Advances in Numerical Analysis
Volume 2009 (2009), Article ID 164519, 15 pages
Research Article

Minimization of Functional Majorant in a Posteriori Error Analysis Based on H(div) Multigrid-Preconditioned CG Method

Department of Mathematics, University of Bergen, Johannes Bruns Gate 12, 5008 Bergen, Norway

Received 15 October 2008; Revised 5 May 2009; Accepted 8 June 2009

Academic Editor: Serge Nicaise

Copyright © 2009 Jan Valdman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a Poisson boundary value problem and its functional a posteriori error estimate derived by S. Repin in 1999. The estimate majorizes the H1 seminorm of the error of the discrete solution computed by FEM method and contains a free ux variable from the H(div) space. In order to keep the estimate sharp, a procedure for the minimization of the majorant term with respect to the ux variable is introduced, computing the free ux variable from a global linear system of equations. Since the linear system is symmetric and positive definite, few iterations of a conjugate gradient method with a geometrical multigrid preconditioner are applied. Numerical techniques are demonstated on one benchmark example with a smooth solution on a unit square domain including the computation of the approximate value of the constant in Friedrichs' inequality.