A high-frequency boundary element method for scattering by a class of multiple obstacles

Gibbs, Andrew; Chandler-Wilde, Simon N.; Langdon, Stephen; Moiola, Andrea

doi:10.1093/imanum/draa025

Cited by 5 publications

(3 citation statements)

References 31 publications

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“…The frequency-dependence of the norms of both Helmholtz boundary-integral operators and their inverses has been studied since the work of Kress and Spassov [59,58] and Amini [2], who studied the case when the obstacle is a ball. Over the last 15 years there has been renewed interest in this dependence at high-frequency [20,10,34,30,24,23,14,12,77,66,13,75,26,42,47,78,11,43,31], motivated mainly by its importance in the analysis of associated boundary-element methods [29,44,63,25,51,27,46,50,41,45]. Almost all of the analysis of boundary-integral operators for the high-frequency Helmholtz equation has been for the exterior Dirichlet problem.…”

Section: Motivation and Informal Discussion Of The Main Results And T...mentioning

confidence: 99%

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

Galkowski¹,

Marchand²,

Spence³

2021

Preprint

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We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L 2 (Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp, and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is curved. Together, these results give bounds on the condition number of the operator on L 2 (Γ); this is the first time L 2 (Γ) condition-number bounds have been proved for this operator for obstacles other than balls.

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Section: Motivation and Informal Discussion Of The Main Results And T...mentioning

confidence: 99%

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

Galkowski¹,

Marchand²,

Spence³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Over the last 15 years there has been renewed interest in this dependence at high-frequency [11][12][13][14][15]21,26,27,29,33,34,37,44,45,51,70,81,83,84], motivated mainly by its importance in the analysis of associated boundaryelement methods [28,30,32,43,[48][49][50]54,55,67]. Almost all of the analysis of boundary-integral operators for the high-frequency Helmholtz equation has been for the exterior Dirichlet problem.…”

Section: Introduction 1motivation and Informal Discussion Of The Main...mentioning

confidence: 99%

High-Frequency Estimates on Boundary Integral Operators for the Helmholtz Exterior Neumann Problem

Galkowski

Marchand

Spence

2022

Integr. Equ. Oper. Theory

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We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $$\Gamma $$ Γ for the boundary of the obstacle, the relevant integral operators map $$L^2(\Gamma )$$ L 2 ( Γ ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $$\Gamma $$ Γ and are sharp up to factors of $$\log k$$ log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $$\Gamma $$ Γ and are observed to be sharp at least when $$\Gamma $$ Γ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on $$L^2(\Gamma )$$ L 2 ( Γ ) ; this is the first time $$L^2(\Gamma )$$ L 2 ( Γ ) condition-number bounds have been proved for this operator for obstacles other than balls.

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“…A closely-related activity is the design and analysis of numerical methods for the Helmholtz equation based on proving new results about the k → ∞ asymptotics of Helmholtz solutions for polygonal obstacles; see[20,21,58,[65][66][67] …”

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confidence: 99%

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

2021

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For the h-finite-element method (h-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth h must decrease with the frequency k to maintain accuracy as k increases has been studied since the mid 80’s. Nevertheless, there still do not exist in the literature any k-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, p, equal to one), the condition “$$h^2 k^3$$ h 2 k 3 sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of k) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for $$p\ge 2$$ p ≥ 2 . A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.

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A high-frequency boundary element method for scattering by a class of multiple obstacles

Cited by 5 publications

References 31 publications

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

High-Frequency Estimates on Boundary Integral Operators for the Helmholtz Exterior Neumann Problem

A sharp relative-error bound for the Helmholtz h-FEM at high frequency

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