An extension of the approximate component mode synthesis (ACMS) method to the heterogeneous Helmholtz equation is proposed. 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. This complements related results for elliptic problems where the focus is on the refinement of the domain decomposition instead. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the interface problems. In certain, practically relevant cases, even exponential convergence for the local Helmholtz problems can be achieved without oversampling.

翻译：本文提出了一种将近似分量模态综合（ACMS）方法扩展到非均匀亥姆霍兹方程的方案。ACMS方法最初由Hetmaniuk和Lehoucq提出，是一种用于求解椭圆型偏微分方程的多尺度方法。该方法通过区域分解，将变分问题拆分为两个独立部分：局部亥姆霍兹问题和全局界面问题，从而分离数值逼近过程。局部亥姆霍兹问题天然具有局部性和解耦性，易于并行求解；而全局界面问题则需要基于局部特征模态及适当扩展构造合适的局部基函数。我们针对该方法的完整误差分析展开研究，重点关注区域分解固定而特征函数数量增加的情形。这补充了椭圆问题中侧重于区域分解细化的相关研究结果。工作中的理论结果得到数值实验的支持，验证了界面问题具有代数收敛性。在实际相关案例中，局部亥姆霍兹问题甚至无需过采样即可实现指数收敛。