On Integral Equation and Least Squares Methods for Scattering by Diffraction Gratings (Communications in Computational Physics, 2020)
In this paper we consider the scattering of a plane acoustic or electromagneticwave by a one-dimensional, periodic rough surface. We restrict the discussion to the case whenthe boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in theEM case, so that the total field vanishes on the boundary. We propose a uniquely solvablefirst kind integral equation formulation of the problem, which amounts to a requirementthat the normal derivative of the Green's representation formula for the total field vanishon a horizontal line below the scattering surface. We then discuss the numerical solution byGalerkin's method of this (ill-posed) integral equation. We point out that, with two particularchoices 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 termthe 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 sizeof the approximation space increases. Moreover, we make a connection with the classicalleast squares method, in which the coefficients in the Rayleigh expansion of the solution aredetermined by enforcing the boundary condition in a least squares sense, pointing out thatthe linear system to be solved in the SS∗ method is identical to that in the least squaresmethod. Using this connection we show that (reflecting the ill-posed nature of the integralequation solved) the condition number of the linear system in the SS∗ and least squaresmethods approaches infinity as the approximation space increases in size. We also providetheoretical error bounds on the condition number and on the errors induced in the numericalsolution computed as a result of ill-conditioning. Numerical results confirm the convergenceof the SS∗ method and illustrate the ill-conditioning that arises.
Keywords: Helmholtz equation; first kind integral equation; spectral method; condition number.