On Integral Equation and Least Squares Methods for Scattering by Diffraction Gratings
Tilo Arens 1, Simon N. Chandler-Wilde 2*, John A. DeSanto 31 Mathematisches Institut II, Univeristat Karlsruhe, 76128 Karlsruhe, Germany.
2 Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire RG6 6AX, United Kingdom.
3 Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401, USA.
Received 5 December 2005; Accepted (in revised version) 8 May 2006
In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414, 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.
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Key words: Helmholtz equation; first kind integral equation; spectral method; condition number.
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Correspondence to: Simon N. Chandler-Wilde , Department of Mathematics, University of Reading, Whiteknights, PO Box 220, Berkshire RG6 6AX, United Kingdom. Email: S.N.Chandler-Wilde@reading.ac.uk