Robust fast direct integral equation solver for three-dimensional doubly periodic scattering problems with a large number of layers

Abstract

Frequency-domain wave scattering problems that arise in acoustics and electromagnetism can be often described by the Helmholtz equation. This work presents a boundary integral equation method for the Helmholtz equation in 3-D multilayered media with many doubly periodic smooth layers. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to 10−5 accuracy in less than 2 hours on a desktop workstation.