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We present a novel cell-centered finite volume discretization of the heterogeneous and anisotropic diffusion/convection-diffusion problems on polygonal/polyhedral meshes. The unknowns of the resulting linear scheme are the values at the cell centers,
and no auxiliary unknowns are involved. To the best of our knowledge, this is the first linear cell-centered scheme that is interpolation-free and of second order accuracy on arbitrary meshes with arbitrary discontinuities. Numerical experiments show the efficiency of the new linear finite volume scheme.