A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

TL;DR

This paper introduces a spectrally accurate direct method for solving frequency-domain scattering problems with smoothly varying media, combining recursive boundary operator merges and spectral collocation for high precision and efficiency.

Contribution

It develops a stable, high-accuracy direct solution technique using impedance-to-impedance maps and a quadtree structure, effectively handling variable media without internal resonance issues.

Findings

Achieves 9-digit accuracy for large problems with 4 million unknowns

Solves challenging scattering problems efficiently in under 5 minutes

Each additional incident wave solution takes only 0.04 seconds

Abstract

This paper presents a direct solution technique for the scattering of time-harmonic waves from a bounded region of the plane in which the wavenumber varies smoothly in space.The method constructs the interior Dirichlet-to-Neumann (DtN) map for the bounded region via bottom-up recursive merges of (discretization of) certain boundary operators on a quadtree of boxes.These operators take the form of impedance-to-impedance (ItI) maps. Since ItI maps are unitary, this formulation is inherently numerically stable, and is immune to problems of artificial internal resonances. The ItI maps on the smallest (leaf) boxes are built by spectral collocation on tensor-product grids of Chebyshev nodes. At the top level the DtN map is recovered from the ItI map and coupled to a boundary integral formulation of the free space exterior problem, to give a provably second kind equation.Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 minutes on a desktop workstation. Each additional solve corresponding to a different incident wave (right-hand side) then requires only 0.04 seconds.