to estimate this term in a cheap way.

Hence, we have two representations of the algebraic error, the first one from given by $\etaAk=\res{\uhk}{\zhk}$ and the second one from given by $\estAk:=\J(\uh - \uhk)$. Both representations are “algebraically consistent” which means that if $\uhk\to \uh$ and $\zhk\to \zh$ for $k\to \infty$ then $$\begin{aligned}
  \label{JP4}
  \etaAk \to 0 \quad \mbox{ for } k\to \infty
  \qquad \mbox{ and }\qquad 
  \estAk \to 0 \quad \mbox{ for } k\to \infty.\end{aligned}$$ However, the speed of convergence for $\etaAk$ and $\estAk$ can differ substantially, see numerical experiments in Section \[sec:numerA\].

Moreover, for $\uhk\to \uh$ and $\zhk\to \zh$, both discretization errors $\errSk$ and $\estSk$ are closer and closer since (cf. and ) $$\begin{aligned}
  \label{JP4a}
  | \errSk - \estSk| = | \estAk - \errAk| \leq | \estAk| + |\errAk| \to 0.\end{aligned}$$

Finally, we present the dual analogue of – in the form $$\begin{aligned}
  \label{JP5}
  \ell_h(\z - \zhk) &= \ell_h(\z - \zh) + \ell_h(\zh - \zhk) 
  =: \estDSk + \estDAk,\end{aligned}$$ where the term $\estDAk$ can be estimated in the same manner as $\estAk$ in , cf. Section \[sec:algeb\].

Solution of primal and dual discretized problems {#sec:algeb}
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In this section, we introduce the algebraic representation of the goal-oriented error 