Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Abstract

We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions. Following rA. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer. Math., 100 (2005), pp. 1-19s we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of rX. Claeys and R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1163-1201s to situations with Dirichlet boundary conditions.

Quasi‐local multitrace boundary integral formulations

“…Numerical Methods for Partial Differential Equations DOI 10.1002/num iii) are verified with K j replaced by K j * . Since we also know that ii) and iii) are verified with K j 0 (see e.g., Lemmas 3.6 and 3.7 in [16]), this concludes the proof.…”

“…Besides the kernel of K j * is C ∞ as K and K 0 coincide in a neighborhood of 0. This proves the continuity properties announced for K j , and shows in addition (see e.g., Lemmas 3.6 and 3.7 in [16]), this concludes the proof.…”

“…Using this to evaluate the real part of (16), and according to Proposition 6.2, we see that there exists a compact operator R : H( ) → H( ) such that e [[(−˜ + R )u, (u)]] ≥ 0 which concludes that proof.…”

Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

“…setting κ 2 = κ 0 , and defining Ω 1 to be impenetrable, too. In this setting, in accordance with the theory of [8], we get rid of the spurious resonances in the case of the first-kind formulation. In the case of the second-kind formulation, spurious resonances persist.…”

Essential spectrum of local multi-trace boundary integral operators

“…The Boundary Element Tearing and Interconnecting method (BETI) was developed in this spirit, more than a decade ago, as an integral equation counterpart of the FETI method, see [8,13,14,17,23]. An alternative approach dubbed Multi-Trace formalism, leading to different solvers, was introduced a few years ago [18,19,10,11,2,3,4,5], providing other boundary integral formulations adapted to multi-domain geometrical configurations. Multi-trace boundary integral formulations seem well adapted to block preconditioners for domain decomposition but still very little is known about associated iterative global solvers.…”