The Burton–Miller method is a widely used approach in acoustics to enhance the stability of the boundary element method for exterior Helmholtz problems at so-called critical frequencies. This method depends on a coupling parameter and it can be shown that as long as has an imaginary part different from 0, the boundary integral formulation for the Helmholtz equation has a unique solution at all frequencies. A popular choice for this parameter is , where is the wavenumber. It can be shown that this choice is quasi optimal, at least in the high frequency limit. However, especially in the low frequency region, where the critical frequencies are still sparsely distributed, different choices for this factor result in a smaller condition number and a smaller error of the solution. In this work, alternative choices for this factor are compared based on numerical experiments. Additionally, a way to enhance the Burton–Miller solution with for a sound hard scatterer in the low frequency region by an additional step of a modified Richardson iteration is introduced.

中文翻译：

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关于低频区域的 Burton-Miller 因子


Burton-Miller 方法是声学中广泛使用的方法，用于增强边界元方法在所谓的临界频率下处理外部亥姆霍兹问题的稳定性。该方法依赖于耦合参数，并且可以证明，只要虚部不为0，亥姆霍兹方程的边界积分公式在所有频率下都具有唯一解。此参数的常用选择是 ，其中 是波数。可以证明这种选择是准最优的，至少在高频限制下是这样。然而，特别是在低频区域，临界频率分布仍然稀疏，对该因子的不同选择会导致条件数更小，解的误差也更小。在这项工作中，根据数值实验比较了该因素的替代选择。此外，还介绍了一种通过改进的理查森迭代的附加步骤来增强低频区域中的声音硬散射体的 Burton-Miller 解决方案的方法。