Explicit error bounds for the α-quasi-periodic Helmholtz problem

Natacha H. Lord, Anthony J. Mulholland

Research output: Contribution to journalArticle

2 Citations (Scopus)
69 Downloads (Pure)

Abstract

This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

Original languageEnglish
Pages (from-to)2111-2123
Number of pages13
JournalJournal of the Optical Society of America A
Volume30
Issue number10
DOIs
Publication statusPublished - 1 Oct 2013

Fingerprint

A Priori Error Estimates
Explicit Bounds
Hermann Von Helmholtz
Error Bounds
estimates
Dirichlet-to-Neumann Map
Quasi-periodic Solutions
Diffraction Grating
Polynomial Basis
Diffraction gratings
Variational Formulation
uniqueness
gratings (spectra)
Electromagnetic Wave
regularity
continuity
Well-posedness
Electromagnetic waves
electromagnetic radiation
finite element method

Keywords

  • periodic diffraction grating
  • electromagnetic waves
  • Dirichlet-to-Neumann map

Cite this

@article{4c28498065934bbcacb7442351969d71,
title = "Explicit error bounds for the α-quasi-periodic Helmholtz problem",
abstract = "This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.",
keywords = "periodic diffraction grating, electromagnetic waves, Dirichlet-to-Neumann map",
author = "Lord, {Natacha H.} and Mulholland, {Anthony J.}",
year = "2013",
month = "10",
day = "1",
doi = "10.1364/JOSAA.30.002111",
language = "English",
volume = "30",
pages = "2111--2123",
journal = "Journal of the Optical Society of America A",
issn = "1084-7529",
publisher = "Optical Society of America",
number = "10",

}

Explicit error bounds for the α-quasi-periodic Helmholtz problem. / Lord, Natacha H.; Mulholland, Anthony J.

In: Journal of the Optical Society of America A, Vol. 30, No. 10, 01.10.2013, p. 2111-2123.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Explicit error bounds for the α-quasi-periodic Helmholtz problem

AU - Lord, Natacha H.

AU - Mulholland, Anthony J.

PY - 2013/10/1

Y1 - 2013/10/1

N2 - This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

AB - This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

KW - periodic diffraction grating

KW - electromagnetic waves

KW - Dirichlet-to-Neumann map

UR - http://www.scopus.com/inward/record.url?scp=84885358176&partnerID=8YFLogxK

U2 - 10.1364/JOSAA.30.002111

DO - 10.1364/JOSAA.30.002111

M3 - Article

VL - 30

SP - 2111

EP - 2123

JO - Journal of the Optical Society of America A

JF - Journal of the Optical Society of America A

SN - 1084-7529

IS - 10

ER -