Multi‐Trace Boundary Integral Formulation for Acoustic Scattering by Composite Structures

Abstract

We study the scattering of an acoustic wave by an object composed of several adjacent parts with different material properties. For this problem we derive a new integral equation formulation of the first kind. This formulation involves two Dirichlet data and two Neumann data at each point of each material interface of the diffracting object. It is immune to spurious resonances, and it enjoys a stability property that ensures quasi‐optimal convergence of conforming Galerkin boundary element discretization. In addition, the operator of this formulation satisfies a relation similar to the standard Calderón identity. © 2012 Wiley Periodicals, Inc.

Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

“…We successfully apply them to the new multi-trace formulation and establish asymptotic quasi-optimality of Galerkin solutions along with efficacy of our Calderón preconditioner. Thus, we hope to convince the reader that the techniques presented in [19] are relevant beyond acoustic scattering. We shall also tackle the case where the domain of propagation contains metallic parts or screens in a forthcoming article.…”

“…This missing preconditioner motivated the present article. The main ideas have first been elaborated for acoustic scattering in [19] and now we adapt them to Maxwell's equations. Again, we start with the observation that the single-trace formulation admits straightforward Calderón preconditioning in the case of scattering at a single homogeneous object (n = 1 in (1.1)).…”

“…Big parts of it run parallel to the developments in the companion paper [19]. However, extra difficulties arise due to the use of electromagnetic trace spaces and the lack of coercivity of the electric field integral operator.…”

Multitrace Formulations and Dirichlet-Neumann Algorithms

“…Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [4] and [1,2] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as indicated in [5], but very little is known so far about such associated iterative solvers.…”

Closed Form Inverse of Local Multi-Trace Operators

“…This formulation was introduced in [9] to tackle transmission problems for the Helmholtz equation, where the material properties are constant in each subdomain, see also [4,5], and [6] for associated boundary integral methods. Multi-trace formulations lead naturally to block preconditioners, see [10].…”