Wavenumber-Explicit hp-BEM for High Frequency Scattering

For the Helmholtz equation (with wavenumber k) and analytic curves or surfaces Γ we analyze the Galerkin discretization of classical combined field integral equations in an L-setting. We give abstract conditions on the approximation properties of the ansatz space that ensure stability and quasi-optimality of the Galerkin method. Special attention is paid to the hp-version of the boundary element method (hp-BEM). Under the assumption of polynomial growth of the solution operator we show stability and quasi-optimality of the hp-BEM if the following scale resolution condition is satisfied: the polynomial degree p is at least O(log k) and kh/p is bounded by a number that is sufficiently small, but independent of k. Under this assumption, the constant in the quasioptimality estimate is independent of k. Numerical examples in 2D illustrate the theoretical results and even suggest that in many cases quasi-optimality is given under the weaker condition that kh/p is sufficiently small.