An abstract analysis framework for nonconforming approximations of diffusion problems on general meshes

In this work, we propose a unified analysis framework encompassing a wide range of nonconforming discretizations of anisotropic heterogeneous diffusion operators on general meshes. The analysis relies on two discrete function analytic tools for piecewise polynomial spaces, namely a discrete Sobolev-Poincaré inequality and a discrete Rellich theorem. The convergence requirements are grouped into seven hypotheses, each of them characterizing one salient ingredient of the analysis. Finite volume schemes as well as the most common discontinuous Galerkin methods are shown to fit in the analysis. A new finite volume cell-centered method is also introduced.