Abstract

We consider solving the acoustic scattering problem for the Helmholtz equation (-\Delta-k^{2)u=0 with constant wave speed and bounded Dirichlet or Neumann scatterer using the standard second-kind BIEs. We discuss various methods (including Galerkin, Collocation, and Nystrom) for the numerical approximation solutions of these BIEs;  addressing the fundamental question: how quickly must N, the dimension of the approximation space, grow with k to maintain accuracy as k →\infty? We give sufficient conditions on N to maintain accuracy as k\to \infty. Strikiningly, we show that these conditions are optimal for the Galerkin method with piecewise polynomials, and hence that, despite the common belief to the contrary, the boundary element method suffers from the pollution effect in many geometries (i.e. N growing like k}{d-1} is not sufficient to maintain accuracy).