A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Abstract

Abstract. In this paper we consider the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data, a problem which models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies with a relatively low number of degrees of freedom, we propose a novel Galerkin boundary element method, using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that, on each element, the approximation space contains polynomials (of degree ν) multiplied by traces of plane waves on the boundary. We prove stability and convergence and show that the error in computing the total acoustic field is O(N −(ν+1) log 1/2 N ), where the number of degrees of freedom is proportional to N log N . This error estimate is independent of the wavenumber, and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity.

A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions

“…In order to deduce more detailed regularity estimates we combine ideas from [6] (for the related sound soft problem) and [7,14] (for an impedance half-plane problem). These more detailed estimates are bounds on derivatives of all orders related to the trace of the total field γ + u t , relevant to the analysis of boundary element methods based on a direct integral equation formulation obtained from Green's theorem (see §3 below).…”

“…In [7] a method in the spirit of the geometrical theory of diffraction was applied to obtain a representation of the solution, with the known leading order behaviour being subtracted off, leaving only the remaining scattered field due to the discontinuities in the impedance boundary conditions to be approximated. This diffracted field was expressed as a product of oscillatory and non-oscillatory functions, with a rigorous error analysis, supported by numerical experiments, demonstrating that the number of degrees of freedom required to maintain accuracy as k→∞ grows only logarithmically with respect to k. This approach was improved in [14], where derivation of sharper regularity estimates regarding the rate of decay of the scattered field away from impedance discontinuities led to error estimates independent of k.…”

“…The approach we will take in this paper combines ideas from our earlier work for sound soft convex polygons [6] with ideas developed for solving a two-dimensional problem of high frequency scattering by an inhomogeneous half-plane of piecewise constant impedance [7,14]. In [7] a method in the spirit of the geometrical theory of diffraction was applied to obtain a representation of the solution, with the known leading order behaviour being subtracted off, leaving only the remaining scattered field due to the discontinuities in the impedance boundary conditions to be approximated.…”

Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems

“…Bruno et al [12] presented an approach with complexity independent of wavelength by restricting the interval over which boundary integrals are performed to small regions in the immediate vicinity of stationary points; Langdon and Chandler-Wilde [13] have shown that this approach is suitable for polygonal scatterers; Domínguez et al [14] demonstrated that, for problems of asymptotically large wavenumbers, the required number of degrees of freedom increases only with O(k 1/9 ), for a fixed error bound; Anand et al [15] extended this approach for problems of multiple scatterers.…”

An Efficient Method for Computing Highly Oscillatory Physical Optics Integral

“…Hence, the PO integral kernel is getting more oscillatory as the electrical size of object becomes larger compared with the incident wavelength λ. Consequently, the computational cost by a direct numerical integration scheme [12,20] for the PO integral is extremely high. Due to the importance and challenges of acoustic, elastic and electromagnetic waves in high frequency applications, efficient numerical methods have attracted much attentions from mathematicians [23][24][25][26][27][28][29]. Bruno et al employed the full-wave combined-field boundary integral formulation and asymptotic theories to solve this type of problems.…”