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

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 alpha-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of e(iax) and an unknown function called the alpha-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 alpha 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 alpha 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 alpha-quasi-periodic transformation with a lattice sum technique is then presented. (C) 2013 Optical Society of America