# A hybrid numerical-asymptotic boundary element method for high frequency scattering by penetrable convex polygons

## TL;DR

This paper introduces a hybrid boundary element method combining geometrical optics and diffraction for high frequency scattering by convex polygons, achieving high accuracy with fewer degrees of freedom.

## Contribution

The paper presents a novel hybrid numerical-asymptotic boundary element method that efficiently captures high frequency oscillations in scattering problems using a combined geometrical optics and diffraction approach.

## Key findings

- Achieves fixed accuracy with frequency-independent degrees of freedom.
- Including diffraction improves accuracy by an order of magnitude.
- Effective for acoustic and electromagnetic scattering by convex polygons.

## Abstract

We present a novel hybrid numerical-asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.