In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods. * Colour online for monochrome figures available at journals.cambridge.org/anu.

There has been much recent research on preconditioning discretisations of the Helmholtz operator + k 2 (subject to suitable boundary conditions) using a discrete version of the so-called "shifted Laplacian" + (k 2 + iε) for some ε > 0. This is motivated by the fact that, as ε increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large ε can be so that the shifted problem provides a preconditioner that leads to k-independent convergence of GMRES, and our main result is a sufficient condition on ε for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large ε should be so that the preconditioner can easily be inverted by standard iterative methods.Mathematics Subject Classification 35J05 · 65F08 · 65F10 · 65N30 · 78A45

Abstract. The usual variational (or weak)

Abstract. In this paper we give new results on domain decomposition preconditioners for GM-RES when computing piecewise-linear finite-element approximations of the Helmholtz equation −∆u − (k 2 + iε)u = f , with absorption parameter ε ∈ R. Multigrid approximations of this equation with ε = 0 are commonly used as preconditioners for the pure Helmholtz case (ε = 0). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation (ε = 0), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left-or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a k-and ε-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if |ε| ∼ k 2 , then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. While the theory applies to a certain weighted variant of GMRES, the experiments for both weighted and classical GMRES give comparable results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case ε = 0. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about O(n 4/3 ) for solving finite element systems of size n = O(k 3 ), where we have chosen the mesh diameter h ∼ k −3/2 to avoid the pollution effect. Experiments on problems with h ∼ k −1 , i.e. a fixed number of grid points per wavelength, are also given.

We consider three problems for the Helmholtz equation in interior and exterior domains in R d , (d = 2, 3): the exterior Dirichlet-to-Neumann and Neumannto-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.