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 recent 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] for pure transmission problems and extend it to settings with essential boundary conditions. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in L²(Σ), where Σ 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.