Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes

Agélas, Léo; Masson, Roland

doi:10.1016/j.crma.2008.07.015

Cited by 47 publications

(92 citation statements)

References 13 publications

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“…However, due to the lack of simple finite element methods for elasticity and the Biot equations, of relevance for the current work is the analysis within the hybrid FV framework [14] and the recent analysis of the MPFA [3] and MPSA methods [29].…”

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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Abstract. In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and the convergence of the method. 1. Introduction. Deformable porous media are becoming increasingly important in applications. In particular, the emergence of strongly engineered geological systems such as CO 2 storage [31, 33], geothermal energy [35], and shale-gas extraction all require analysis of the coupling of fluid flow and deformation. Beyond the subsurface, physiological processes are increasingly simulated, exploiting the Biot models [8].With this motivation, we consider the following model problem poroelastic media [9]:

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Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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“…The face σ is completely contained either in one element of the partition P Ω or in an interface of this partition; using then either the regularity of ϕ inside each element of the partition or the continuity of its tangential derivatives through the interfaces of P Ω , we can re-write τ σ under the form τ σ = (∇ϕ(x K ) − ∇ϕ(y σ ))·t σ + (∇ϕ(y σ ) − ∇ϕ(x L ))·t σ and the proof is complete since ϕ is C 2 on K and L. and, ∇ϕ and Λ being C 1 on the closure of each control volume, we deduce from (B.7) that the real ζ σ (ϕ) = 8) where C 17 > 0 depends only on ϕ, Λ.…”

Section: A Proof Of Lemma 32mentioning

confidence: 89%

“…The convergence of the MPFA O method has been theoretically proved in [4] on two-dimensional quadrilateral grids and in [7,8] on general two-and three-dimensional polyhedral meshes. In [25], the equivalence of multipoints methods with the lowest-order Mixed Finite Element method on matching triangular grids has been pointed out, and local coercivity conditions have been proposed.…”

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confidence: 99%

The G method for heterogeneous anisotropic diffusion on general meshes

Agélas¹,

Pietro²,

Droniou³

2010

ESAIM: M2AN

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Abstract.In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method. A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method. Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity, we introduce a non-standard test space in H 1 0 (Ω) and prove its density. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with classical multi-point Finite Volume methods is provided.Mathematics Subject Classification. 65N08, 65N12.

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“…Considering a patch of (sub)elements, supposing a piecewise affine, possibly nonconforming, potential approximation, imposing the normal flux continuity, and solving a local linear system on the patch is also the principle of the MPFA method [1,3,19], cf. also [28,41,14,56,42].…”

Section: Definition Of the Local Problems By A Geometrical Interpretamentioning

confidence: 99%

“…The last goal of the present paper is to recall known and show new relations between MFE methods and other discretization schemes, namely the two-point finite volume (FV) [29], mimetic finite difference (MFD) [17,13], hybrid finite volume [31], mixed finite volume [26], multi-point flux-approximation (MPFA) [1,3,19], and related methods [28,41,14,56,42]. We in particular prove in Section 7 that the RT0 MFE method on arbitrary polygonal meshes is a particular example of the MFD method.…”

Section: Introductionmentioning

confidence: 99%

Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

Vohralı́k

Wohlmuth

2013

Math. Models Methods Appl. Sci.

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In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart-Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart-Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

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Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes

Cited by 47 publications

References 13 publications

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Stable Cell-Centered Finite Volume Discretization for Biot Equations

The G method for heterogeneous anisotropic diffusion on general meshes

Mixed Finite Element Methods: Implementation With One Unknown Per Element, Local Flux Expressions, Positivity, Polygonal Meshes, and Relations to Other Methods

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