the pioneering works summarized in [@RannacherBook; @BeckerRannacher01; @GileSuli02]. This approach was further developed for many problems, let us mention [@OdenPrudhomme_CAMWA01; @Korotov_JCAM06; @SolDemko04] dealing with linear elliptic problems, [@KuzminMoller_JCAM10] dealing with steady linear hyperbolic problems. Moreover, extensions to nonlinear problem were presented, e.g., in [@Rey2Gosselet_CMAME14; @Rey2Gosselet_CMAME15] for elasticity problems and, e.g., in [@LoseilleDervieuxAlauzet_JCP10; @Richter_IJNMF10; @HH06:SIPG2; @GeHaHo09; @Hartman08; @BalanWoopenMay16] for computational fluid dynamics, for a survey see [@FidkowskiDarmofal_AIAA11].

The goal-oriented error estimates require, except the solution of the original ([*primal*]{}) problem , also to solve the [*dual*]{} (or adjoint) problem $$\begin{aligned}
  \label{eq:DP}
  \krl^* \z = \J,\end{aligned}$$ where $\krl^*$ is the [*dual operator*]{} to $\krl$, $\J$ is the target functional and $\z:\Om\to\IR$ is the dual solution. The problem has to be accompanied by suitable boundary conditions as well. It is necessary to approximate the solution of by $\zhp\in\Vhp$ where $\Vhp$ is a richer space than $\Vh$. One possibility is to discretize directly on $\Vhp$ (e.g., [@Korotov_JCAM06]) but then we have to solve two different algebraic problems. Another way is to discretize on $\Vh$ and define $\zhp:= \krr(\zh)$, where $\zh\in\Vh$ is a numerical approximation of and $\krr:\Vh\to\Vhp$ is a suitable higher order reconstruction (e.g., [@Richter_IJNMF10; @CarpioPrietoBermejo_SISC13]).

The advantage of the latter approach is that 