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

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Exact reflected-mode error from a finite Dirichlet-terminated PML

Statement

For $q > 0$ , $A \neq 0$ , and a finite layer $0 \leq x \leq L$ with $S (L) > 0$ , let a one-dimensional mode be $u (x) = A e^{i q X (x)} + B e^{- i q X (x)}$ , with the PML stretch from Cartesian complex-stretch PML for a homogeneous Helmholtz mode. Imposing $u (L) = 0$ gives the reflected amplitude at the interface $\frac{B}{A} = - e^{2 i q X (L)}, ∣ \frac{B}{A} ∣ = e^{- 2 q S (L)} .$ This is the exact continuous one-mode error for a Dirichlet termination.

Facts & Assumptions

Given: A propagating mode $q > 0$ and finite stretch data [F1].

[F1]

Cartesian complex-stretch PML for a homogeneous Helmholtz mode: $X (L) = L + i S (L)$ ; the positive and negative normal modes in the stretched coordinate are $e^{i q X}$ and $e^{- i q X}$ .

[F2]

Perfect matching and attenuation for non-grazing propagating modes: the outgoing mode is matched at the interface and attenuated by $e^{- q S}$ in one pass.

Proof

Given: $q > 0$ , $S (L) > 0$ , and [F1]–[F2].

1.1givenF1algebra

The condition $u (L) = 0$ reads $A e^{i q X (L)} + B e^{- i q X (L)} = 0$ . Since exponentials never vanish, if $A \neq 0$ division gives $B / A = - e^{2 i q X (L)}$ .

2.1F1F2step 1.1algebra∎

With $X (L) = L + i S (L)$ , $∣ e^{2 i q X (L)} ∣ = ∣ e^{2 i q L - 2 q S (L)} ∣ = e^{- 2 q S (L)}$ . The result measures the reflected-to-incident coefficient at $x = 0$ , where $X (0) = 0$ ; it is the two-way attenuation through the continuous layer.

Depends on

Used by

A weakly damped truncated layer can produce a resonant cavity response Example

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Sources

Steven G. Johnson, Notes on Perfectly Matched Layers (PMLs), §3.3 (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)