We propose a boundary element method for problems of time-harmonic acoustic scattering by multiple obstacles in two dimensions, at least one of which is a convex polygon. By combining a hybrid numerical-asymptotic (HNA) approximation space on the convex polygon with standard polynomial-based approximation spaces on each of the other obstacles, we show that the number of degrees of freedom required in the HNA space to maintain a given accuracy needs to grow only logarithmically with respect to the frequency, as opposed to the (at least) linear growth required by standard polynomial-based schemes. This method is thus most effective when the convex polygon is many wavelengths in diameter and the small obstacles have a combined perimeter comparable to the problem wavelength.

中文翻译：

---

一类多障碍物散射的高频边界元方法

针对二维障碍物中至少一个是凸多边形的多个障碍物的时谐声散射问题，我们提出了一种边界元方法。通过将凸多边形上的混合数值渐近（HNA）逼近空间与每个其他障碍物上的基于标准多项式的逼近空间相结合，我们证明了维持给定精度所需的HNA空间中的自由度数相对于频率仅对数增长，这与基于标准多项式的方案所需的（至少）线性增长相反。因此，当凸多边形的直径为许多波长并且小障碍物的组合周长与问题波长相当时，此方法最有效。