Minimization of Functional Majorant in a Posteriori Error Analysis Based on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi><mml:mtext>(div)</mml:mtext></mml:math> Multigrid-Preconditioned CG Method

Abstract

We consider a Poisson boundary value problem and its functional a posteriori error estimate derived by S. Repin in 1999. The estimate majorizes the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math> seminorm of the error of the discrete solution computed by FEM method and contains a free ux variable from the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math>(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.