The Validity of Johnson--Nédélec's BEM--FEM Coupling on Polygonal Interfaces

Sayas, Francisco–Javier

doi:10.1137/120892283

Cited by 38 publications

(57 citation statements)

References 13 publications

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“…The other two methods we will discuss in this work are very well known in the applied mathematics literature and often referred to as the Johnson-Nédélec coupling JN [42,28,35] and the Bielak-MacCamy coupling BMC [5]. From our point of view JN might be the most natural way to deal with the unbounded domain problem in the framework of a finite element method.…”

Section: Introductionmentioning

confidence: 99%

FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains

Faugeras

Heumann

2017

Journal of Computational Physics

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Incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing. The combination of finite element methods (FEM) and boundary element methods (BEM) is the obvious instrument, and we adapt here for the first time the two standard FEM-BEM coupling approaches to the free-boundary equilibrium problem: the Johnson-Nédélec coupling and the Bielak-MacCamy coupling. We recall also the classical approach for fusion applications, dubbed according to its first appearance von-Hagenow-Lackner coupling and present the less used alternative introduced by Albanese, Blum and de Barbieri in [2]. We show that the von-Hagenow-Lackner coupling suffers from undesirable non-optimal convergence properties, that suggest that other coupling schemes, in particular Johnson-Nédélec or Albanese-Blum-de Barbieri are more appropriate for non-linear equilibrium problems. Moreover, we show that any of such coupling methods requires Newton-like iteration schemes for solving the corresponding non-linear discrete algebraic systems.

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Section: Introductionmentioning

confidence: 99%

FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains

Faugeras

Heumann

2017

Journal of Computational Physics

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“…• Based on an argument of Sayas [Say09], Steinbach [Ste11] showed that the nonsymmetric coupling of the elliptic-elliptic interface problem with a lowest order term in the interior domain in fact leads to a coercive variational formulation; see also [EOS17]. This allows us to extend the results of [MS87,CES90] to the non-symmetric coupling method on non-smooth domains.…”

Section: Introductionmentioning

confidence: 98%

On the Nonsymmetric Coupling Method for Parabolic-Elliptic Interface Problems

Egger¹,

Erath²,

Schorr³

2018

SIAM J. Numer. Anal.

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We consider the numerical approximation of parabolic-elliptic interface problems by the non-symmetric coupling method of MacCamy and Suri [Quart. Appl. Math., 44 (1987), pp. 675-690]. We establish well-posedness of this formulation for problems with non-smooth interfaces and prove quasi-optimality for a class of conforming Galerkin approximations in space. Therefore, error estimates with optimal order can be deduced for the semi-discretization in space by appropriate finite and boundary elements. Moreover, we investigate the subsequent discretization in time by a variant of the implicit Euler method. As for the semi-discretization, we establish well-posedness and quasi-optimality for the fully discrete scheme under minimal regularity assumptions on the solution. Error estimates with optimal order follow again directly. Our analysis is based on estimates in appropriate energy norms. Thus, we do not use duality arguments and corresponding estimates for an elliptic projection which are not available for the non-symmetric coupling method. Additionally, we provide again error estimates under minimal regularity assumptions. Some numerical examples illustrate our theoretical results.

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“…The review article [43] describes some theoretical validations of the coupling approaches considered in the earlier literature and delicate choices of the coupling interface. The coupling methods in [5,6,31,32,43] lead to very large algebraic systems with both dense and sparse structures. For wave propagation models, given the complexity involved in even separately solving the FEM and BEM algebraic systems, it is efficient to avoid large combined dense and sparse structured systems arising from the coupling methods in [5,6,31,32,43].…”

Section: Introductionmentioning

confidence: 99%

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

Domínguez

Ganesh

Sayas

2020

Journal of Computational Physics

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A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model.The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to setup a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.

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The Validity of Johnson--Nédélec's BEM--FEM Coupling on Polygonal Interfaces

Cited by 38 publications

References 13 publications

FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains

FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains

On the Nonsymmetric Coupling Method for Parabolic-Elliptic Interface Problems

An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

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