Explicit error bounds for the α-quasi-periodic Helmholtz problem
Lord, Natacha H. and Mulholland, Anthony J. (2013) Explicit error bounds for the α-quasi-periodic Helmholtz problem. Journal of the Optical Society of America A, 30 (10). pp. 2111-2123. ISSN 1084-7529 (https://doi.org/10.1364/JOSAA.30.002111)
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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.
ORCID iDs
Lord, Natacha H. and Mulholland, Anthony J. ORCID: https://orcid.org/0000-0002-3626-4556;-
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Item type: Article ID code: 45016 Dates: DateEvent1 October 2013PublishedSubjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 26 Sep 2013 12:35 Last modified: 03 Nov 2024 01:50 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/45016