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

Xavier Claeys (1, 2) , Ralf Hiptmair (3) , Elke Spindler (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 ALPINES - Algorithms and parallel tools for integrated numerical simulations
3 SAM - Seminar for Applied Mathematics [Zürich]

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] and extend it to this setting. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in $L^2(\Sigma)$, where $\Sigma$ denotes the union of material interfaces. To suppress spurious resonances, we introduce a combined-field version (CFIE) of our new method. Thorough numerical tests highlight the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

Mots clés: Galerkin boundary element methods; second-kind boundary integral equations; acoustic scattering

Domaines: Analyse numérique [math.NA]

Xavier Claeys, Ralf Hiptmair, Elke Spindler. Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects. 2015. ⟨hal-01251240⟩