discrete solutions $\uh$ and $\zh$ are not available, we have only their approximation $\uhk\in\Vh$ and $\zhk\in\Vh$, $k=1,2,\dots$. However, they do not fulfil the Galerkin orthogonalities and .

Abstract goal-oriented error estimates {#sec:GOEE}
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Obviously, for adjoint consistent discretization we have (due to and ) the equivalence between the quantity of interest $\J(\u)$ and the right-hand side of evaluated for the dual solution $\z,$ i.e. $$\begin{aligned}
  \label{eq:eqi}
 \ell_h(\z) = \ah(\u, \z) = \J(\u).\end{aligned}$$ Similarly, from and , we obtain the discrete variant of as $$\begin{aligned}
  \label{eq:eqi_h}
 \ell_h(\zh) = \ah(\uh, \zh) = \J(\uh),\end{aligned}$$ where $\uh$ and $\zh$ are the discrete solutions of the primal and dual problems, respectively. Consequently, we have an [*error equivalence*]{} $$\begin{aligned}
  \label{eq:eqi_err}
  \J(\u -\uh) =  \J(\u) - \J(\uh) = \ell_h(\z) - \ell_h(\zh) = \ell_h(\z -\zh).\end{aligned}$$ Therefore, the difference $\ell_h(\z -\zh)$ can be used as an error estimate of the quantity of interest as well.

First, we present the primal and dual error identities for the error of the quantity of interest for the algebraically exact discrete solution $\uh$ and $\zh$ of and , respectively. Using the adjoint consistency , the Galerkin orthogonality and relation , we get the [*primal error identity*]{} $$\begin{aligned}
 \label{eq:JP}
 \J(\u - \uh) &=  \ah(\u 