A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods

Droniou, Jérôme; Eymard, Robert; Gallouët, Thierry; Herbin, Raphaèle

doi:10.1142/s0218202510004222

Cited by 200 publications

(215 citation statements)

References 22 publications

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“…The fact that W D tends to 0 may be proved in a similar way to that of the proof of Lemma 3.1 below. Note that the SUSHI scheme is also part of the mimetic mixed hybrid family [16]; however, in the general form of mimetic schemes, we do not know how to include the stabilisation term (which is needed for the coercivity of the scheme) in the gradient term in order to write the scheme under the form (1.2).…”

Section: Examplesmentioning

confidence: 99%

Small-stencil 3D schemes for diffusive flows in porous media

Eymard¹,

Guichard²,

Herbin³

2011

ESAIM: M2AN

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149

185

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Abstract. In this paper, we study some discretization schemes for diffusive flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall into this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes show the efficiency of the new schemes, compared to existing ones.

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Section: Examplesmentioning

confidence: 99%

Small-stencil 3D schemes for diffusive flows in porous media

Eymard¹,

Guichard²,

Herbin³

2011

ESAIM: M2AN

Self Cite

149

185

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“…The SUSHI method. As a last example we consider two variants of the SUSHI scheme of Eymard, Gallouët, and Herbin [24]; see also Droniou, Eymard, Gallouët, and Herbin [21] for a discussion on the link with the MFD methods of Brezzi, Lipnikov, and coworkers [8,9]. This method is based on the gradient reconstruction (2.12), but stabilization is achieved in a rather 7 different manner with respect to (2.14).…”

Section: Remark 23 (Numerical Integration)mentioning

confidence: 99%

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

2012

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Abstract. In this work we propose an original implementation of a large family of lowest-order methods for diffusive problems including standard and hybrid finite volume methods, mimetic finite difference-type schemes, and cell centered Galerkin methods. The key idea is to regard the method at hand as a (Petrov-)Galerkin scheme based on possibly incomplete, broken affine spaces defined from a gradient reconstruction and a point value. The resulting unified framework serves as a basis for the development of a FreeFEM-like domain specific language targeted at defining discrete linear and bilinear forms. Both the back-end and the front-end of the language are extensively discussed, and several examples of applications are provided. The overhead of the language is evaluated by comparing with a more traditional implementation. A benchmark including the comparison with more classical finite element methods on standard meshes is also proposed.Key words. Domain specific embedded language, finite volume methods, cell centered Galerkin methods, Petrov-Galerkin methods AMS subject classifications. 65N08, 65N30, 65Y051. Introduction. Lowest-order methods possibly featuring conservation of physical quantities are traditionally employed in industrial applications where computational cost is a crucial issue. In this context, the use of general polyhedral, possibly nonconforming meshes commends itself for a number of reasons. To cite a few: (i) remeshing can be avoided or postponed in problems that involve mesh deformation -e.g. in sedimentary basin modeling non-standard elements and nonconformities can appear due to the erosion of geological layers; -(ii) the number of degrees of freedom can be reduced by aggregative coarsening techniques -cf. [7] for an application in the context of discontinuous Galerkin (dG) methods; -(iii) geometrical features can be represented more accurately without unduly increasing the number of mesh elements.Handling general polyhedral meshes requires numerical schemes that possess the usual properties of stability and consistency. In the context of cell centered finite volume methods, a popular way to achieve consistency on general polyhedral meshes is provided by Multipoint Finite Volume schemes independently introduced by Aavatsmark, Barkve, Bøe and Mannseth [2] and Edwards and Rogers [22]. The main advantage of multipoint schemes is that they can be easily fitted into existing simulators based on standard finite volume schemes. A major drawback is their lack of stability in some configurations. Two ways of overcoming this difficulty by designing discretizations based on the variational formulation of the problem and featuring cell-and face-unknowns have been proposed by Brezzi, Lipnikov and coworkers [8,9] (Mimetic Finite Difference methods) and by Droniou and Eymard [20] (Mixed/Hybrid Finite Volume methods). In this context, Eymard, Gallouët and Herbin [24] have shown that face unknowns can be selectively used as Lagrange multipliers to enforce flux continuity, or eliminated using a consisten...

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“…The Gradient Discretization method (GDM) [5,3] provides a common mathematical framework for a number of numerical schemes dedicated to the approximation of elliptic or parabolic problems, linear or nonlinear, coupled or not; these include conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic schemes [4] and some Multi-Point Flux Approximation [1] and Discrete Duality finite volume schemes [2] : we refer to [3, Part III] for more on this (note that in the present proceedings, it is shown that in some way the Discontinuous Galerkin schemes may also enter this framework [6]). Let us recall this framework in the case of the following linear elliptic problem:…”

Section: Introductionmentioning

confidence: 99%

An Error Estimate for the Approximation of Linear Parabolic Equations by the Gradient Discretization Method

Droniou

Eymard

Gallouët

et al. 2017

Springer Proceedings in Mathematics &Amp; Statistics

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We establish an error estimate for fully discrete time-space gradient schemes on a simple linear parabolic equation. This error estimate holds for all the schemes within the framework of the gradient discretisation method: conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume scheme and some discontinuous Galerkin schemes.

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A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods

Cited by 200 publications

References 22 publications

Small-stencil 3D schemes for diffusive flows in porous media

Small-stencil 3D schemes for diffusive flows in porous media

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

An Error Estimate for the Approximation of Linear Parabolic Equations by the Gradient Discretization Method

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