### Abstract

Original language | English |
---|---|

Pages (from-to) | 1187–1205 |

Number of pages | 19 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 36 |

Issue number | 10 |

Early online date | 13 Sep 2012 |

DOIs | |

Publication status | Published - 15 Jul 2013 |

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### Keywords

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

### Cite this

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**An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating.** / Lord, Natacha; Mulholland, Anthony.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

AU - Lord, Natacha

AU - Mulholland, Anthony

N1 - change of date

PY - 2013/7/15

Y1 - 2013/7/15

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.

KW - periodic diffraction grating;

KW - helmholtz

KW - scattering

KW - finite element

KW - priori error estimate

KW - electromagnetic waves

U2 - 10.1002/mma.2671

DO - 10.1002/mma.2671

M3 - Article

VL - 36

SP - 1187

EP - 1205

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 10

ER -