Effective Minimization of Functional Majorant in A Posteriori Error Analysis

We consider a Poisson problem and its functional a posteriori estimate derived in [20]. The estimate majorizes the L2 norm of the error of the discrete solution computed by FEM method and contains a free variable from a H(div) space. In order to keep the estimate sharp, the majorant term is minimized with respect to the free variable. A minimization procedure is introduced, containing a solution of linear system of equations as its computationally most expensive part. The linear system is efficiently solved using a conjugate gradient method with a multigrid as a preconditioner. All numerical techniques including the computation of the constant from the Korn’s inequality as a part of majorant estimate are demonstrated on one benchmark example.