PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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BDDC is the constrained partial-energy minimizer

Statement

For every assembled residual $f \in {\hat{W}}^{'}$ , the unique minimizer over $w \in \tilde{W}$ of

$J_{f} (w) = \frac{1}{2} ⟨ \tilde{S} w, w ⟩ - ⟨ f, E_{0} w ⟩$

is $w_{*} = {\tilde{S}}^{- 1} E_{0}^{T} f$ . The conforming correction returned by BDDC is $E_{0} w_{*} = E_{0} {\tilde{S}}^{- 1} E_{0}^{T} f$ .

Facts & Assumptions

Given: The positive definite partial Schur operator $\tilde{S}$ , the averaging map $E_{0}$ , and an assembled residual $f$ .

[F1]

Coefficient-scaled averaging and BDDC: $E_{0} : \tilde{W} \to \hat{W}$ is linear and $M_{B D D C}^{- 1} = E_{0} {\tilde{S}}^{- 1} E_{0}^{T}$ .

[F2]

Primal constraints remove local floating kernels: under the stated primal anchoring, $\tilde{S}$ is positive definite.

Proof

1.1givenF1algebra

Differentiating the finite-dimensional quadratic in a direction $z \in \tilde{W}$ gives $J_{f}^{'} (w) [z] = ⟨ \tilde{S} w, z ⟩ - ⟨ f, E_{0} z ⟩$ . Thus its critical point solves $\tilde{S} w = E_{0}^{T} f$ , whose unique solution is $w_{*} = {\tilde{S}}^{- 1} E_{0}^{T} f$ .

2.1givenF1F2step 1.1algebra∎

For any $z \in \tilde{W}$ , expansion about the critical point yields $J_{f} (w_{*} + z) = J_{f} (w_{*}) + \frac{1}{2} ⟨ \tilde{S} z, z ⟩$ . By [F2] the added term is positive unless $z = 0$ , so $w_{*}$ is the unique global minimizer. Applying $E_{0}$ gives the displayed BDDC correction by [F1].

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Coefficient-scaled averaging and BDDC.

Dependency tree · two levels

5 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

Jan Mandel and Bedřich Sousedík, BDDC and FETI-DP under Minimalist Assumptions, §3.2 (standard reference, not scraped)
Olof B. Widlund, Introduction to the Design and Theory of Domain Decomposition Algorithms, DD20 Tutorial, §§87–88 (standard reference, not scraped)