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

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.