DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Cartesian complex-stretch PML for a homogeneous Helmholtz mode

Definition

Let $k > 0$ , $x \in R$ be the Cartesian normal coordinate, and $y \in R^{2}$ the tangential coordinate. Let $σ$ be nonnegative, continuous, piecewise $C^{1}$ , and zero for $x \leq 0$ . Define $S (x) = \int_{0}^{x} σ (t) d t, X (x) = x + i S (x), s (x) = X^{'} (x) = 1 + i σ (x) .$ The pulled-back constant-coefficient Helmholtz operator in the layer is $\frac{1}{s (x)} \partial_{x} (\frac{1}{s (x)} \partial_{x}) + Δ_{y} + k^{2} .$ This definition concerns the homogeneous Cartesian branch only; it makes no claim for heterogeneous media, curved stretches, grazing or evanescent uniform attenuation, discretizations, or finite-element convergence.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Steven G. Johnson, Notes on Perfectly Matched Layers (PMLs), §§3.1–3.2 (standard reference, not scraped)
Michler, Demkowicz, Kurtz, and Pardo, Improving the performance of Perfectly Matched Layers by means of hp-adaptivity, §§2.1–2.2 (standard reference, not scraped)