Direct/iterative hybrid solver for scattering by inhomogeneous media

TL;DR

This paper introduces a fast, high-order hybrid solver combining volumetric and boundary integral methods for efficiently solving 2D scattering problems in inhomogeneous media, including cases with discontinuous refractivities.

Contribution

It presents a novel hybrid direct/iterative approach with theoretical guarantees for unique solvability, enabling efficient high-frequency scattering simulations with complex media.

Findings

Achieves high-order accuracy for smooth refractivities

Maintains second-order accuracy with low dispersion for discontinuous media

Provides an efficient solver with near-linear computational complexity

Abstract

This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation. The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $O(N^{\alpha})$ operations, with $\alpha \approx 1.07$, for an $N$-point discretization), together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing strong refractivity contrasts and discontinuities.