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This paper proposes well-conditioned boundary integral equations based on theBurton-Miller method for solving transmission problems. The Burton-Millermethod is widely accepted as a highly accurate numerical method based on theboundary integral equation for solving exterior wave problems. While thismethod can also be applied to solve the transmission problems, astraightforward formulation may yield ill-conditioned integral equations.Consequently, the resulting linear algebraic equations may involve acoefficient matrix with a huge condition number, leading to poor convergence ofKrylov-based linear solvers. To address this challenge, our study enhancesBurton-Miller-type boundary integral equations tailored for transmissionproblems by exploiting the Calderon formula. In cases where a single materialexists in an unbounded host medium, we demonstrate the formulation of theboundary integral equation such that the underlying integral operator ${\calA}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed ina simple manner that ensures ${\cal A}^2$ is bounded and has only a singleeigenvalue accumulation point. Furthermore, we extend our analysis to themulti-material case, proving that the square of the proposed operator has onlya few eigenvalues except for a compact perturbation. Through numerical examplesof several benchmark problems, we illustrate that our formulation reduces theiteration number required by iterative linear solvers, even in the presence ofmaterial junction points; locations where three or more sub-domains meet on theboundary.

Calderon-preconditioned boundary integral equations of the Burton-Miller type for transmission problems