Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

Nicholls, David P.; Pérez‐Arancibia, Carlos; Turc, Catalin

doi:10.1007/s10915-020-01133-z

Cited by 8 publications

(10 citation statements)

References 28 publications

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“…Remark As far as we are aware, it is still an open problem to determine which particular choices of operators

{Y^{(m)}, Z^{(m)}}

deliver unique solutions. However, it is known that if

Im \{\int_{0}^{d} ((), Y^{(m)}, φ) \bar{φ} d x\} > 0, Im \{\int_{0}^{d} ((), Z^{(m)}, φ) \bar{φ} d x\} > 0,

then (9), (11), and (13) are well‐posed (see, e.g., 58 ). However, we will provide more precise and readily verified, though more complicated, characterizations in (), (), and ().…”

Section: A Nonoverlapping Domain Decomposition Methodsmentioning

confidence: 99%

“…Principally what we have in mind are the “Dirichlet eigenvalues” which arise when Dirichlet data are selected as a boundary unknown on an interior layer. To fix this, we follow the lead of Després 58,66–68 by pursuing IIOs which can be constructed to exist at all values of

k^{false(m false)}

.…”

Section: A Nonoverlapping Domain Decomposition Methodsmentioning

confidence: 99%

“…One approach to eliminating this artificial source of singularity is to utilize “Impedance–Impedance Operators” (IIOs) as advocated by Gillman et al 54 . On interior layers, these IIOs can be constructed to be unitary so that not only are their eigenvalues nonzero, they are restricted to the unit circle in the complex plane giving a very well‐conditioned algorithm 57,58 …”

Section: Introductionmentioning

confidence: 99%

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On analyticity of scattered fields in layered structures with interfacial graphene

Nicholls

2021

Stud Appl Math

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Two‐dimensional materials such as graphene have become crucial components of most state‐of‐the‐art plasmonic devices. The possibility of not only generating plasmons in the terahertz regime, but also tuning them in real time via chemical doping or electrical gating make them compelling materials for engineers seeking to build accurate sensors. Thus, the faithful modeling of the propagation of linear waves in a layered, periodic structure with such materials at the interfaces is of paramount importance in many branches of the applied sciences. In this paper, we present a novel formulation of the problem featuring surface currents to model the two‐dimensional materials which not only is free of the artificial singularities present in related approaches, but also can be used to deliver a proof of existence, uniqueness, and analytic dependence of solutions. We advocate for a surface integral formulation which is phrased in terms of well‐chosen Impedance–Impedance Operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet–Neumann Operators that appear in classical formulations. With a High‐Order Perturbation of Surfaces approach we are able to give a straightforward demonstration of this new well‐posedness result which only requires the verification that a finite collection of explicitly stated transcendental expressions be nonzero. We further illustrate the utility of this formulation by displaying results of a High‐Order Spectral numerical implementation which is flexible, rapid, and robust.

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“…Remark As far as we are aware, it is still an open problem to determine which particular choices of operators

{Y^{(m)}, Z^{(m)}}

deliver unique solutions. However, it is known that if

Im \{\int_{0}^{d} ((), Y^{(m)}, φ) \bar{φ} d x\} > 0, Im \{\int_{0}^{d} ((), Z^{(m)}, φ) \bar{φ} d x\} > 0,

then (9), (11), and (13) are well‐posed (see, e.g., 58 ). However, we will provide more precise and readily verified, though more complicated, characterizations in (), (), and ().…”

Section: A Nonoverlapping Domain Decomposition Methodsmentioning

confidence: 99%

k^{false(m false)}

.…”

Section: A Nonoverlapping Domain Decomposition Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On analyticity of scattered fields in layered structures with interfacial graphene

Nicholls

2021

Stud Appl Math

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“…Wave-scattering by periodic media, including RW anomalous configurations, at which the quasiperiodic Green function ceases to exist, has continued to attract significant attention in the fields of optics [17,22,33,34,35,36,39,45,50] and computational electromagnetism [3,8,4,9,10,31,14,26,42,39,18]. Classical boundary integral equations methods [43,49,52] have relied on the quasi-periodic Green function (denoted throughout this work as G q κ ), which is defined in terms of a slowly converging infinite series (equation (27)).…”

Section: Introductionmentioning

confidence: 99%

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

Bruno

Fernandez-Lado

2020

Journal of Computational Physics

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This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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“…This paper investigates this question for the case of stratified media. Our problem setting differs significantly from the case of quasiperiodic Helmholtz transmission problems as recently considered in [9], for instance, because it allows for a complete reflection of waves at the domain boundaries. As a concrete example we consider a problem from [7,8] in which spherical coordinates fr; u; hg are used.…”

Section: Introductionmentioning

confidence: 99%

Sweeping preconditioners for stratified media in the presence of reflections

Preuß

Hohage

Lehrenfeld³

2020

SN Partial Differ. Equ. Appl.

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In this paper we consider sweeping preconditioners for time harmonic wave propagation in stratified media, especially in the presence of reflections. In the most famous class of sweeping preconditioners Dirichlet-to-Neumann operators for half-space problems are approximated through absorbing boundary conditions. In the presence of reflections absorbing boundary conditions are not accurate resulting in an unsatisfactory performance of these sweeping preconditioners. We explore the potential of using more accurate Dirichlet-to-Neumann operators within the sweep. To this end, we make use of the separability of the equation for the background model. While this improves the accuracy of the Dirichlet-to-Neumann operator, we find both from numerical tests and analytical arguments that it is very sensitive to perturbations in the presence of reflections. This implies that even if accurate approximations to Dirichlet-to-Neumann operators can be devised for a stratified medium, sweeping preconditioners are limited to very small perturbations.

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Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

Cited by 8 publications

References 28 publications

On analyticity of scattered fields in layered structures with interfacial graphene

On analyticity of scattered fields in layered structures with interfacial graphene

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

Sweeping preconditioners for stratified media in the presence of reflections

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