In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional 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 super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.

中文翻译：

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近似分量模合成方法对异质亥姆霍兹方程的扩展


在这项工作中，我们提出并分析了近似分量模态合成 （ACMS） 方法对二维异质亥姆霍兹方程的扩展。ACMS 方法最初由 Hetmaniuk 和 Lehoucq 引入，作为求解椭圆偏微分方程的多尺度方法。ACMS 方法使用域分解通过将变分问题拆分为两个独立的部分来分离数值近似：局部亥姆霍兹问题和全局界面问题。前者自然是局部的和解耦的，因此可以很容易地并行求解，而后者需要依靠局部特征模态和合适的扩展来构建合适的局部基函数。我们对这种方法进行了全面的误差分析，重点关注域分解保持固定但特征函数数量增加的情况。这项工作的理论结果得到了验证该方法代数收敛性的数值实验的支持。在某些实际相关的情况下，即使是局部亥姆霍兹问题的超代数收敛也可以在不过度采样的情况下实现。