Abstract.A postprocessing technique, developed earlier for spectral methods, is extended here to Galerkin finite-element methods for dissipative evolution partial differential equations. The postprocessing amounts to solving a linear elliptic problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. This technique increases the convergence rate of the finite-element method to which it is applied, and this is done at almost no additional computational cost. The numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied (say, quadratic finite elements) as well as standard methods of similar order of convergence as the postprocessed one (say, cubic finite elements). The error analysis of the new method is performed in L 2 and in L ∞ norms. , an inexpensive novel technique to increase the accuracy and computational efficiency of Fourier spectral methods was developed. In this paper, we present a general technique to improve the convergence rate of Galerkin methods which applies to finite-element methods (and in the particular case of Fourier-Galerkin methods it coincides with that in [21]; see also [37]). We do this with no direct address to [13].Let Ω ⊂ R d be a domain with a smooth boundary. We consider dissipative PDEs (see, for instance, [8], [24], [25], [41]) which can be written as evolution systems of the form