A fast direct solver for quasi-periodic scattering problems

Abstract

We consider the numerical solution of the scattering of time-harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium, in two dimensions. Boundary integral formulations are ideal since they reduce the problem to $N$ unknowns on the obstacle boundary. However, for complex geometries and/or higher frequencies the resulting dense linear system becomes large, ruling out dense direct methods, and often ill-conditioned (despite being 2nd-kind), rendering fast multipole-based iterative schemes also inefficient. We present an integral equation based solver with O(N) complexity, which handles such ill-conditioning, using recent advances in "fast" direct linear algebra to invert hierarchically the isolated obstacle matrix. This is combined with a recent periodizing scheme that is robust for all incident angles, including Wood's anomalies, based upon the free space Green's function kernel. The resulting solver is extremely efficient when multiple incident angles are needed, as occurs in many applications. Our numerical tests include a complicated obstacle several wavelengths in size, with $N=10^5$ and solution error of $10^{-10}$, where the solver is 66 times faster per incident angle than a fast multipole based iterative solution, and 600 times faster when incident angles are chosen to share Bloch phases.

References in corpus (2)

• Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials
• High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane

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• Windowed Green function method for wave scattering by periodic arrays of 2D obstacles
• A Fast Summation Method for Oscillatory Lattice Sums
• An efficient preconditioner for the fast simulation of a 2D Stokes flow in porous media
• Linearly scalable fast direct solver based on proxy surface method for two-dimensional elastic wave scattering by cavity
• Two spectral methods for 2D quasi-periodic scattering problems