In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.

翻译：本文讨论了适用于腔内（即不包含出射波条件的封闭域）时间谐波亥姆霍兹方程求解的非重叠Schwarz方法中的不同传输算子。此类问题受到反向传播波的严重影响，而在设计Schwarz方法的优化传输算子时，这些波往往被忽略。本文探索了考虑这些反向传播波的新算子，并将其与忽略这些贡献的成熟算子进行了比较。特别地，本文聚焦于矩形腔的情况，因为此时可以简便地确定最优（非局部）传输算子。尽管如此，本文也考虑了偏离这一理想几何形态的情形。具体而言，文中讨论了利用优化Schwarz方案对某束线低温恒温器氦容器三维模型进行声学噪声计算。这些计算表明，与针对无界问题优化的算子相比，针对空腔优化的算子在迭代次数上减少了46%。