E.g., for [*discontinuous Galerkin method*]{}, we employ the so-called [*broken Sobolev space*]{} and put $$\begin{aligned}
  \label{HsTh}
  \Wh:= H^s(\Th) = \{ v \in L^{2}(\Om); \, \restr{v}{K} \in H^{s}(K) \,
  \forall K \in \Th\},\end{aligned}$$ where $\Th$ is a mesh partition of $\Om$ and $s>0$ is a suitable Sobolev index, e.g., for the second-order operator $\krl$, we have $s=2$.

Let $$\begin{aligned}
  \label{ah}
  \ah:\Wh\times\Wh\to\IR\qquad \mbox{and} \qquad  \ell_h:\Wh\to\IR\end{aligned}$$ be a bilinear and a linear forms corresponding to the discretization of the left-hand and right-hand sides of , respectively, by a particular numerical method.

We say that $\uh\in\Vh$ is the [*discrete solution*]{} of the [*primal problem*]{} if $$\begin{aligned}
  \label{eq:PPh}
    \ah(\uh,\vp_h) = \ell_h(\vp_h)\qquad \forall \vp_h\in\Vh.\end{aligned}$$ We assume that the numerical scheme is [*consistent*]{}, i.e., $$\begin{aligned}
  \label{eq:cons}
  \ah(\u,\vp) = \ell_h(\vp)\qquad \forall \vp\in\Wh\end{aligned}$$ where $\u\in\V$ is the weak solution of . This implies the [*Galerkin orthogonality of the error*]{} of the primal problem $$\begin{aligned}
  \label{eq:GO}
  \ah(\uh - u, \vp_h) = 0 \qquad\forall \vp_h\in\Vh.\end{aligned}$$ Finally, we define the [*residual of the primal problem*]{} by $$\begin{aligned}
  \label{res}
  \res{\uh}{\vp}:= \ell_h(\vp) - \ah(\uh, \vp) = \ah(\u-\uh,\vp) ,\quad \vp\in\Wh,\end{aligned}$$ where the last equality follows from the consistency .

Quantity of interest and the dual problem
-----------------------------------------

As mentioned in the introduction, we are interested 