When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?

We consider solving the sound-soft scattering problem for the Helmholtz equation with the -version of the boundary element method using the standard second-kind combined-field integral equations. We obtain sufficient conditions for the relative best approximation error to be bounded independently of . For certain geometries, these rigorously justify the commonly-held belief that a fixed number of degrees of freedom per wavelength is sufficient to keep the relative best approximation error bounded independently of . We then obtain sufficient conditions for the Galerkin method to be quasi-optimal, with the constant of quasi-optimality independent of . Numerical experiments indicate that, while these conditions for quasi-optimality are sufficient, they are not necessary for many geometries.