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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,βquasi periodic solution are used to transform the associated Helmholtz problem. The wellposedness 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,βquasi periodic method converges at a far lower number of degrees of freedom than the α,0quasi 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.
Original language  English 

Pages (fromto)  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 
Keywords
 periodic diffraction grating;
 helmholtz
 scattering
 finite element
 priori error estimate
 electromagnetic waves
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Projects
 1 Finished

PGII: Generation, Detection & Analysis of Optimally Coded Ultrasonic Waveforms
Gachagan, A., Hayward, G., Mulholland, A. & Pierce, G.
EPSRC (Engineering and Physical Sciences Research Council)
9/06/08 → 9/09/12
Project: Research