\z-w_h + w_h,\ w_h\in\Vh$ and relation , we get the [*dual error identity*]{} including [*algebraic errors*]{} $$\begin{aligned}
  \label{eq:JD_alg0}
  \ell_h(\z - \zhk) &=  \ah(\u , \z-\zhk)=  \ah(\u - w_h, \z - \zhk) + \ah( w_h, \z - \zhk)
  \\
  &   = \resD{\zhk}{\u -  w_h} + \resD{\zhk}{w_h} 
  \quad \forall w_h \in \Vh,\notag\end{aligned}$$ where $\resD{\zhk}{\cdot}$ is given by . Putting $w_h:=\uhk$ in , we obtain $$\begin{aligned}
  \label{eq:JD_alg}
  \ell_h(\z - \zhk) = \resD{\zhk}{\u -  \uhk} + \resD{\zhk}{\uhk} =: \errDSk + \errDAk,\end{aligned}$$ where, similarly as the quantities $\errSk$ and $\errAk$ in , the quantities $\errDSk$ and $\errDAk$ represent the discretization and algebraic error of the dual problem. Let us note that and hold without the Galerkin orthogonality .

Computable goal-oriented error estimates {#sec:GOEE_com}
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Whereas the algebraic errors $\errAk$ and $\errDAk$ from and , respectively, are computable quantities, the discretization errors $\errSk$ and $\errDSk$ require the knowledge of the exact dual and primal and solutions $\z$ and $\u$, respectively. In practical computations, they should be approximated by a higher-order reconstruction denoted here by $$\begin{aligned}
  \label{krr}
  \huh=\krr(\uhk),\qquad \hzh=\krr(\zhk),\end{aligned}$$ where $\uhk\in\Vh$ and $\zhk\in\Vh$ denote approximations of $\uh$ and $\zh$, $\krr:\Vh\to\Vhp$ denotes a reconstruction operator and $\Vhp$ is a “richer” finite dimensional space, for examples, see the papers 