Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

Abstract

Abstract. We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, or (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-toDirichlet map when condition (i) and (ii) hold. We also prove bounds on the solutions of the interior and exterior impedance problems when the obstacle is a general Lipschitz domain. These bounds are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber. One motivation for proving these collection of bounds is that they can then be used to prove wavenumber-explicit bounds on the inverses of the standard second-kind integral operators used to solve the exterior Dirichlet, Neumann, and impedance problems for the Helmholtz equation.

Is the Helmholtz Equation Really Sign-Indefinite?

“…Regarding bounds on the solution in terms of the data, the currently best available bounds for the interior impedance problem in general Lipschitz domains have positive powers of k on the right-hand sides [69,Theorem 1.6]. Whether these bounds are sharp is not yet known; if they are sharp, then any formulation that is coercive for general Lipschitz domains would have the coercivity constant decreasing at least polynomially in k (assuming b(·, ·) and G(·) are normalized such that G V f L 2 (Ω) + g L 2 (∂Ω) with the omitted constant independent of k).…”

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$hp$$ h p -Version

“…Following the theory developed in [20,30,31] we prove approximation bounds for finite-dimensional spaces made of circular and plane wave functions. The main ingredients are three: (i) the explicit approximation bounds for harmonic functions and harmonic polynomials proved in [20] (improving on [25]) and reported in Proposition 5.1; (ii) the Vekua operators, which permit to transfer these approximation properties to Helmholtz solutions and circular waves (see a detailed discussion [32] and the continuity bounds in Lemma 5.2 below); (iii) the approximate inversion of the Jacobi-Anger expansion, which allows to prove bounds for plane waves (see (39) below, which was proved in [31,Lemma 4.3]). The interplay of these ingredients is outlined in Fig.…”

“…where = R and D = ∅), k-explicit stability bounds have been proved in [9,Theorem 2.4] and improved in their k-dependence in [39,Theorem 1.6], without assuming to be star-shaped. If the scatterer D is Lipschitz but trapping, thus not star-shaped, the constants in the stability bounds may grow exponentially in k, as shown in [4, Theorem 2.8] (note in particular Eq.…”

The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter

“…1 Recent investigations confirmed for a large group of domains, so-called nontrapping domains, that choosing |η| proportional to 1/k minimizes the condition number of the system of equations for which the sign of η is not relevant. [43][44][45][46][47] According to Spence, 43 the only property of the system matrix which strongly depends on the sign of η is coercivity. The system matrix for the Dirichlet problem (discretized by a Galerkin method) is coercive for η = i/k but not for η = −i/k, 48 of course assuming αβ = 1.…”