Adaptive BEM with optimal convergence rates for the Helmholtz equation

Alex Bespalov; Timo Betcke; Alexander Haberl; Dirk Praetorius

doi:10.1016/j.cma.2018.12.006

Adaptive BEM with optimal convergence rates for the Helmholtz equation

Alex Bespalov, Timo Betcke, Alexander Haberl, Dirk Praetorius

Mathematics

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7 Citations (Scopus)

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Abstract

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

Original language	English
Pages (from-to)	260-287
Number of pages	34
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	346
Early online date	12 Dec 2018
DOIs	https://doi.org/10.1016/j.cma.2018.12.006
Publication status	Published - 1 Apr 2019

Keywords

boundary element method
Helmholtz equation
a posteriori error estimate
adaptive algorithm
convergence
optimality

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10.1016/j.cma.2018.12.006

Bespalov_et_al_Adaptive_BEM-with_optimal_convergence_CMAME_2018Accepted author manuscript, 1.92 MBLicence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

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title = "Adaptive BEM with optimal convergence rates for the Helmholtz equation",

abstract = "We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.",

keywords = "boundary element method, Helmholtz equation, a posteriori error estimate, adaptive algorithm, convergence, optimality",

author = "Alex Bespalov and Timo Betcke and Alexander Haberl and Dirk Praetorius",

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T1 - Adaptive BEM with optimal convergence rates for the Helmholtz equation

AU - Bespalov, Alex

AU - Betcke, Timo

AU - Haberl, Alexander

AU - Praetorius, Dirk

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

AB - We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

KW - boundary element method

KW - Helmholtz equation

KW - a posteriori error estimate

KW - adaptive algorithm

KW - convergence

KW - optimality

UR - https://arxiv.org/abs/1807.11802

U2 - 10.1016/j.cma.2018.12.006

DO - 10.1016/j.cma.2018.12.006

M3 - Article

SN - 0045-7825

VL - 346

SP - 260

EP - 287

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -

Adaptive BEM with optimal convergence rates for the Helmholtz equation

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Keywords

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