cited in Introduction. Replacing $\u$ and $\z$ in and by $\huh$ and $\hzh$, we obtain the computable approximations $\etaSk$ and $\etaDSk$ of the discretization errors $\errSk$ and $\errDSk$ by $$\begin{aligned}
  \label{etas}
  &\errSk = \res{\uhk}{\z -  \zhk} \approx \res{\uhk}{\hzh -  \zhk} =: \etaSk, \\
  &\errDSk = \resD{\zhk}{\u -  \uhk} \approx \res{\zhk}{\huh -  \uhk} =: \etaDSk,\qquad
  k=1,2,\dots.
  \notag\end{aligned}$$ The quantities $\etaSk$ and $\etaDSk$ are called the estimates of the primal and dual discretization errors, respectively.

The approximations together with and lead to the computable estimate of the total error by $$\begin{aligned}
  \label{EEP}
  \J(\u - \uhk) & \approx   \etaSk + \etaAk, \qquad 
       \ell_h(\z - \zhk)  \approx \etaDSk + \etaDAk, \quad k=1,2,\dots,\end{aligned}$$ where, in order to have a consistent notation, we put $$\begin{aligned}
  \label{etasA}
  \etaAk := \errAk = \res{\uhk}{\zhk}, \quad
  \etaDAk := \errDAk = \resD{\zhk}{\uhk}.\end{aligned}$$ The terms $\etaAk$ and $\etaDAk$ are equal to the [*algebraic errors*]{} of the primal and dual problems and , respectively, and they are independent on the used higher-order reconstruction $\krr$ from .

\[rem:guaranteed\] The approximations do not give a guaranteed error estimate since, as usual, we neglected the terms $\res{\uhk}{\z -  \hzh} $ and $\res{\zhk}{\u -  \huh} $. Hence, the guaranteed error estimate requires an estimation of these terms. 