We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a G{\aa}rding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.

翻译：本文对由分段均匀部分和任意非均匀部分构成的物体进行时谐声波散射建模。我们提出并分析了一种新的耦合公式，采用Costabel型方法，将均匀子域的边界积分方程与非均匀子域的体变分公式相结合。这是Costabel有限元-边界元耦合方法在多域构型上的推广，允许存在交叉点（即三个或更多子域相邻的点）。虽然通常仅对外部无界子域采用边界元法处理，但本文旨在尽可能利用边界元法的优势——即对所有均匀部分的散射物体均适用。我们的公式基于最初为分段均匀物体声波散射引入的多迹形式，而本文进一步允许波数在域内任意变化。我们证明了所提公式相关双线性形式满足G{\aa}rding强制性不等式，这保证了在解唯一存在时变分问题的稳定性。我们识别了单射条件，并构造了可避免虚假谐振的改进版本。