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 domain variational formulations for the heterogeneous subdomain. This is an extension of Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains abut. 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; 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：我们研究了由分段均匀部分与任意非均匀部分构成的物体对时谐声波的散射建模问题。提出并分析了一种新型耦合算法——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域变分公式相结合。这是Costabel有限元-边界元耦合法向多域构型的拓展，允许跨点（即三个及以上子域邻接点）存在。通常仅对无界外部子域采用边界元法处理，但本研究旨在充分利用边界元法在所有适用场景（即散射体所有均匀部分）的技术优势。本算法基于多迹线形式体系构建——该体系最初是针对分段均匀物体的声散射问题提出，现允许波数在域内部分区域任意变化。我们证明，当该变分问题具有唯一可解性时，所提形式的双线性形式满足G{\aa}rding不等式，确保了变分问题的稳定性。进一步确定了单射性条件，并构造了能有效规避虚假共振的修正版本。