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

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.