Convergence of adaptive boundary element methods

Carstensen, Carsten; Praetorius, Dirk

doi:10.1216/jie-2012-24-1-1

Cited by 11 publications

(7 citation statements)

References 46 publications

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“…First, we prove that (A2) implies boundedness of ( ℓ ) ℓ∈ℕ 0 . We recall that nestedness X ℓ ⊆ X ℓ+1 for all ℓ ∈ ℕ 0 in combination with the Céa lemma (2.5) implies that the limit lim ℓ ℓ ( ) =: ∞ ( ) exists in H, see, e.g., [7,17,48] or even the pioneering work [8]. For ℓ = 0 and = 1, assumption (A2) implies…”

Section: Abstract Convergence Analysismentioning

confidence: 98%

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: Abstract Convergence Analysismentioning

confidence: 98%

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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“…Put differently, the set M ' satisfies the Do¨rfler marking (16) To see this, it remains to verify that the obstacle problem leads to a priori convergence lim ' U ' ¼ u 1 with a certain limit u 1 2 H. For linear problems, such a result is found in [25][26][27], and we refer to [24,Lemma 3.3.8] for the proof of the a priori convergence in our non-linear setting. Then, (17) takes the form…”

Section: Convergent Adaptive Algorithmmentioning

confidence: 99%

Convergence of adaptive FEM for some elliptic obstacle problem

Page¹,

Praetorius²

2013

Applicable Analysis

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In this work, we treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator from [D. Braess, C. Carstensen, and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), pp. 455-471]. We extend recent ideas from [J. Cascon, C. Kreuzer, R. Nochetto, and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008Anal. 46 ( ), pp. 2524Anal. 46 ( -2550 for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum of energy error, edge residuals and data oscillations satisfies a contraction property within each step of the adaptive feedback loop. This result is superior to a prior result from Braess et al. (2007) in two ways: first, it is unnecessary to control the decay of the data oscillations explicitly; second, our analysis avoids the use of some discrete local efficiency estimate so that the local mesh-refinement is fairly arbitrary.

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“…The first work on convergence of ABEM known to the authors is [16], where convergence is guaranteed by use of a feedback control which occasionally leads to uniform refinements by a numerical check of the saturation assumption. This result is somewhat unsatisfactory, since the feedback control is computationally expensive and seemed to be unnecessary in practice.…”

Section: Introductionmentioning

confidence: 99%

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

Feischl

Karkulik²,

Melenk³

et al. 2013

SIAM J. Numer. Anal.

106

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For the simple layer potential V that is associated with the 3D Laplacian, we consider the weakly singular integral equation V φ = f. This equation is discretized by the lowest order Galerkin boundary element method. We prove convergence of an h-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorithm converges quasi-optimally with respect to the number of elements. In particular, we prove that adaptive mesh refinement is superior to uniform mesh refinement.

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Convergence of adaptive boundary element methods

Cited by 11 publications

References 46 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

Convergence of adaptive FEM for some elliptic obstacle problem

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

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