Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces

Ruming Zhang

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Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces

Ruming Zhang

Numerical Analysis · Submitted 5 Jul 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The main task in this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter, for scattering problems with periodic surfaces.

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arXiv ID2107.02032 (version 2)

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CitationRuming Zhang. "Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces." arXiv:2107.02032 (2021). https://pith.science/paper/AVVB7ADC

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,407 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.38 · compares with prior results 0.96 · mentions code or data availability 0.03.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The main task in this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter, for scattering problems with periodic surfaces. In [5], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is if exponential convergence holds locally. In our paper, we answer this question for a special case, which is scattering problems with periodic surfaces. The result can also be easily extended to locally perturbed periodic surfaces or periodic layers. Due to technical reasons, we have to exclude all the half integer valued wavenumbers. The main idea of the proof is to apply the Floquet-Bloch transform to write the problem into an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet-Bloch parameters. Then the Cauchy integral formula is applied for piecewise analytic functions to avoid linear convergent points. Finally the exponential convergence is proved from the inverse Floquet-Bloch transform. Numerical results are also presented at the end of this paper.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The main task in this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter, for scattering problems with periodic surfaces.	contribution 0.79	0.88	0.92	asserted 0.99
s1	In [5], a linear convergence is proved for the PML method for scattering problems with rough surfaces.	background 0.97	0.03	0.02	asserted 1.00
s2	At the end of that paper, three important questions are asked, and the third question is if exponential convergence holds locally.	gap 0.55	0.08	0.02	asserted 0.95
s3	In our paper, we answer this question for a special case, which is scattering problems with periodic surfaces.	contribution 0.96	0.92	0.12	asserted 1.00
s4	The result can also be easily extended to locally perturbed periodic surfaces or periodic layers.	interpretation 0.67	0.59	0.04	asserted 0.67
s5	Due to technical reasons, we have to exclude all the half integer valued wavenumbers.	limitation 1.00	0.81	0.02	asserted 1.00
s6	The main idea of the proof is to apply the Floquet-Bloch transform to write the problem into an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet-Bloch parameters.	method 0.99	0.88	0.04	asserted 1.00
s7	Then the Cauchy integral formula is applied for piecewise analytic functions to avoid linear convergent points.	method 1.00	0.90	0.02	asserted 1.00
s8	Finally the exponential convergence is proved from the inverse Floquet-Bloch transform.	method 0.84	0.94	0.42	asserted 1.00
s9	Numerical results are also presented at the end of this paper.	contribution 0.50	0.92	0.02	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

The main task in this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter, for scattering problems with periodic surfaces. In [5], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is if exponential convergence holds locally. In our paper, we answer this question for a special case, which is scattering problems with periodic surfaces. The result can also be easily extended to locally perturbed periodic surfaces or periodic layers. Due to technical reasons, we have to exclude all the half integer valued wavenumbers. The main idea of the proof is to apply the Floquet-Bloch transform to write the problem into an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet-Bloch parameters. Then the Cauchy integral formula is applied for piecewise analytic functions to avoid linear convergent points. Finally the exponential convergence is proved from the inverse Floquet-Bloch transform. Numerical results are also presented at the end of this paper.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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The relative error is defined as [equation] Note that the meshes are exactly the same as for the “exact solutions” and the discretization method of [ref] is introduced in [ref] with also $80$ points.

Numerical method for scattering problems in periodic waveguides · Numerical results · 0.97

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