A spectral boundary element method for acoustic problems with analytic interfaces

In this paper we consider high-frequency acoustic problems modelled by Helmholtz equations with piecewise constant coefficients with discontinuities along properly separated, closed analytic interfaces. The solution then is highly oscillatory and, in addition, may be localized in a very small vicinity of interfaces (whispering gallery modes). For the reliable numerical approximation (a) the partial differential equation is transformed into a classical single trace integral equation on the interfaces and (b) a spectral Galerkin boundary element method is employed for its solution. We show that the resulting integral equation is well posed and analyse the convergence of the boundary element method for the particular case of concentric circular interfaces. We prove a condition on the number of degrees of freedom for quasi-optimal convergence. Numerical experiments confirm the efficiency of our method and the sharpness of the theoretical estimates.