A new conservative numerical scheme for flow problems on unstructured grids and unbounded domains

Erath, Christoph

doi:10.1016/j.jcp.2013.03.055

Cited by 9 publications

(19 citation statements)

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“…These two formulations are than coupled with the aid of (1d) and (1e). For more details we refer again to [15]. If we write all unknowns on the left-hand side, the discrete problem for the cell-centered FVM-BEM coupling reads for a ∞ = 0:…”

Section: The Discrete Systemmentioning

confidence: 99%

“…We use the Morley interpolation of [18] and extend it in an appropriate way for the coupling problem. We stress that the analysis differs substantially from that in [15,16]. More precisely, the proof for reliability, i.e., to show an upper bound for a certain error, relies on two Helmholtz decompositions of the discontinuous interior Morley error which also affects the exterior part.…”

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confidence: 96%

“…The normal vector n on Γ is pointing outward with respect to Ω. In this work we consider the same discrete coupling approach as in [15], but in contrast we focus on nonconforming a posteriori estimates of Morley-type for the cell-centered FVM part; i.e., we will build a nonconforming Morley-type interpolant based on the piecewise constant FVM solution in Ω. The BEM part, which basically approximates the model problem in Ω e , involves the usual quantities.…”

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confidence: 99%

“…A tangential derivative jump on the coupling boundary measures the error of the tangential derivative between the trace solution of the exterior problem and a piecewise affine and globally continuous solution on the coupling boundary, which is generated from the interior solution. We remark, that the coupling of the cell-centered FVM and the BEM [13,15] has an additional block in the discrete system compared to the vertex centered FVM-BEM [13,14] or a finite element boundary element coupling [4]. This results from further continuous ansatz functions on the interface to link the discontinuous displacement field from the cell-centered FVM to necessarily continuous boundary ansatz functions from the BEM on the coupling boundary.…”

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confidence: 99%

“…The numerical examples confirm the theoretical results. In the last problem we even compare our new approach with some results of [15], where an adaptive coupling is done with a conforming Morley-type interpolation for the FVM part. Some conclusions close the paper.…”

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A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

Erath

2014

Numer. Math.

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The coupling of the cell-centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain, where local flux conservation is important. Based on the piecewise constant interior finite volume solution we define a Morley-type interpolant built on a nonconforming finite element. Together with the Cauchy data of the exterior boundary element solution this allows us to define a residual-based a posteriori error estimator. With respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh-refining algorithm. In two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming Morley interpolant.Mathematics Subject Classification 65N08 · 65N38 · 65N15 · 65N50 · 76M12 · 76M15

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Section: The Discrete Systemmentioning

confidence: 99%

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confidence: 96%

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A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

Erath

2014

Numer. Math.

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A non symmetric FVM‐BEM coupling method

Erath

Sayas

2016

Proc Appl Math and Mech

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We consider the prototype model for flow and transport of a concentration in porous media in an interior domain and couple it with a diffusion process in the corresponding unbounded exterior domain. To guarantee mass conservation and stability with respect to dominating convection also for a discrete solution we introduce a non symmetric coupling of the vertex‐centered finite volume method (FVM) and the boundary element method (BEM). BEM approximates the unbounded exterior problem which avoids truncation of the domain. One can also interpret that the (unbounded) exterior problem “replaces” the boundary conditions of the interior problem. We aim to provide a first rigorous analysis of the discrete system for different model parameters; existence and uniqueness, convergence, and a priori estimates. Numerical examples illustrate the strength of the chosen method which is computational cheaper than the previous three field FVM‐BEM couplings. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Comparison of Two Couplings of the Finite Volume Method and the Boundary Element Method

Erath

2014

Springer Proceedings in Mathematics &Amp; Statistics

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A new conservative numerical scheme for flow problems on unstructured grids and unbounded domains

Cited by 9 publications

References 13 publications

A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

A non symmetric FVM‐BEM coupling method

Comparison of Two Couplings of the Finite Volume Method and the Boundary Element Method

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