Summary: We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in \(L^{\infty} (\Omega)\) and the right-hand side belongs to \(L^1 (\Omega)\); we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in \(W_0^{1, q} (\Omega)\) for every \(q\) with \(1 \leq q < d / (d - 1)\) (\(d = 2\) or \(d = 3\)) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in \(W_0^{1, q} (\Omega)\) when the right-hand side \(f\) belongs to \(L^r (\Omega)\) verifying \(T_k (f) \in H^1 (\Omega)\) for every \(k > 0\), for some \(r > 1\).