In this paper we present an abstract analysis framework for nonconforming approximations of heterogeneous and anisotropic elliptic operators on general 2D and 3D meshes. The analysis applies to discontinuous Galerkin methods, to the popular MPFA O-method, as well as to hybrid finite volume methods. A number of examples is provided in the paper. The guidelines of the analysis can be summarized as follows: (i) each method is re-written in weak formulation by introducing two discrete gradient operators. Penalty terms may be required for non $H^1$-conformal spaces to ensure coercivity; (ii) a discrete $H^1$-norm is introduced such that a discrete version of the Rellich theorem holds; (iii) an a priori estimate on the discrete solution in the discrete $H^1$-norm is derived using the coercivity of the bilinear form; (iv) the convergence of the discrete problem to the continuous one is eventually deduced, thus proving the convergence of the approximation. In order for the above analysis to hold, the gradient operator applied to the discrete solution should be consistent, whereas the one applied to the test function should be weakly convergent in $L^2$. According to the method, further assumptions on the mesh may be required. This analysis framework easily extends to non-linear problems like the ones encountered in the study of multi-phase Darcy flows, and it allows to weaken regularity assumptions on the coefficients of the problem as well as on the exact solution. From a practical viewpoint, its main interest lies in the possibility to analyze and to design new methods as well as to improve existing ones.


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