- \uh, \z)=  \ah(\u - \uh, \z - v_h)  \\
   &= \res{\uh}{\z -  v_h}  \quad \forall v_h \in \Vh, \notag\end{aligned}$$ where $\res{\zh}{\cdot}$ denotes the residual of the primal problem given by .

Similarly, exploiting in addition the Galerkin orthogonality and relation , we get the [*dual error identity*]{} $$\begin{aligned}
\label{eq:JD}
\J(\u - \uh) &=  \ah(\u - \uh, \z - \zh)
=   \ah(\u - w_h, \z - \zh ) \\
&=   \resD{\zh}{\u -  w_h}  \quad \forall w_h \in \Vh, \notag\end{aligned}$$ where $\resD{\zh}{\cdot}$ denotes the residual of the dual problem given by .

Abstract goal-oriented error estimates including algebraic errors {#sec:GOEE_alg}
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As mentioned above, the discrete solutions $\uh$ and $\zh$ fulfilling and , respectively, are non-available but we have only $\uhk\in\Vh$ and $\zhk\in\Vh$, $k=1,2,\dots$ denoting their approximations given by an algebraic iterative solver. Hence, the error representations and are useless and we have to estimate the error $J(\u-\uhk)$.

In the same way as in [@Meidner2009Goal; @DolRoskovec_AM17], we employ the adjoint consistency , identity $\z = \z-v_h + v_h,\ v_h\in\Vh$ and relation , we get the [*primal error identity*]{} including [*algebraic errors*]{} $$\begin{aligned}
 \label{eq:JP_alg0}
 \J(\u - \uhk) &=  \ah(\u - \uhk, \z)=  \ah(\u - \uhk, \z - v_h) + \ah(\u - \uhk, 