Spectral Analysis of Discretized Boundary Integral Operators in 3D: a High-Frequency Perspective

TL;DR

This paper investigates the spectral properties of discretized boundary integral operators in 3D at high frequencies, revealing discrepancies that challenge the effectiveness of standard discretization practices as frequency increases.

Contribution

It provides a spectral analysis showing how discretized operators deviate from continuous ones at high frequencies, questioning common boundary element method assumptions.

Findings

Spectral discrepancies grow with frequency.

Standard mesh sizes may be insufficient at high frequencies.

Challenges to existing discretization practices.

Abstract

When modeling propagation and scattering phenomena using integral equations discretized by the boundary element method, it is common practice to approximate the boundary of the scatterer with a mesh comprising elements of size approximately equal to a fraction of the wavelength $\lambda$ of the incident wave, e.g., $\lambda/10$. In this work, by analyzing the spectra of the operator matrices, we show a discrepancy with respect to the continuous operators which grows with the simulation frequency, challenging the common belief that the aforementioned widely used discretization approach is sufficient to maintain the accuracy of the solution constant when increasing the frequency.