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

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Perfect matching and attenuation for non-grazing propagating modes

Statement

Let $k > 0$ , $η \in R^{2}$ satisfy $∣ η ∣ < k$ , and let $q = \sqrt{k^{2} - ∣ η ∣^{2}} > 0$ be the outgoing normal branch. In the stretch of Cartesian complex-stretch PML for a homogeneous Helmholtz mode, $u (x, y) = e^{i q X (x) + i η \cdot y}$ solves the pulled-back Helmholtz equation. At $x = 0$ , its value and normal flux $(1 / s) \partial_{x} u$ agree with those of the physical mode $e^{i q x + i η \cdot y}$ . Its modulus is $e^{- q S (x)}$ . If a family of modes satisfies $q \geq q_{0} > 0$ , then $∣ u (x, y) ∣ \leq e^{- q_{0} S (x)}$ uniformly; whenever $S (x) \geq σ_{0} x$ for some $σ_{0} > 0$ , this is at most $e^{- q_{0} σ_{0} x}$ . No uniform bound is asserted as $q ↓ 0$ .

Facts & Assumptions

Given: The constant-coefficient Cartesian operator and stretch data in [F1].

[F1]

Cartesian complex-stretch PML for a homogeneous Helmholtz mode: $X = x + i S$ , $s = X^{'} = 1 + i σ$ , and the operator is $s^{- 1} \partial_{x} (s^{- 1} \partial_{x}) + Δ_{y} + k^{2}$ .

Proof

Given: $k > 0$ , $∣ η ∣ < k$ , $q = \sqrt{k^{2} - ∣ η ∣^{2}}$ , and [F1].

1.1givenF1algebra

For the stated mode, $\partial_{x} u = i q s (x) u$ , so $s^{- 1} \partial_{x} u = i q u$ and $s^{- 1} \partial_{x} (s^{- 1} \partial_{x} u) = s^{- 1} \partial_{x} (i q u) = - q^{2} u$ . Also $Δ_{y} u = - ∣ η ∣^{2} u$ . Since $q^{2} + ∣ η ∣^{2} = k^{2}$ , substitution into [F1] gives zero.

2.1F1step 1.1algebra∎

At $x = 0$ , $S (0) = 0$ and $σ (0) = 0$ by continuity and the definition, so $X (0) = 0$ , $s (0) = 1$ , and the stretched mode and its normal flux equal $e^{i η \cdot y}$ and $i q e^{i η \cdot y}$ , respectively, exactly as in the physical outgoing mode. Since $∣ e^{i q (x + i S) + i η \cdot y} ∣ = e^{- q S}$ , the bounds follow from $q \geq q_{0}$ and, when assumed, $S (x) \geq σ_{0} x$ . As $q ↓ 0$ , the attenuation bound tends to one, so this proof gives no uniform grazing decay.

Depends on

Cartesian complex-stretch PML for a homogeneous Helmholtz mode

Used by

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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)