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A family of mimetic finite difference methods on polygonal and polyhedral meshes. (English) Zbl 1083.65099
For a diffusion problem written as a system of two partial differential equations, the authors propose a family of mimetic finite difference methods (MFD) on unstructured polygonal and polyhedral meshes. Such unstructured polygonal and polyhedral meshes appear, for example, in geoscience problems simulating the flow and transport in porous media and using dual or median meshes. The key element of the MFD method is to define a scalar product in the space of discrete velocities which should satisfy the stability assumption and the consistency assumption.
In this paper, a technique to give a rigorous mathematical description of a family of acceptable scalar product for very general meshes in two and three dimensions, are proposed. The discretization method developed is computationally much cheaper and easier to implement, than the methods presented in previous papers. The resulting MFD methods have optimal convergence rates for a wide variety of problems. Three numerical examples are considered: two Dirichlet boundary value problems in two dimensions and one Dirichlet boundary value problem in three dimensions.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
35J25 Boundary value problems for second-order elliptic equations
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