the discretizations of the primal and dual problems and using the same space $\Vh$ are equivalent to two mutually transposed linear algebraic systems which can be beneficial in practical solutions. Namely, we obtain $$\begin{aligned}
  \label{i0}
  \mA\bx = \bb\qquad\mbox{and}\qquad \mA^{\T}\by=\bc,\end{aligned}$$ where $\mA\in\IR^{\Nh\times\Nh}$ is the matrix arising from the discretization of $\krl$ from , $\mA^{\T}$ is the transpose matrix of $\mA$, $\bb\in\IR^{\Nh}$ and $\bc\in\IR^{\Nh}$ represent the discretization of the right-hand sides of and , respectively, and $\bx\in\IR^{\Nh}$ and $\by\in\IR^{\Nh}$ are the vectors corresponding to the discrete solutions $\uh\in\Vh$ and $\zh\in\Vh$ of and , respectively.

In many situations, it is advantageous to solve systems iteratively since (i) approximate solutions satisfying the prescribed tolerance are sufficient as an output of the computation; (ii) very good initial approximations of $\bx$ and $\by$ are typically available from previous level of mesh adaptation. Let $\uhk\in\Vh$ and $\zhk\in\Vh$ denote functions corresponding to the $k$-th iterations $\bxk$ and $\byk$, respectively. Then the computed approximations $\uhk$ and $\zhk$ are influenced also by the algebraic error arising from the inexact solution of systems . Let us note that even if a direct solver for is used then these systems are not solved exactly (only on the level of machine accuracy) and 