Automatically well-conditioned collocation boundary element method for transmission problems based on the Burton--Miller formulation

TL;DR

This work addresses efficient solution of 2D transmission problems with boundary element methods by tackling spectral ill-conditioning in Burton–Miller formulations. It introduces Calderón-based rearrangements and parameter choices that make the square of the integral operator, $A^2$, exhibit eigenvalue clustering at a small number of points, enabling fast GMRES convergence when discretised via collocation. The authors develop a single-material Calderón BM formulation with $β=-α$ yielding a single accumulation point, and extend this to multi-material cases with $β=-α$, $γ=α/ε_3$, achieving similarly favorable spectra; numerical experiments on single and multi-material configurations confirm fast convergence and robustness to resonances. The method offers an easy-to-implement, spectrally well-conditioned approach for transmission problems with material junctions and has potential for extension to 3D and vector waves, broadening its practical impact for wave-scattering analyses.

Abstract

This paper proposes a collocation boundary element method based on the Burton--Miller method for solving transmission problems, which is rapidly convergent within the Krylov subspace solver framework. Our study enhances Burton--Miller-type boundary integral equations tailored for transmission problems by exploiting the Calderon formula. In cases where a single material exists in an unbounded host medium, we demonstrate the formulation of the boundary integral equation such that the underlying integral operator ${\cal A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed such that ${\cal A}^2$ has only a single eigenvalue accumulation point. Furthermore, we extend this to the multi-material case, proving that the square of the proposed operator has only a few eigenvalues clustering points. When the collocation method is used to discretise the proposed boundary integral equations, the good spectral properties of the integral operator are naturally inherited to the coefficient matrix $\mathsf{A}$ similarly to the Nystöm methods; Almost all eigenvalues of $\mathsf{A}^2$ cluster at a few points in the complex plane ensuring the small condition number for $\mathsf{A}$. Through numerical examples of several benchmark problems, we illustrate that our formulation reduces the iteration number required by iterative linear solvers, even in the presence of material junction points.

Figures (15)

• Figure 1: Illustrative sketch for multi-material scattering. The scatterer is composed of $\Omega_2$ and $\Omega_3$ each of which is filled with a different material.
• Figure 2: Benchmark setting for the single material case.
• Figure 3: Relative $\ell_2$-error in $u$ vs the number of boundary elements $N$ for the case of single material scattering. "Conventional BM", "Calderon BM ($\beta=1$)", and "Calderon BM ($\beta=-\alpha$)" respectively solve the integral equations \ref{['eq:naive_bie']}, \ref{['eq:calderon_bie1']}, and \ref{['eq:calderon_bieb']} with $\beta=-\alpha$. The same keys shall be used in the figures to follow.
• Figure 4: The number of iterations for GMRES vs $N$ for the case of single material scattering.
• Figure 5: The number of GMRES iterations against the incident angular frequency $\omega$ (left) and the material constant of the circular inclusion $\varepsilon_2$ (right).