Condition number estimates for combined potential boundary integral operators in acoustic scattering

Abstract

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panič, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single-and double-layer potential operators.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…Time averaged acoustic intensity is given by [62]: (29) Here the first statement is the standard definition, with being acoustic particle velocity. The second statement has the real operator expanded as half the sum of the argument and its conjugate.…”

“…Choosing an optimal value of this parameter has since become an area of study in its own right. Generally the objective is to minimise the condition number of the resulting linear system, with earlier work typically concentrating on idealised obstacles such as spheres and cylinders [26][27][28] and latter work extending results to more complicated classes of obstacle [29,30]. Terai [31] commented that "it is clearly from physical considerations" that the sign of coupling parameter should be inverted depending on whether or time dependence is assumed, a detail that follows naturally from the time domain implementation discussed below, and Marburg showed that this can affect the solution in some cases [32].…”

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

“…These choices have been justified by theoretical studies for the case of a circle or sphere [2,3,40,41], and also on the basis of computational experience [11]. Recently these choices have been shown to be near optimal in terms of minimizing the condition number of A k;Á for more general domains by the analysis and numerical experiments of [8,19].…”

“…are proved using the Riesz-Thorin interpolation theorem in [19]. Mimicking the proof of the bound for kD 0 k k, it is straightforward to obtain…”

Coercivity of Combined Boundary Integral Equations in High‐Frequency Scattering

“…When considering the question of whether or not A 0 k;Á is coercive, it is instructive to also consider two other questions about A k that are sharp in their k-dependence for a wide variety of domains can be obtained just by using general techniques for bounding the norms of oscillatory integral operators; see [9], [10, §5.5], [56, §1.2]. In contrast, to obtain k-explicit bounds on k.A , thus this dependence is natural.)…”