stopping criteria for the BiCG solver. Their computational performance is demonstrated in Section \[sec:numerA\] for the Laplace and convection-dominated problems. We conclude with several remarks and discuss a possible extension of this technique for nonlinear problems in Section \[sec:sum\].

Framework for the goal-oriented error estimates {#sec:GO}
===============================================

We briefly recall a general framework for the goal-oriented error estimates. More details can be found, e.g., in [@BeckerRannacher01; @GileSuli02].

Primal problem
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Let the [*weak formulation*]{} of the primal problem be given by $$\begin{aligned}
  \label{eq:PPw}
  a( \u, \vp) = \ell(\vp)\qquad \forall \vp\in \V,\end{aligned}$$ where $\u\in \V$ is a weak solution, $a(\cdot,\cdot):\V\times \V\to\IR$ is a bilinear form, $\ell(\cdot):\V\to\IR$ is a linear form and $V$ is a Hilbert space. We assume that is well-posed, i.e., it admits a unique weak solution.

For the numerical approximation of , let $\Vh,\ h>0$ be a finite element space of functions defined on $\Om$, typically piecewise polynomial functions related to the partition of $\Om$ onto a set of finite elements $\Th$. Moreover, let $\Wh$ be a functional space such that $\V\subset \Wh$ and $\Vh\subset\Wh$. For conforming finite element methods ($\Vh\subset\V$), we simply put $\Wh:=\V$. However, for nonconforming methods (the case $\Vh\not\subset\V$), the choice of $\Wh$ is more delicate. 