Abstract
We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequencies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attainedviaNyström methods.
Highlights
A vast number of scientific and engineering applications rely on harnessing acoustic and electromagnetic wave diffraction by periodic and/or multilayered domains
We build upon our theoretical review [5] and present a spectral Galerkin method for solving second-kind direct boundary integral equations (BIEs) for the Helmholtz transmission problem for twodimensional, periodic multi-layered gratings with smooth interfaces
We have proposed a fast spectral method for the efficient representation, through surface potentials based on the quasi-periodic Green’s function, for the solution of the Helmholtz equation with transmission boundary conditions on a periodic domain
Summary
A vast number of scientific and engineering applications rely on harnessing acoustic and electromagnetic wave diffraction by periodic and/or multilayered domains. Assuming impinging time-harmonic plane waves, scattered and transmitted fields have been solved by a myriad of mathematical formulations and associated solution schemes These range from volume variational formulations to various boundary integral representations and equations (cf [2, 3, 9, 11, 21, 33]), pure or coupled implementations of finite and boundary element methods
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