Guaranteed a-posteriori error estimation for finite element solutions of nonstationary heat conduction problems based on their elliptic reconstructions

Abstract

We deal with the a-posteriori estimation of the error for finite element solutions of nonstationary heat conduction problems with mixed boundary conditions on bounded polygonal domains. The a-posteriori error estimates are constucted by solving stationary “reconstruction” problems, obtained by replacing the time derivative of the exact solution by the time derivative of the finite element solution. The main result is that the reconstructed solutions, or reconstructions, are superconvergent approximations of the exact solution (they are more accurate than the finite element solution) when the error is measured in the gradient or the energy-norm. Because of this, the error in the gradient of the finite element solution can be estimated reliably, by computing its difference from the gradient of its reconstructions. Numerical examples show that “reconstruction estimates” are reliable for the most general classes of solutions which can occur in practical computations.