RemarkRemark: Literature-sourcedProof: AI-adaptedprecheck pass

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.

Exact DPG stability does not certify quadrature or algebraic solve errors

Statement

The Fortin, boundedness, and Riesz arguments use exact integrals and exact finite solves; they imply no quadrature or floating-point perturbation estimate. For full-column-rank real $A \in R^{m \times n}$ with $m \geq n \geq 1$ , $κ_{2} (A^{T} A) = κ_{2} (A)^{2}$ .

Facts & Assumptions

Given: The exact finite Riesz algorithm and a full-column-rank real $m \times n$ matrix with $m \geq n \geq 1$ .

[F1]

The practical-test algorithm specifies exact integrals and solves. (Exact local finite Riesz solves for practical DPG tests)

[F2]

For a full-column-rank real $m \times n$ matrix with $m \geq n \geq 1$ , the normal-equation condition number is squared. (For full-column-rank A, the normal equations square the spectral condition number)

Proof

1.1F1algebra

Perturbations and rounded arithmetic are absent from [F1], so it supplies no error estimate for them.

2.1F2givenalgebra∎

Under the stated size and rank hypotheses, [F2] gives the displayed identity; this is a conditioning statement, not a solver-specific error bound.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Jay Gopalakrishnan and Weifeng Qiu, An analysis of the practical DPG method (standard reference, not scraped)
Leszek Demkowicz and Jay Gopalakrishnan, Analysis of the DPG method for the Poisson equation (standard reference, not scraped)
Leszek Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)
C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, A Posteriori Error Control for DPG Methods (standard reference, not scraped)