In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for value problems Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These combine conventional piecewise polynomial approximations with high-frequency asymptotics build basis functions suitable representing oscillatory solutions. They have potential solve accurately a computation is...

We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal in 2D and 3D space. For an arbitrary compact scatterer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mi mathvariant="italic">Γ</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> we reformulate the Dirichlet boundary value problem for Helmholtz equation as a first kind integral (IE) on...

In this paper we study, via variational methods, the problem of scattering time harmonic acoustic waves by an unbounded sound soft surface. The boundary $partial D$ is assumed to lie within a finite distance flat plane and incident wave that arising from inhomogeneous term in Helmholtz equation whose support lies some $\partial D$. Via analysis equivalent formulation, provide first proof existence unique solution three-dimensional rough surface for arbitrary number. Our method does not...

In this paper we consider the problem of scattering time-harmonic acoustic waves by a bounded, sound soft obstacle in two and three dimensions, studying dependence on wave number classical formulations problem. The first is standard weak formulation part exterior domain contained large sphere, with an exact Dirichlet-to-Neumann map applied boundary. second as kind boundary integral equation which solution sought combined single- double-layer potential. For variational obtain, case when...

In this paper we consider the problem of time‐harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for problems have a computational cost that grows at least linearly as function frequency incident wave. Here present novel Galerkin method, which uses an approximation space consisting products plane waves with piecewise polynomials supported on graded mesh, smaller elements closer to corners polygon. We prove best from requires number...

Abstract We consider the classical coupled, combined‐field integral equation formulations for time‐harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on L 2 condition numbers these also norms of single‐ double‐layer potential operators. These to some extent make explicit dependence wave number k , geometry scatterer, coupling parameter. For example, with usual choice parameter they show that, while grows like 1/3 as → ∞ when...

This paper provides an overview of interpolation Banach and Hilbert spaces, with a focus on establishing when equivalence norms is in fact equality the key results theory. (In brief, our conclusion for space case that, right normalizations, all hold norms.) In final section we apply to Sobolev spaces , open . We exhibit examples one two dimensions sets which these scales are not scales. cases where they (in particular, if Lipschitz) that show general, norm does coincide intrinsic and, fact,...

We consider the problem of scattering time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance some plane. The paper concerned with study equivalent variational formulation this set in scale weighted Sobolev spaces. prove well-posedness energy space weights extends previous results unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] more general inhomogeneous terms Helmholtz equation. In...

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

Consider the Dirichlet boundary value problem for Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This models time-harmonic electromagnetic scattering transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced problem, which generalization of usual one used study diffraction gratings when solution quasi-periodic, and allows variety incident fields including plane wave...

We consider a two-dimensional problem of scattering time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on perfectly plate. The magnetic permeability is assumed to be fixed positive constant in the media. material properties media are characterized completely index refraction, which bounded measurable function and above corresponding homogeneous medium. In this paper, we only examine TM (transverse magnetic) polarization case. A radiation condition...

Abstract In this paper, we consider the Dirichlet and impedance boundary value problems for Helmholtz equation in a non‐locally perturbed half‐plane. These arise study of time‐harmonic acoustic scattering an incident field by sound‐soft, infinite rough surface where total vanishes (the problem) or infinite, satisfies homogeneous condition problem). We propose new integral formulation problem, utilizing combined double‐ single‐layer potential half‐plane Green's function. For problem two...

In this paper we consider the impedance boundary value problem for Helmholtz equation in a half-plane with piecewise constant data, which models, example, outdoor sound propagation over inhomogeneous flat terrain. To achieve good approximation at high frequencies relatively low number of degrees freedom, propose novel Galerkin element method, using graded mesh smaller elements adjacent to discontinuities and special set basis functions so that, on each element, space contains polynomials (of...

Abstract A new boundary integral operator is introduced for the solution of soundsoft acoustic scattering problem, i.e., exterior problem Helmholtz equation with Dirichlet conditions. We prove that this coercive in L 2 (Γ) (where Γ surface scatterer) all Lipschitz star‐shaped domains. Moreover, coercivity uniform wavenumber k = ω/ c , where ω frequency and speed sound. The operator, which we call “star‐combined” potential a slight modification standard combined shown to be as easy implement...

Journal Article A frequency-independent boundary element method for scattering by two-dimensional screens and apertures Get access D. P. Hewett, Hewett * Department of Mathematics Statistics, University Reading, UK *Corresponding author: hewett@maths.ox.ac.uks.langdon@reading.ac.uks.n.chandler-wilde@reading.ac.uk Search other works this author on: Oxford Academic Google Scholar S. Langdon, Langdon N. Chandler-Wilde IMA Numerical Analysis, Volume 35, Issue 4, October 2015, Pages 1698–1728,...

We consider the Dirichlet boundary–value problem for Helmholtz equation in a non–locally perturbed half–plane. This models time–harmonic electromagnetic scattering by one–dimensional infinite rough perfectly conducting surface; same arises acoustic sound–soft surface. Chandler–Wilde and Zhang have suggested radiation condition this problem, generalization of Rayleigh expansion diffraction gratings, uniqueness solution has been established. Recently, an integral formulation also proposed and,...