An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

Natacha Lord, Anthony Mulholland

Research output: Contribution to journalArticle

Abstract

An a priori error estimate using a so called α,β- periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β-quasi periodic solution are used to transform the associated Helmholtz problem. The well-posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β-quasi periodic method converges at a far lower number of degrees of freedom than the α,0-quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer.
LanguageEnglish
Pages1187–1205
Number of pages19
JournalMathematical Methods in the Applied Sciences
Volume36
Issue number10
Early online date13 Sep 2012
DOIs
Publication statusPublished - 15 Jul 2013

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Diffraction Grating
A Priori Error Estimates
Finite Element Modeling
Diffraction gratings
Electromagnetic Wave
Electromagnetic waves
Scattering
Numerical methods
Quasi-periodic Solutions
Hermann Von Helmholtz
Finite Element Discretization
Scattering Problems
Small Perturbations
Well-posedness
Analytical Solution
Approximate Solution
Uniqueness
Degree of freedom
Numerical Methods
Transform

Keywords

  • periodic diffraction grating;
  • helmholtz
  • scattering
  • finite element
  • priori error estimate
  • electromagnetic waves

Cite this

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title = "An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating",
abstract = "An a priori error estimate using a so called α,β- periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β-quasi periodic solution are used to transform the associated Helmholtz problem. The well-posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β-quasi periodic method converges at a far lower number of degrees of freedom than the α,0-quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer.",
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N2 - An a priori error estimate using a so called α,β- periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β-quasi periodic solution are used to transform the associated Helmholtz problem. The well-posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β-quasi periodic method converges at a far lower number of degrees of freedom than the α,0-quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer.

AB - An a priori error estimate using a so called α,β- periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β-quasi periodic solution are used to transform the associated Helmholtz problem. The well-posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β-quasi periodic method converges at a far lower number of degrees of freedom than the α,0-quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer.

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