ExampleConstruction: AI-generatedVerification: 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.

A uniform two-level 1D Poisson correction bound

Example

Let $N = 2 M$ with $M \geq 2$ on the uniform fine P1 Dirichlet mesh of $(0, 1)$ . Take the coarse P1 space on the even nodes and one local space $span {ϕ_{2 k - 1}}$ for each odd node. The exact two-level additive Schwarz operator has $C_{0} \leq 2$ , $C_{1} \leq 2$ , and hence $κ_{A} (B A) \leq 4$ , independently of $M$ .

Facts & Assumptions

Given: The uniform mesh with step $h = 1 / N$ , the fine stiffness energy $a (u, u) = h^{- 1} \sum_{j = 1}^{N} (u_{j} - u_{j - 1})^{2}$ with $u_{0} = u_{N} = 0$ , the coarse space through the even nodes, and odd-node local spaces.

[F1]

The additive Schwarz condition theorem gives $κ_{A} (B A) \leq C_{0} C_{1}$ for the stable and synthesis constants of the chosen subspaces (Additive Schwarz condition bound from stable decomposition and synthesis).

[F2]

Finite-dimensional Schwarz subspace-correction data: a subspace prolongation $R_{i}$ has local energy matrix $A_{i} = R_{i}^{T} A R_{i}$ and exact additive preconditioner $B = \sum_{i} R_{i} A_{i}^{- 1} R_{i}^{T}$ .

Verification

1.1givenconstruct

Given a fine vector $u$ , let $u_{H}$ be its coarse nodal interpolant: $(u_{H})_{2 k} = u_{2 k}$ and $(u_{H})_{2 k - 1} = (u_{2 k - 2} + u_{2 k}) / 2$ . Define $η_{k} = u_{2 k - 1} - (u_{2 k - 2} + u_{2 k}) / 2$ and $u_{k} = η_{k} ϕ_{2 k - 1}$ . Nodal values give $u = u_{H} + \sum_{k = 1}^{M} u_{k}$ .

1.2givenalgebra

On each fine interval, at most one odd nodal hat has nonzero derivative. Thus only the coarse derivative and at most one local derivative can be nonzero there. The elementary inequality $(s + t)^{2} \leq 2 (s^{2} + t^{2})$ , which follows from $(s - t)^{2} \geq 0$ , and integration give ${∥ v_{0} + \sum_{k} v_{k} ∥}_{A}^{2} \leq 2 (∥ v_{0} ∥_{A}^{2} + \sum_{k} ∥ v_{k} ∥_{A}^{2}) .$ Hence $C_{1} \leq 2$ .

2.1givenstep 1.1algebra

On each pair of fine intervals set $a_{k} = u_{2 k - 1} - u_{2 k - 2}$ and $b_{k} = u_{2 k} - u_{2 k - 1}$ . The coarse energy on that coarse interval is $(a_{k} + b_{k})^{2} / (2 h) \leq (a_{k}^{2} + b_{k}^{2}) / h$ , exactly its fine energy contribution. Summing over $k$ gives $∥ u_{H} ∥_{A}^{2} \leq ∥ u ∥_{A}^{2}$ .

2.2step 1.1algebra

Since $η_{k} = (a_{k} - b_{k}) / 2$ and $(a_{k} - b_{k})^{2} \leq 2 (a_{k}^{2} + b_{k}^{2})$ , the local energy is $∥ u_{k} ∥_{A}^{2} = 2 η_{k}^{2} / h \leq (a_{k}^{2} + b_{k}^{2}) / h$ . Summing gives $\sum_{k} ∥ u_{k} ∥_{A}^{2} \leq ∥ u ∥_{A}^{2}$ . The decomposition in step 1.1 therefore has total energy at most $2 ∥ u ∥_{A}^{2}$ , so $C_{0} \leq 2$ .

3.1F1F2step 2.2step 1.2algebra∎

Apply [F1] to the exact Schwarz data [F2] with $C_{0} \leq 2$ and $C_{1} \leq 2$ to obtain $κ_{A} (B A) \leq 4$ . This finite one-dimensional split has a global coarse space; higher-dimensional overlap estimates are handled separately on the A page.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Jinchao Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34(4) (1992) (standard reference, not scraped)
Peter Bastian, Lecture Notes on Parallel Solution of Large Sparse Linear Systems, University of Heidelberg (2015) (standard reference, not scraped)