Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem

In this paper, a class of cell centered finite volume schemes, on general u nstructured 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 W-2,W-p (for p > 2) reg ularity 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 i n [12].