A priori error estimates for a time‐dependent boundary element method for the acoustic wave equation in a half‐space

Gimperlein, Heiko; Nezhi, Zouhair; Stephan, Ernst P.

doi:10.1002/mma.3340

Cited by 25 publications

(33 citation statements)

References 16 publications

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“…Together with the a priori estimates for the time domain boundary element methods on screens [23,24], our results imply convergence rates for the p-version Galerkin approximations which are twice those observed for the quasi-uniform h-method in [22].…”

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confidence: 58%

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hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Gimperlein

Özdemir

Stark

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R 3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.This article initiates the study of high-order boundary elements in the time domain. For elliptic problems, p-and hp-versions of the finite element method give rise to fast approximations of both smooth solutions and geometric singularities. These methods converge to the solution by increasing the polynomial degree p of the elements, possibly in combination with reducing the mesh size h of the quasi-uniform mesh. They were first investigated in the group of Babuska [3,4,17,18]. See [49] for a comprehensive analysis for 2d problems.The analogous p-and hp-versions of the boundary element method go back to [2,53,54]. More recent optimal convergence results for boundary elements on screens and polyhedral surfaces covering 3d problems have been obtained, for example, in [9,10,11,12,13].Boundary element methods for time dependent problems have recently become of interest [48]. In this article we introduce a space-time hp-version of the time domain boundary element method for the wave equation with non-homogeneous Dirichlet or Neumann boundary conditions.

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confidence: 58%

“…These operators are studied in space-time anisotropic Sobolev spaces H r σ (R + , H s (Γ)), see [23] or [29]. To define the spaces for ∂Γ = ∅, extend Γ to a closed, orientable Lipschitz manifold Γ.…”

Section: Boundary Integral Operators and Sobolev Spacesmentioning

confidence: 99%

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Gimperlein

Özdemir

Stark

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…(2.14), p. 174 in [12]. For the half-space or when ∂Γ = ∅, a) is shown in [9]; the proof of b) is obtained by extending Ha Duong's proof in [13] for ∂Γ = ∅, using the modifications from [9]. The mapping and coercivity properties give a basic well-posedness theorem for the integral equations (2.3) and (2.6).…”

Section: Boundary Integral Equationsmentioning

confidence: 99%

“…Cruse and Rizzo [4]. A first mathematical analysis of time dependent boundary element methods goes back to Bamberger and Ha-Duong [1,12], see also [9] for Dirichlet and acoustic boundary problems in a half-space. First numerical experiments for integral equations of the second kind in the full space were reported by Ding et al [5], and the practical realization of the numerical marching-on-in-time scheme include the Ph.D. thesis of Terrasse [19] as well as [14].…”

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Time Domain Boundary Element Methods for the Neumann Problem: Error Estimates and Acoustic Problems

Gimperlein¹,

Oezdemir

Stephan³

2018

JCM

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We investigate time domain boundary element methods for the wave equation in R 3 , with a view towards sound emission problems in computational acoustics. The Neumann problem is reduced to a time dependent integral equation for the hypersingular operator, and we present a priori and a posteriori error estimates for conforming Galerkin approximations in the more general case of a screen. Numerical experiments validate the convergence of our boundary element scheme and compare it with the numerical approximations obtained from an integral equation of the second kind. Computations in a half-space illustrate the influence of the reflection properties of a flat street.Mathematics subject classification: 65N38, 65R20, 74J05.

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Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

Gimperlein,

Meyer,

Özdemir

2024

Numerical Meth Engineering

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SummaryAcoustic emission or scattering problems naturally involve uncertainties about the sound sources or boundary conditions. This article initiates the study of time domain boundary elements for such stochastic boundary problems for the acoustic wave equation. We present a space‐time stochastic Galerkin boundary element method which is applied to sound‐hard, sound‐soft and absorbing scatterers. Uncertainties in both the sources and the boundary conditions are considered using a polynomial chaos expansion. The numerical experiments illustrate the performance and convergence of the proposed method in model problems and present an application to a problem from traffic noise.

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A priori error estimates for a time‐dependent boundary element method for the acoustic wave equation in a half‐space

Cited by 25 publications

References 16 publications

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Time Domain Boundary Element Methods for the Neumann Problem: Error Estimates and Acoustic Problems

Space‐time stochastic Galerkin boundary elements for acoustic scattering problems

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