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**Relationships among some locally conservative discretization methods which handle discontinuous coefficients.**
*(English)*
Zbl 1124.76030

Summary: This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control-volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76S05 | Flows in porous media; filtration; seepage |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N06 | Finite difference methods for boundary value problems involving PDEs |

### Keywords:

relationships, mixed finite element method (MFEM); expanded mixed finite element method (EMFEM); enhanced cell-centered finite difference method (ECCFDM); control-volume mixed finite element method (CVMFEM); support operator method (SOM); multi-point flux-approximation (MPFA) control-volume method
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\textit{R. A. Klausen} and \textit{T. F. Russell}, Comput. Geosci. 8, No. 4, 341--377 (2005; Zbl 1124.76030)

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### References:

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