Postprocessing the Galerkin method: The finite-element case

A postprocessing technique, developed earlier for spectral methods, is exte nded here to Galerkin finite-element methods for dissipative evolution part ial differential equations. The postprocessing amounts to solving a linear elliptic problem on a finer grid (or higher-order space) once the time inte gration on the coarser mesh is completed. This technique increases the conv ergence rate of the finite-element method to which it is applied, and this is done at almost no additional computational cost. The numerical experimen ts presented here show that the resulting postprocessed method is computati onally more efficient than the method to which it is applied (say, quadrati c finite elements) as well as standard methods of similar order of converge nce as the postprocessed one (say, cubic finite elements). The error analys is of the new method is performed in L-2 and in L-infinity norms.