Energy norm based a posteriori error estimation for boundary element methods in two dimensions

Erath, Christoph; Ferraz-Leite, Samuel; Funken, Stefan A.; Praetorius, Dirk

doi:10.1016/j.apnum.2008.12.024

Cited by 29 publications

(38 citation statements)

References 19 publications

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“…where the constant depends only on Γ , see also [18], Lemma 2.1. The combination of the last three estimates thus yields η μ and concludes the proof.…”

Section: Lemma 38mentioning

confidence: 99%

“…In [20], localized variants of η were introduced. In [18,19] the equivalence of η to hierarchical two-level error estimators from [24,28] and averaging error estimators from [9][10][11] has been analyzed.…”

Section: Introduction and Overviewmentioning

confidence: 99%

“…We apply a symmetric coupling method [13,14] and the lowest-order Galerkin scheme to obtain a (nonlinear) system of coupled FEM-BEM equations. For linear problems, the ideas from [18,19] can be used to prove that the introduced error estimators are equivalent to the two-level error estimator from [27].…”

Section: Introduction and Overviewmentioning

confidence: 99%

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Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

2012

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Abstract.We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear) interface problem for the 2D Laplacian. We introduce some new a posteriori error estimators based on the (h − h/2)-error estimation strategy. In particular, these include the approximation error for the boundary data, which allows to work with discrete boundary integral operators only. Using the concept of estimator reduction, we prove that the proposed adaptive algorithm is convergent in the sense that it drives the underlying error estimator to zero. Numerical experiments underline the reliability and efficiency of the considered adaptive mesh-refinement.Mathematics Subject Classification. 65N30, 65N15, 65N38.

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“…where the constant depends only on Γ , see also [18], Lemma 2.1. The combination of the last three estimates thus yields η μ and concludes the proof.…”

Section: Lemma 38mentioning

confidence: 99%

Section: Introduction and Overviewmentioning

confidence: 99%

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Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

2012

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“…Since the H 1/2 -norm is non-local, the a posteriori error analysis requires appropriate localization techniques. These have recently been developed in the context of adaptive boundary element methods [2,10,11,15,16,20]: Under certain orthogonality properties of g − g ℓ ∈ H 1 (Γ D ), the natural trace norm g − g ℓ H 1/2 (Γ D ) is bounded by a locally weighted H 1 -seminorm…”

mentioning

confidence: 99%

Convergence and quasi‐optimality of adaptive FEM with inhomogeneous Dirichlet data

Feischl

Page

Praetorius

2011

Proc Appl Math and Mech

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Abstract. We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with quasi-optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L 2 -projection or the Scott-Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott-Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.

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“…Based on certain localization techniques, the works [8,9,15,16,21], for instance, give suitable error estimators μ 2 = E∈E μ (E) 2 which satisfy η μ , but whose contributions μ (E), at least heuristically, measure the local error. Moreover, based on recent results from [5], a compound error estimator ρ which additionally controls data oscillations, is used for all problem settings discussed in this paper.…”

Section: (H − H/2)-type Error Estimatorsmentioning

confidence: 99%

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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Energy norm based a posteriori error estimation for boundary element methods in two dimensions

Cited by 29 publications

References 19 publications

Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems

Convergence and quasi‐optimality of adaptive FEM with inhomogeneous Dirichlet data

HILBERT — a MATLAB implementation of adaptive 2D-BEM

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