In recent years spectral methods for the numerical solution of nonlinear parabolic equations have received increasingly attention. The p and h-p versions of finite element method allow to treat complex geometries with spectral accuracy. Spectral-type approximations have been proven to be suitable for parabolic equations due to their regularization properties. In this talk we focus on the p- version of the finite element method. The theory of a posteriori error estimation for elliptic problems has been extensively treated in the literature. For parabolic equations much fewer results are available, see [1]. In [1], [2], it is proven that an a posteriori error estimator for evolutionary problems can be obtained by using any elliptic a posteriori error estimator. In these papers, the analysis is done in a special time-weighted energy norm in the space-time cylinder. In the present talk we propose a similar procedure that can be used for nonlinear dissipative equations. In our approach the error estimation is done at fixed times by means of only one elliptic problem with right hand side the residual of the numerical solution with time frozen. Furthermore, the main advantage of the procedure we propose is that it gives a hierarchy of corrections to the numerical solution that allow to improve the spatial accuracy (if it is needed) without increasing the computational cost of the overall time integration, including the cost of error estimation. The results are based in the theory of the so called postprocessed Galerkin methods [3], [4]. Postprocessed methods have been used in a different context to get improved solutions

of dissipative equations. The main hypothesis needed is that the solutions of the equations are smooth in time, a property that is shared for most dissipative partial differential equations including, reaction-convection-diffusion equations, Navier-Stokes equations, etc. The computation of the a posteriori error estimator, and a fortiori the correction, is far less expensive than the computation of the numerical solution and can be carried out at every time when it is needed. The results can be used as a basis for an adaptive numerical procedure that controls adaptive enrichment through h or p version of finite element method. Some numerical results are presented.