A New Frequency-uniform Coercive Boundary Integral Equation for Acoustic Scattering

A new boundary integral operator is introduced for the solution of the sound-soft acoustic scattering problem, i.e. for the exterior problem for Helmholtz equation with Dirichlet bound-ary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined ” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to imple-ment as the standard one. Additionally, to the authors ’ knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approxima-tion space. In particular these error estimates apply to several hybrid asymptotic-numerical methods developed recently which provide robust approximations in the high frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first in-troduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains