In this paper, a class of cell centered finite volume schemes,
on general unstructured meshes, for a linear convection-diffusion
problem, is studied. The convection and the diffusion are respectively
approximated by means of an upwind scheme and the so called diamond
cell method [4]. Our main result is an error estimate of
order h, assuming only the W2,p (for p>2) regularity of the
continuous solution, on a mesh of quadrangles. The proof is based on an
extension of the ideas developed in [12]. Some new
difficulties arise here, due to the weak regularity of the solution, and the
necessity to approximate the entire gradient, and not only its normal
component, as in [12].