Wave-number estimates for regularized combined field boundary integral operators in acoustic scattering problems with Neumann boundary conditions

We study the coercivity properties and the norm dependence on the wave-number k of certain regularized combined field boundary integral operators that we recently introduced for the solution of two-and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral operators are L² coercive for large enough values of the coupling parameter, and that the norms of these operators are bounded by constant multiples of the coupling parameter. We establish that the norms of the regularized combined field boundary integral operators grow modestly with the wave-number k for smooth boundaries and we provide numerical evidence that these operators are L² coercive for two-dimensional starlike boundaries. We present and analyse a fully discrete collocation (Nyström) method for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions based on regularized combined field integral equations. In particular, for analytic boundaries and boundary data, we establish pointwise superalgebraic convergence rates of the discrete solutions.