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 \(h\)-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 \(k\). 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 \(k\). We then obtain sufficient conditions for the Galerkin method to be quasi-optimal, with the constant of quasi-optimality independent of \(k\). Numerical experiments indicate that, while these conditions for quasi-optimality are sufficient, they are not necessary for many geometries.