Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

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

doi:10.1007/s00466-012-0779-6

Cited by 49 publications

(110 citation statements)

References 58 publications

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“…Starting from an initial mesh T 0 , the triangulations T ℓ are successively re ned by means of the following realization of (1.1), where ℓ ( ) := ℓ ( ; T ℓ ) with ℓ ( ; E ℓ ) := ∈E ℓ ℓ ( ; ) 2 …”

Section: Adaptive Algorithmmentioning

confidence: 99%

“…ℓ ( ) being the two-level error estimator for BEM [4, 24-26, 39, 41, 45, 50, 51] and the FEM-BEM coupling [5,37,49]. In either case, ℓ ( ) denotes some weighted-residual error estimator, see [11,14,15,19,20] for BEM and [2,18,37] for the FEM-BEM coupling.…”

Section: Remarksmentioning

confidence: 99%

“…Following [18], it is proved in [5] for linear problems and in [2] for strongly monotone problems that…”

Section: Weighted-residual Error Estimatormentioning

confidence: 99%

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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: Adaptive Algorithmmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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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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“…Finally, the near field integrals have to be stored, which is at most νM, ν = max j |I NF (S j )| numbers. Overall, the storage requirements 5 are…”

Section: Errors Complexity and Storagementioning

confidence: 99%

“…Within this setting the existence of a unique solution to (1) has been proved. For details we refer the reader to [5,6] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

Exl

Schrefl

2014

Journal of Computational Physics

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We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M + N + N log N) for N nodes and M surface triangles. We confirm our approach by several numerical tests.

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Coupling of Boundary Element Methods and Finite Element Methods

Stephan

2017

Encyclopedia of Computational Mechanics Second Edition

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For the numerical treatment of elliptic boundary value problems, the coupling of boundary element methods (BEM) and finite element methods (FEM) is widely used. Usually, the FEM is applied in a bounded subdomain, whereas the BEM subdomain may be unbounded, but without inhomogeneous or nonlinear material behavior. This article gives a survey of FE/BE coupling for elliptic interface problems from a mathematical perspective. We concentrate on the symmetric coupling method because of its general applicability and give a priori and a posteriori error estimates. Further, we address implementational aspects such as suitable preconditioning for appropriate solvers of the Galerkin equations. As new developments, we consider a least squares coupling method with first‐order systems in the FEM domain and consider both primal and dual FE/BE coupling methods for interface problems with Signorini contact condition. Besides standard formulations, mixed FEM formulations are also considered. Applications in elasticity are discussed. We present proofs of essential theorems because we feel that these proofs are essential for understanding the mathematical aspects. We also present illustrative numerical examples and provide a list of references to the current literature. The new version contains sections on convergence of an adaptive coupling algorithm and on nonsymmetric coupling methods

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Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity

Cited by 49 publications

References 58 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

Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

Coupling of Boundary Element Methods and Finite Element Methods

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