A non-symmetric coupling of the finite volume method and the boundary element method

Erath, Christoph; Sayas, Francisco–Javier

doi:10.1007/s00211-016-0820-3

Cited by 12 publications

(39 citation statements)

References 27 publications

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“…For the analysis of the system (28) we employ some results from the stationary FVM-BEM coupling [EOS17]. The main idea is to measure the discrete difference between the right-hand sides and the bilinear forms (10) and (26):…”

Section: Semi-discretizationmentioning

confidence: 99%

“…Remark 15. The restriction b·n ∈ P 0 (E in Γ ) in [EOS17,Lemma 7], where E in Γ denotes the set of all edges on the inflow boundary Γ in , results from the estimate [EOS17, Lemma 6]. However, this is not necessary.…”

Section: Semi-discretizationmentioning

confidence: 99%

“…see [Era10] for details. Hence φ − φ n h L 2 (0,T ;V) is an equivalent norm to φ − φ n h L 2 (0,T ;H −1/2 (Γ)) .…”

Section: Numerical Illustrationmentioning

confidence: 99%

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Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Erath

Schorr²

2019

Computational Methods in Applied Mathematics

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Many problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [EES17]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability.A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

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Section: Semi-discretizationmentioning

confidence: 99%

Section: Semi-discretizationmentioning

confidence: 99%

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Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Erath

Schorr²

2019

Computational Methods in Applied Mathematics

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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: 97%

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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“…We can also interpret the model that the (unbounded) exterior problem "replaces" the (unknown) boundary conditions of the interior problem [Era12, Remark 2.1]. Recently, the non-symmetric vertex-centered FVM-BEM coupling approach was introduced in [EOS15], which results in a smaller system of linear equations than the previous three field coupling approach cited above. However, a posteriori estimators for this kind of FVM-BEM coupling were not developed.…”

Section: Introduction and Model Problemmentioning

confidence: 99%

An Adaptive Nonsymmetric Finite Volume and Boundary Element Coupling Method for a Fluid Mechanics Interface Problem

Erath¹,

Schorr²

2017

SIAM J. Sci. Comput.

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Abstract. We consider an interface problem often arising in transport problems: a coupled system of partial differential equations with one (elliptic) transport equation on a bounded domain and one equation (in this case the Laplace problem) on the complement, an unbounded domain. Based on the non-symmetric coupling of the finite volume method and boundary element method of [EOS15] we introduce a robust residual error estimator. The upper bound of the error in an energy (semi)norm is robust against variation of the model data. The lower bound, however, additionally depends on the Péclet number. In several examples we use the local contributions of the a posteriori error estimator to steer an adaptive mesh-refining algorithm. The adaptive FVM-BEM coupling turns out to be an efficient method especially to solve problems from fluid mechanics, mainly because of the local flux conservation and the stable approximation of convection dominated problems.

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A non-symmetric coupling of the finite volume method and the boundary element method

Cited by 12 publications

References 27 publications

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

On the Nonsymmetric Coupling Method for Parabolic-Elliptic Interface Problems

An Adaptive Nonsymmetric Finite Volume and Boundary Element Coupling Method for a Fluid Mechanics Interface Problem

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