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.

翻译：本文提出了一种求解可穿透非均匀介质二维散射问题的高阶快速方法，适用于包含（可能）不连续折射率的高频构型。该方法基于以下两种技术的直接/迭代混合组合：1）基于拉普拉斯算子作用的契比雪夫微分矩阵的微分体积公式，以及2）第二类边界积分公式。该方案对光滑折射率具有低色散和高阶精度，对不连续折射率情况（保持低色散）具有二阶精度。求解过程通过应用阻抗-阻抗（ItI）映射耦合体积离散与边界离散。体积线性代数解通过多波前求解器获得，而边界积分公式的耦合则通过迭代线性代数求解器GMRES实现。特别地，本文提出的存在唯一性定理为所考虑的整体散射问题是否存在唯一可解的第二类ItI公式这一开放问题提供了肯定答案。该算法基于适度依赖散射体的预计算阶段（实际计算复杂度约为$O(N^{\alpha})$，其中$\alpha \approx 1.07$，$N$为离散点数），并结合每个入射场快速（$O(N)$复杂度）的单核计算，能够有效求解包含强折射率对比与不连续性的大型复杂物体散射问题。