A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Chandler-Wilde, Simon N.; Rahman, Mizanur; Ross, C. R.

doi:10.1007/bf02679436

Cited by 26 publications

(39 citation statements)

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“…For the special case of a at surface, e cient boundary element techniques for the impedance problem have recently been proposed and analysed in Reference [29].…”

Section: Introductionmentioning

confidence: 99%

Integral equation methods for scattering by infinite rough surfaces

Zhang

Chandler-Wilde

2003

Math Methods in App Sciences

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SUMMARYIn this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane. These boundary value problems arise in a study of timeharmonic acoustic scattering of an incident ÿeld by a sound-soft, inÿnite rough surface where the total ÿeld vanishes (the Dirichlet problem) or by an inÿnite, impedance rough surface where the total ÿeld satisÿes a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double-and single-layer potential and a Dirichlet half-plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half-plane impedance Green's function, the ÿrst derived from Green's representation theorem, and the second arising from seeking the solution as a single-layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident ÿelds including an incident plane wave, the impedance boundary value problem for the scattered ÿeld has a unique solution under certain constraints on the boundary impedance.

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“…For the special case of a at surface, e cient boundary element techniques for the impedance problem have recently been proposed and analysed in Reference [29].…”

Section: Introductionmentioning

confidence: 99%

Integral equation methods for scattering by infinite rough surfaces

Zhang

Chandler-Wilde

2003

Math Methods in App Sciences

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“…We note that the approximate and numerical solution of this integral equation has been extensively studied; see, for example, [42,30,17,21,20].…”

Section: Conversely If U| γ ∈ Bc(γ) (The Space Of Bounded and Continmentioning

confidence: 99%

“…In the case of boundary element methods, a great deal of effort has focused on the fast solution of the large systems which arise, using preconditioned iterative methods (e.g., [22]) combined with fast multipole (e.g., [24,25]) or fast Fourier transform based methods (e.g., [9,21]) to carry out the matrix-vector multiplications efficiently. The reduction in the computing cost achieved by the use of these schemes increases the upper limit on the frequency for which accurate results can be obtained in a reasonable time.…”

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confidence: 99%

“…This problem has attracted much attention in the literature (see, for example, [17,30,31,16,21,6,48]), both in its own right and also as a model of the scattering of an incident acoustic or electromagnetic wave by an infinite rough surface [8,47,36,7]. In the case in which there is no variation in the acoustical properties of the surface or the incident field in some fixed direction parallel to the surface, the problem is effectively twodimensional.…”

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confidence: 99%

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A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Langdon¹,

Chandler-Wilde²

2006

SIAM J. Numer. Anal.

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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.

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“…Chandler-Wilde et al . [4] recently considered the numerical treatment of an equation of the form (2.3) and an application to acoustic scattering. In the problem considered by Biggs and Porter [2], a particular version of the system (2.1) arises in which λ m = 0 for m = 1, 2, .…”

Section: Formulationmentioning

confidence: 99%

Systems of Integral Equations With Weighted Difference Kernels

Porter

Biggs

2004

Proceedings of the Edinburgh Mathematical Society

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Explicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding.

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A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Cited by 26 publications

References 39 publications

Integral equation methods for scattering by infinite rough surfaces

Integral equation methods for scattering by infinite rough surfaces

A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem

Systems of Integral Equations With Weighted Difference Kernels

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