In this work we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even exponential convergence for the local Helmholtz problems can be achieved without oversampling.

翻译：本文提出并分析了一种将近似分量模式综合（ACMS）方法推广至非均匀亥姆霍兹方程的方案。ACMS方法最初由Hetmaniuk和Lehoucq提出，作为一种求解椭圆型偏微分方程的多尺度方法。该方法采用区域分解，将变分问题分离为两个独立部分——局部亥姆霍兹问题与全局界面问题——以实现数值近似。前者天然具有局域性和解耦性，便于并行求解；后者则需要构建依赖于局部本征模式及其适定延拓的局部基函数。我们针对区域分解固定但本征函数数量增加的情形，对该方法进行了完整误差分析。本文的理论结果通过数值实验得到验证，证实该方法具有代数收敛性。在某些实际相关情形下，局部亥姆霍兹问题甚至可在无过采样条件下实现指数收敛。