Abstract
We consider the scattering of time-harmonic acoustic waves at objects composed of several homogeneous parts with different material properties. In Claeys (A single trace integral formulation of the second kind for acoustic scattering, 2011), a novel second-kind boundary integral formulation for this scattering problem was proposed, that relies on skeleton Cauchy data as unknowns. We recast it into a variational problem set in \(L^{2}\) and investigate its Galerkin boundary element discretization from a theoretical and algorithmic point of view. Empiric studies demonstrate the competitive accuracy and superior conditioning of the new approach compared to a widely used Galerkin boundary element approach based on a first-kind boundary integral formulation.
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Notes
Capital letters are used to refer to functions defined over a volume domain.
Notations for function spaces (Sobolev spaces) follow the usual conventions, see [7, 17]. In particular, we write \(H_\mathrm{{loc}}^{s}({{\mathbb {R}}^{d}})\) for functions that belong to \(H^{s}(K)\) for any compact subset \(K\) of \({\mathbb {R}}^{d}\), see [21, Definition 2.6.1]. \(H_\mathrm{{comp}}^{s}({\Omega })\) contains all distributions in \(H_\mathrm{{loc}}^{s}({\Omega })\) that have compact support in \(\Omega \), see [21, Definition 2.6.5].
\(H_\mathrm{{loc}}^{1}({\Delta ,\overline{\Omega }}):=\{U\in H_\mathrm{{loc}}^{1}({\overline{\Omega }})\,|\, \Delta U \in L_\mathrm{{comp}}^{2}({\overline{\Omega }}) \}\), see [21, Equation (2.108)].
Fraktur font is used to designate functions in the Cauchy trace space, whereas Roman typeface is reserved for Dirichlet traces, and Greek symbols for Neumann traces.
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Communicated by Ulrich Langer.
The work of E. Spindler was partially supported by SNF under grant 20021_137873/1.
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Claeys, X., Hiptmair, R. & Spindler, E. A second-kind Galerkin boundary element method for scattering at composite objects. Bit Numer Math 55, 33–57 (2015). https://doi.org/10.1007/s10543-014-0496-y
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DOI: https://doi.org/10.1007/s10543-014-0496-y