Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

TL;DR

This paper investigates the occurrence of spurious quasi-resonances in boundary integral equations for the Helmholtz transmission problem, explaining their causes and proposing modified equations to avoid these artifacts.

Contribution

It provides a rigorous analysis of spurious quasi-resonances and introduces modified boundary integral equations that prevent their occurrence.

Findings

Spurious quasi-resonances cause unbounded inverse norms at certain frequencies.

Modified boundary integral equations eliminate the spurious resonances.

The analysis clarifies when and why these artifacts occur.

Abstract

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.