Wave-Number-Explicit Bounds in Time-Harmonic Scattering

Abstract

In this paper we consider the problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions, studying dependence on the wave number in two classical formulations of this problem. The first is the standard variational/weak formulation in the part of the exterior domain contained in a large sphere, with an exact Dirichletto-Neumann map applied on the boundary. The second formulation is as a second kind boundary integral equation in which the solution is sought as a combined single-and double-layer potential. For the variational formulation we obtain, in the case when the obstacle is starlike, explicit upper and lower bounds which show that the inf-sup constant decreases like k −1 as the wave number k increases. We also give an example where the obstacle is not starlike and the inf-sup constant decreases at least as fast as k −2. For the boundary integral equation formulation, if the boundary is also Lipschitz and piecewise smooth, we show that the norm of the inverse boundary integral operator is bounded independently of k if the coupling parameter is chosen correctly. The methods we use also lead to explicit bounds on the solution of the scattering problem in the energy norm when the obstacle is starlike in which the dependence of the norm of the solution on the wave number and on the geometry are made explicit.

Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

“…In [12], this question has been investigated in much more generality and, hence, will not be discussed here. The Fourier analysis which we developed in Section 3.2 gives explicit bounds on this constant provided the right-hand side is in L 2 (Ω).…”

“…Existence and uniqueness. Existence, uniqueness, and well-posedness of problem (2.6) has been studied in much more generality (concerning the assumption on the domain Ω) in [12] by using different techniques. The main goal of the estimates which we have derived in the previous sections is their application to the proof of the discrete stability for the finite element discretization and the convergence rates.…”

Reverse time migration for extended obstacles: acoustic waves

“…We refer to the recent work of Chandler-Wilde and Monk [12] and the references therein for results in that direction. In this paper we are interested in the case when the wavelength λ = 2π/k is of comparable size of the scatterer, that is, the case when k is not very large.…”

Foreign Ownership Restrictions: A Numerical Approach

“…Note that the proof of Theorem 3.6 does not provide how the constant C (Ω, k) depends on the wave number. In [12], this question has been investigated in much more generality and, hence, will not be discussed here. The Fourier analysis which we developed in Section 3.2 give explicit bounds on this constant provided the right-hand side is in L 2 (Ω).…”

“…Existence, uniqueness, and well-posedness of problem (2.5) has been studied in much more generality (concerning the assumption on the domain Ω) in [12] by using different techniques.…”