AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge 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.

Ideal Poisson DPG trial-to-test Riesz map

Definition

Input: A real trial vector $w \in U$ .

Output: The ideal test vector $T w \in V$ .

Admissible conditions: $U$ and $V$ are the real Hilbert spaces in Ultraweak Poisson DPG spaces and plan-consistent bilinear form, and $b$ is the bounded real bilinear form on them, with boundedness supplied by The ultraweak Poisson DPG bilinear form is bounded. The definition uses exact Hilbert-space operations and makes no claim of practical computability.

Instructions: Form the bounded linear functional $ℓ_{w} (v) = b (w, v)$ on $V$ . Apply the Hilbert-space Riesz representation procedure to obtain the unique $T w \in V$ satisfying

(T w, v)_{V} = ℓ_{w} (v) = b (w, v) (v \in V),

and return $T w$ . For a finite-dimensional trial subspace $U_{h}$ , the ideal test space is $T (U_{h})$ .

Depends on

Used by

Dependency tree · two levels

28 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)