Postprocessing the Galerkin method: the finite-element case

Abstract

A postprocessing technique, developed earlier for spectral methods, is extended here to Galerkin nite-element methods for dissipative evolution partial di erential equations. The postprocessing amounts to solving a linear elliptic problem on a ner grid (or higher-order space) once the time integration on the coarser mesh is completed. This technique increases the convergence rate of the nite-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 e cient than the method to which it is applied (say, quadratic nite elements) as well as standard methods of similar order of convergence as the postprocessed one (say, cubic nite elements). The error analysis of the new method is performed in L2 and in L1 norms.