Adaptive BEM with optimal convergence rates for the Helmholtz equation

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 suffi...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering 2019-04, Vol.346, p.260-287
Main Authors: Bespalov, Alex, Betcke, Timo, Haberl, Alexander, Praetorius, Dirk
Format: Article
Language:English
Subjects:
A posteriori error estimate
Adaptive algorithm
Adaptive algorithms
Boundary element method
Convergence
Helmholtz equation
Helmholtz equations
Integral equations
Mathematical analysis
Nonlinear programming
Operators (mathematics)
Optimality
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Summary: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. •Adaptive mesh-refinement stabilizes BEM computations for indefinite problems.•Local inverse-type estimates for the integral operators of the Helmholtz equation.•The analysis avoids any assumption on the initial mesh.•The algorithm guarantees optimal convergence rates.•Numerical experiments for 3D acoustic scattering problems.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2018.12.006