A spectral boundary element method for acoustic interference problems

TL;DR

This work tackles high-frequency acoustic transmission with piecewise-constant coefficients by combining a single-trace boundary integral formulation with a spectral Galerkin boundary element method. The concentric-circles model enables explicit Fourier-diagonalization of the boundary operators, allowing sharp convergence analysis and a DOF condition that scales linearly with the wavenumber $\kappa$ for quasi-optimality. Theoretical results on well-posedness and spectral approximability are complemented by numerical experiments that confirm whispering-gallery localization near interfaces and the efficiency of the discretization. The approach offers a scalable framework for simulating high-frequency interference phenomena in Helmholtz transmission problems.

Abstract

In this paper we consider high-frequency acoustic transmission problems with jumping coefficients modelled by Helmholtz equations. The solution then is highly oscillatory and, in addition, may be localized in a very small vicinity of interfaces (whispering gallery modes). For the reliable numerical approximation a) the PDE is tranformed in a classical single trace integral equation on the interfaces and b) a spectral Galerkin boundary element method is employed for its solution. We show that the resulting integral equation is well posed and analyze the convergence of the boundary element method for the particular case of concentric circular interfaces. We prove a condition on the number of degrees of freedom for quasi-optimal convergence. Numerical experiments confirm the efficiency of our method and the sharpness of the theoretical estimates.

Figures (5)

• Figure 1: Model problem geometrical setting.
• Figure 2: Example 1. Behaviour of the real and imaginary part of the solution in the radial direction (first two left plots) and in the whole domain (last two right plots), for $\bm = 40$ and $\kappa \approx 90.11$.
• Figure 3: Example 2. Behaviour of the real and imaginary part of the solution in the radial direction (first two left plots) and in the whole domain (last two right plots), for $\bm = 60$ and $\kappa \approx 131.97$.
• Figure 4: Example 3. Behaviour of the traces $u_{D;1}, u_{N;1}$ (first two rows and columns) and $u_{D;0}, u_{N;0}$ (first two rows and second and last columns) with respect to the angle $\vartheta$. Log-log scale of the trace errors (last row).
• Figure 5: Example 4. Behaviour of the traces $u_{D;1}, u_{N;1}$ (first two rows and columns) and $u_{D;0}, u_{N;0}$ (first two rows and second and last rows) with respect to the angle $\vartheta$. Log-log scale of the trace errors (last row).

Theorems & Definitions (9)

• Lemma 4.1
• Proposition 4.2
• Lemma 4.3
• Proposition 4.4
• Remark 4.5
• Lemma 4.6
• Theorem 4.7
• Lemma A.1
• Lemma A.2