Wavenumber-Explicit $hp$-BEM for High Frequency Scattering

Löhndorf, Maike; Melenk, Jens Markus

doi:10.1137/100786034

Cited by 37 publications

(80 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This bound shows that the number of degrees of freedom only needs to grow slighter faster than k 1=9 in order to maintain accuracy as k ! 1; this is to be contrasted with the linear growth required in conventional boundary element methods in two dimensions as proved in [42]. Preliminary results on implementing the star-combined formulation show that this property is realized in practice [39].…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

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

Spence

Chandler-Wilde

Graham

et al. 2011

Comm Pure Appl Math

View full text Add to dashboard Cite

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L 2 ./ (where is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k D !=c, where ! is the frequency and c is the speed of sound. The new boundary integral operator, which we call the "star-combined" potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L 2 ./ is proved without smoothness assumptions on except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

show abstract

Section: Discussionmentioning

confidence: 99%

“…"/, where N 0 . "/ depends only on ", and secondly that the polynomial degree is carefully chosen to depend logarithmically on k [42]. This result was obtained using a novel splitting of the operator A k;Á [45] motivated by a related numerical analysis for a domain-based (finite element) formulation [46].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

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

Spence

Chandler-Wilde

Graham

et al. 2011

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…Despite its lack of sharpness, the bound (1.36) is sufficient for the following numerical analysis application: Löhndorf and Melenk have recently performed a kexplicit convergence analysis of the Galerkin method applied to the integral equation (1.25) using piecewise-polynomial subspaces (the so-called hp-boundary-element method) [31], [34]. An underlying assumption in this analysis is that, when |η| ∼ k,…”

Section: Conditioning Of Boundary Integral Operatorsmentioning

confidence: 99%

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

Spence¹

2014

SIAM J. Math. Anal.

View full text Add to dashboard Cite

Abstract. We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, or (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-toDirichlet map when condition (i) and (ii) hold. We also prove bounds on the solutions of the interior and exterior impedance problems when the obstacle is a general Lipschitz domain. These bounds are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber. One motivation for proving these collection of bounds is that they can then be used to prove wavenumber-explicit bounds on the inverses of the standard second-kind integral operators used to solve the exterior Dirichlet, Neumann, and impedance problems for the Helmholtz equation.

show abstract

“…These issues are very well understood; see e.g. [9,10,37,30,12] and the many references therein.The problem of 'bridging the gap' between conventional numerical methods and fully asymptotic approaches has received a great deal of attention in recent years. Significant progress has been made in developing numerical methods which can achieve a prescribed level of accuracy at high frequencies with fewer degrees of freedom than conventional approaches.…”

mentioning

confidence: 99%

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

Hewett¹,

Langdon²,

Melenk³

2013

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. In this paper we propose and analyse a hybrid hp boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.Key words. High frequency scattering, Boundary Element Method, hp-method AMS subject classifications. 65N12, 65N38, 65R201. Introduction. Conventional numerical schemes for time-harmonic acoustic scattering problems, with piecewise polynomial approximation spaces, become prohibitively expensive in the high frequency regime where the scatterer is large compared to the wavelength of the incident wave. For two-dimensional problems the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to frequency. On the other hand, approximation via high frequency asymptotics alone is often insufficiently accurate when the frequency lies within ranges of practical interest. These issues are very well understood; see e.g. [9,10,37,30,12] and the many references therein.The problem of 'bridging the gap' between conventional numerical methods and fully asymptotic approaches has received a great deal of attention in recent years. Significant progress has been made in developing numerical methods which can achieve a prescribed level of accuracy at high frequencies with fewer degrees of freedom than conventional approaches. A key idea underpinning much recent work is to express the scattered field as a sum of products of known oscillatory functions, selected using knowledge of the high frequency asymptotics, with slowly oscillating amplitude functions, and to approximate just the amplitudes by piecewise polynomials (we call this the hybrid numerical-asymptotic approach). Applying this idea within a boundary element method (BEM) context is particularly attractive since in this case one need only understand the high frequency behaviour on the boundary of the scatterer, rather than throughout the whole propagation domain. Computational methods implementing this approach have been applied successfully to problems of scattering by both smooth [8, 21, 22, 26] and non-smooth [15, 20, 2, 29, 16] convex scatterers, the latter building on previous work on hybrid h-version BEM methods for the special problem of acoustic scat...

show abstract

Wavenumber-Explicit $hp$-BEM for High Frequency Scattering

Cited by 37 publications

References 16 publications

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

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

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

A High Frequency $hp$ Boundary Element Method for Scattering by Convex Polygons

Contact Info

Product

Resources

About