A cell-centered Finite Volume method is proposed to approximate numerically the solution to the steady convection-diffusion equation. The method is developed for unstructured meshes of d-simplexes, where d ≥ 2 is the spatial dimension and is formally second-order accurate by means of a piecewise linear reconstruction within the mesh cells and at the mesh vertices. The face gradients that are required to discretize the diffusive flux are defined by a non-linear strategy that makes it possible to

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... demonstrate the existence of a discrete Maximum Principle. Finally, a set of numerical results illustrates the performance of the method in treating problems with internal layers and strong gradient solutions.