Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

TL;DR

This paper develops two classes of Galerkin boundary element methods for high-frequency sound-hard scattering in 2D, achieving frequency-independent accuracy with minimal degrees of freedom increase, validated by numerical results.

Contribution

Introduction of two novel Galerkin boundary element methods that maintain accuracy at high frequencies with minimal additional computational cost.

Findings

Methods require only a small increase in degrees of freedom with frequency.

Numerical results confirm theoretical frequency-independent accuracy.

Incorporating asymptotic expansion terms improves solution precision.

Abstract

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. Both methods require a small increase in the order of $k^{\epsilon}$ (for any $\epsilon >0$) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber $k$. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validate our theoretical findings.