Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation

Feischl, Michael; Führer, Thomas; Karkulik, Michael; Melenk, Jens Markus; Praetorius, Dirk

doi:10.1007/s10092-013-0100-x

Cited by 33 publications

(62 citation statements)

References 19 publications

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“…Recently, convergence of Algorithm 2.2 has been shown even with quasi-optimal rates, if ℓ ( ) = ℓ ( ) is used for marking (2.6); see [31,34]. We stress that our approach with ℓ ( ) = ℓ ( ) = ℓ ( ) would also give convergence ℓ ( ) → 0 as ℓ → ∞.…”

Section: Weighted-residual Error Estimatormentioning

confidence: 84%

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Feischl¹,

Führer²,

Mitscha-Eibl

et al. 2014

Computational Methods in Applied Mathematics

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Abstract:We analyze adaptive mesh-re ning algorithms in the frame of boundary element methods (BEM) and the coupling of nite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

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Section: Weighted-residual Error Estimatormentioning

confidence: 84%

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Feischl¹,

Führer²,

Mitscha-Eibl

et al. 2014

Computational Methods in Applied Mathematics

Self Cite

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show abstract

“…Moreover, even quasi-optimal rates can be guaranteed for polynomial ansatz of arbitrary, but fixed polynomial order [17,18].…”

Section: State Of the Artmentioning

confidence: 99%

“…The validity of such an estimate is crucial for proving quasi-optimal convergence rates for residual-based ABEM in [19,36] as well as [3,17,18].…”

Section: Boundedness Of K-mesh Constantmentioning

confidence: 99%

“…which is an equivalent formulation of (18). The exact solution φ ∈ H −1/2 ( ) of (21) is the normal derivative φ = ∂ n u of the solution u ∈ H 1 ( ) of (18 Formally, the Dirichlet data satisfy g ∈ H 1/2 ( ).…”

Section: Continuous Model Problemmentioning

confidence: 99%

“…The exact solution φ ∈ H −1/2 ( ) of (21) is the normal derivative φ = ∂ n u of the solution u ∈ H 1 ( ) of (18 Formally, the Dirichlet data satisfy g ∈ H 1/2 ( ). However, we assume additional regularity g ∈ H 1 ( ) ⊂ H 1/2 ( ) so that g is, in particular, continuous.…”

Section: Continuous Model Problemmentioning

confidence: 99%

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HILBERT — a MATLAB implementation of adaptive 2D-BEM

et al. 2013

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We report on the MATLAB program package HILBERT. It provides an easily-accessible implementation of lowest-order adaptive Galerkin boundary element methods for the numerical solution of the Poisson equation in 2D. The library was designed to serve several purposes: The stable implementation of the integral operators may be used in research code. The framework of MATLAB ensures usability in lectures on boundary element methods or scientific computing. Finally, we emphasize the use of adaptivity as general concept and for boundary element methods in particular. In this work, we summarize recent analytical results on adaptivity in the context of BEM and illustrate the use of HILBERT. Various benchmarks are performed to empirically analyze the performance of the proposed adaptive algorithms and to compare adaptive and uniform mesh-refinements. In particular, we do not only focus on mathematical convergence behavior but also on the usage of critical system resources such as memory consumption and computational time. In any case, the superiority of the proposed adaptive approach is empirically supported.

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A-Bem

Gwinner¹,

Stephan²

2018

Springer Series in Computational Mathematics

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Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation

Cited by 33 publications

References 19 publications

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

HILBERT — a MATLAB implementation of adaptive 2D-BEM

A-Bem

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