Goal-oriented error control of the iterative solution of finite element equations

This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems.