Postprocessing the finite element method for semilinear parabolic problems

In this paper we consider postprocessing of the finite element method for semilinear parabolic problems. 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. The convergence rate is increased at almost no additional computational cost. This procedure was introduced and analyzed in Garcia-Archilla and Titi [SIAM J. Numer. Anal., 37 (2000), pp. 470-499]. We extend the analysis to the fully discrete case and prove error estimates for both space and time discretization. The analysis is based on error estimates for the approximation of time derivatives by difference quotients.