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A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. (English) Zbl 1191.65142
Problems with anisotropic heterogeneous diffusion operators are investigated from a numerical point of view. There exist several numerical methods for solving these problems. The authors present the connection between the probably most important of them. The main result of this paper proves that the mimetic finite difference scheme, the hybrid finite volume scheme and the mixed finite volume scheme are identical up to some slight generalization even on general grids and they can be unified in a family of schemes – HMM (hybrid mimetic mixed).
First, each proposed method is briefly described and then the generalized definition of the scheme is presented. The equivalence for these generalized methods is proved. The straightforward consequence of this result contains in an extension of some mathematical results (convergence, error estimates) to each of the presented methods.
Finally, the relationship between this HMM family of schemes and nonconforming finite element schemes or mixed finite element schemes is investigated. Moreover for isotropic operators with so called “super-admissible discretizations” the HMM method boils down to the well-known efficient flux finite volume scheme.
Although the study is performed for the linear equation, this framework can be used for more complex situations as well.

MSC:
65N08 Finite volume methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
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