# A high frequency boundary element method for scattering by a class of multiple obstacles

## TL;DR

This paper introduces a high-frequency boundary element method combining hybrid numerical asymptotics for convex polygons with standard methods for other obstacles, significantly reducing degrees of freedom needed at high frequencies.

## Contribution

The paper develops a novel boundary element method that achieves logarithmic growth in degrees of freedom with frequency for scattering problems involving multiple obstacles, especially with large convex polygons.

## Key findings

- Degrees of freedom grow logarithmically with frequency for the proposed method.
- The method is most effective for large convex polygons and small obstacles with combined perimeter comparable to the wavelength.
- Significant reduction in computational complexity compared to standard polynomial schemes.

## Abstract

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.