We refer to [@NochettoVeeserVerani_IMAJNA09; @Ainsworth2012Guaranteed] where this problem was treated for symmetric elliptic problems discretized by the conforming finite element method. This is also a subject of our further research.

An alternative representation of the algebraic error {#sec:alter}
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In , we expressed the discretization and algebraic parts of the computational error using the approach from [@Meidner2009Goal], namely $$\begin{aligned}
 \label{JP1}
 \J(\u - \uhk) = \res{\uhk}{\z -  \zhk} + \res{\uhk}{\zhk} = \errSk +\errAk.\end{aligned}$$ On the other hand, it is possible to decompose the total error $\J(\u - \uhk)$ in the alternative way as $$\begin{aligned}
  \label{JP2}
  \J(\u - \uhk) = \J(\u - \uh) + \J(\uh - \uhk)=: \estSk + \estAk.\end{aligned}$$ The first term in is the discretization error given by and it satisfies $$\begin{aligned}
  \label{JP3}
   \estSk = \J(\u - \uh) =  \res{\uh}{\z -  v_h}, \quad  v_h \in \Vh\end{aligned}$$ and it is independent of $k$. Obviously, although $\errSk$ in and $\estSk$ in both represent the discretization error, they differ. Moreover, $\estSk$ cannot be evaluated even if we replace $\z$ by $\hzh$ since $\uh$ is unavailable.

The second term $\estAk$ in represents the algebraic error. It cannot be expressed in a residual form but in Section \[sec:algeb\], we present a technique which is able 