VMS split and residual closure. Equip $X_{c}, X_{f}$ and their duals with norms, and use the corresponding induced operator norms. Let $G : = A_{f f}^{- 1}$ , assume the exact Schur system has a solution $u_{c}$ , and put $r_{f} : = F_{f} - A_{f c} u_{c}$ . Suppose the closure map $T$ has $\tilde{S} : = A_{c c} - A_{c f} T A_{f c}$ invertible, and let ${\tilde{u}}_{c}$ solve the approximate Schur system. Then $∥ {\tilde{u}}_{c} - u_{c} ∥ \leq ∥ {\tilde{S}}^{- 1} ∥ ∥ A_{c f} ∥ ∥ T - G ∥ ∥ r_{f} ∥,$ and for the modeled fine component ${\tilde{u}}_{f} = T (F_{f} - A_{f c} {\tilde{u}}_{c})$ , $∥ {\tilde{u}}_{f} - u_{f} ∥ \leq ∥ T - G ∥ ∥ r_{f} ∥ + ∥ T ∥ ∥ A_{f c} ∥ ∥ {\tilde{u}}_{c} - u_{c} ∥ .$ The exact fine scale is $u_{f} = G r_{f}$ . These bounds concern finite-dimensional algebraic blocks; they do not certify a universal PDE subgrid model.

Facts & Assumptions

Given: The direct-sum block system and residual closure in Finite-dimensional resolved–unresolved VMS split and residual closure, an invertible exact fine block $A_{f f}$ , and an invertible approximate Schur operator $\tilde{S}$ .

[F1]

Finite-dimensional resolved–unresolved VMS split and residual closure: exact elimination uses $G = A_{f f}^{- 1}$ and residual $r_{f} = F_{f} - A_{f c} u_{c}$ ; the closure replaces $G$ by $T$ in both the Schur operator and load.

Proof

1.1F1givenalgebra

Write the exact Schur operator and load as $S = A_{c c} - A_{c f} G A_{f c}$ and $g = F_{c} - A_{c f} G F_{f}$ , so $S u_{c} = g$ . Put $Δ : = T - G$ . The approximate quantities satisfy $\tilde{S} = S - A_{c f} Δ A_{f c}$ and $\tilde{g} = g - A_{c f} Δ F_{f}$ .

2.1step 1.1givenalgebra

Since $\tilde{S} {\tilde{u}}_{c} = \tilde{g}$ , step 1.1 gives $\tilde{S} ({\tilde{u}}_{c} - u_{c}) = \tilde{g} - \tilde{S} u_{c} = - A_{c f} Δ (F_{f} - A_{f c} u_{c}) = - A_{c f} Δ r_{f}$ . Apply the induced operator norms and multiply by $∥ {\tilde{S}}^{- 1} ∥$ to prove the coarse bound.

3.1F1step 2.1algebra∎

The exact fine component is $u_{f} = G r_{f}$ . The modeled component is ${\tilde{u}}_{f} = T (r_{f} - A_{f c} ({\tilde{u}}_{c} - u_{c}))$ . Subtracting gives ${\tilde{u}}_{f} - u_{f} = (T - G) r_{f} - T A_{f c} ({\tilde{u}}_{c} - u_{c})$ . The triangle inequality and operator norms give the stated fine-scale bound.

Depends on

Finite-dimensional resolved–unresolved VMS split and residual closure

Used by

A two-scale VMS closure attains its Schur defect bound Example

Dependency tree · one level

1 result within one dependency step 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

Thomas J. R. Hughes, Guglielmo Scovazzi, and Leopoldo P. Franca, Multiscale and Stabilized Methods, Encyclopedia of Computational Mechanics, ICES Report 04-18 (2004) (standard reference, not scraped)
Volker John, Numerical Methods for Convection-Dominated Problems, lecture notes, Chapter 5 (standard reference, not scraped)