an abstract partial differential equation in the form $$\begin{aligned}
  \label{eq:PP}
  \krl \u = f,\end{aligned}$$ where $\u:\Om \to \IR$ is the unknown solution, $\krl$ is a linear differential operator and $f$ is the right-hand side. The problem has to be accompanied by suitable boundary conditions. Further, let $\Vh$ be a finite-dimensional space of functions, ($\dim\Vh=\Nh< \infty$), where an approximation of $\u$ is sought. We note that the space $\Vh$ is not given a priori but it is generated as a sequence of spaces by a suitable mesh adaptive method. By $\uh\in\Vh$ we denote the discrete solution which approximates $u$.

In many practical application, we are not interested in the solution of itself but rather in a [*quantity of interest*]{}, which is the value of a certain, a priori known, solution-dependent [*target functional*]{} $\J(\u)$. For example, the functional $\J$ is a mean value of the solution over a subset of the computational domain or its boundary. Therefore, we need not to estimate the error $\norm{\u-\uh}{}$ in an energy norm (cf. [@AinsworthOden; @verfurth-book2]) but the error $\J(u)-\J(\uh)$. We assume that $J$ is a linear functional.

The necessity to estimate the error of the target functional gives rise to the [*goal-oriented error estimates*]{}, cf. 