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The paper proves the convergence of the finite volume approximate solution of a convection-diffusion equation with anL  1    ${L^{1}}$    right-hand side to the unique renormalized solution. The main difficulties are to handle the noncoercive character of the operator and theL  1    ${L^{1}}$    data. Mixing the techniques of renormalized solutions and the finite volume method allows one to derive estimates for the discrete solutions and in particular a discrete version of the decay of the truncated energy. Indeed, as in the continuous case, the decay of the truncated energy is crucial to show that the limit of the approximate solution is the renormalized solution.

Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with L 1 Data