The p-Laplacian problem -del & sdot; ((mu + |del u|(p-2))del u) = f is considered, where mu is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of mu, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p=3 . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier-Stokes problem arising from aluminium electrolysis.