A local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations is introduced. Departing from classical adaptive algorithms, the proposed method is based on a coarse grid and iteratively improves the accuracy of the solution by solving local elliptic problems identified by an a posteriori error control. An a posteriori error analysis based on fluxes reconstruction shows that the local adaptive method is robust in singularly perturbed regimes. Numerical comparison with a classical adaptive algorithm illustrate the efficiency of the new method.