AI 中文总结
本文针对分段光滑非均匀介质的时谐散射问题，提出了一种结合非结构化网格高阶离散与结构化网格预处理的快速高阶Nyström求解器，其迭代次数与网格尺寸、波数无关，兼具几何灵活性与频率鲁棒性。
AI 中文摘要
本文针对可穿透、分段光滑非均匀介质的时谐散射问题所产生的二维Lippmann–Schwinger方程，提出了一种快速的高阶Nyström求解器。该方法依赖于对适配于间断界面的非结构化网格上的牛顿势进行高阶求值，结合快速多极法等现有快速算法，实现了高阶精度。作为一种迭代方法，当与利用一类结构化网格求解器的预处理策略耦合时，该求解器展现出快速收敛性；这类结构化网格求解器具有准线性时间与内存复杂度，且迭代次数几乎恒定，但长期以来精度受限。该预处理策略通过一对转移算子，将Nyström离散化（由适配空间变化对比度支撑的（弯曲）非结构化网格上的高阶求积节点给出）与底层快速预处理器的均匀笛卡尔网格耦合。所得预处理器继承了其笛卡尔对应物的频率鲁棒性，同时未牺牲非结构化离散化的几何灵活性与高阶精度。我们证明，在网格尺寸和笛卡尔预处理器的明确条件下，所提出的预处理器具有可逆性。数值实验表明，与未预处理的对应系统相比，预处理后的系统所需的GMRES迭代次数显著更少，且迭代次数几乎与网格尺寸和波数无关，验证了该方法对具有分段光滑折射率和界面处跳变间断的非均匀介质的鲁棒性。
英文摘要
This article presents a fast, high-order Nyström solver for the two-dimensional Lippmann--Schwinger equation arising from time-harmonic scattering by penetrable, piecewise-smooth heterogeneous media. Relying on high-order evaluation of the Newtonian potential on unstructured grids adapted to interfaces of discontinuity, the methodology achieves high-order accuracy using existing fast algorithms such as the fast multipole method. As an iterative method the solver exhibits rapid convergence when coupled to a preconditioning strategy that exploits a class of structured-grid solvers---fast solution methods offering quasi-linear time and memory complexity and nearly-constant iteration counts, but long limited in accuracy. The preconditioning strategy couples the Nyström discretization---given by high-order quadrature nodes over a (curved) unstructured mesh conforming to the support of the spatially varying contrast---to a uniform Cartesian grid underlying the fast preconditioner via a pair of transfer operators. The resulting preconditioner inherits the frequency-robust behavior of its Cartesian counterpart without sacrificing the geometric flexibility and high-order accuracy of the unstructured discretization. We prove that invertibility of the proposed preconditioner holds under explicit conditions on the mesh sizes and on the Cartesian preconditioner. Numerical experiments demonstrate that the preconditioned system requires significantly fewer GMRES iterations than its unpreconditioned counterpart, with iteration counts almost independent of mesh size and wavenumber, and illustrate the method's robustness for inhomogeneities with piecewise-smooth refractive indices and jump discontinuities across interfaces.
