AI 中文总结
本文针对三维有限间距球体的时谐声学散射，证明多极展开法（MEM）的谱收敛性，构建收敛分析通用框架，数值实验验证相关特性，为多重散射快速算法的收敛分析铺路。
AI 中文摘要
多重散射是声学与电磁学中一类基础的波相互作用现象，多极展开法（MEM）是快速多极方法（FMM）等多种快速算法的基础。但由于涉及无穷多的波反射，该方法在三维场景下的收敛性尚未得到探究。本文针对三维空间中有限个间距充分大的球体产生的时谐声学散射问题，证明了MEM的谱收敛性。我们采用带对角预条件的单层公式，在自然的球谐能量空间中分析了阶数为N的截断系统，将相互作用截断拆分为目标侧与源侧的高阶部分，并分别为二者选择不同的表示形式：前者采用投影格林核表示，后者采用平移球波族的逐阶估计。基于Parseval恒等式与球谐加法定理的推导，为MEM的收敛性分析提供了通用框架，并揭示了收敛因子的几何与物理起源；我们还通过首次传递分析得到了更精确的估计。数值实验验证了所预测的谱衰减特性与几何收敛因子，为一大类多重散射快速算法的收敛性分析奠定了基础。
英文摘要
Multiple scattering is a fundamental wave interaction phenomenon in acoustics and electromagnetics. The multipole expansion method (MEM) is the basis of many fast algorithms, such as the fast multipole method (FMM), for such problems. However, due to the infinitely many wave reflections involved, its convergence in three dimensions remains unexplored. In this paper, we prove spectral convergence of the MEM for time-harmonic acoustic scattering by finitely many well-separated spheres in three dimensions. Using a diagonally preconditioned single-layer formulation, we analyze the degree-$N$ truncated system in a natural spherical harmonic energy space. We split the interaction truncation into target-side and source-side high-degree parts and choose a different representation for each: a projected Green-kernel representation for the former and degree-wise estimates of a translated spherical wave family for the latter. The resulting argument, based on Parseval's identity and the spherical harmonic addition theorem, provides a general framework for the convergence analysis of MEM and reveals the geometric and physical origins of the convergence factors. We also obtain a sharper estimate through the first-transfer analysis. Numerical experiments confirm the predicted spectral decay and the geometric convergence factor. This paves the way for the convergence analysis of a large class of fast algorithms for multiple scattering.
