the algebraic error estimate was sufficiently smaller than the discretization error estimate. We also mention the recent paper [@MallikVohralikYousef_JCAM20] using a completely deferent approach for elliptic problems. This technique is based on $H(\mathrm{div})$-conforming flux reconstructions and $H^1$-conforming potential reconstructions and yields to a guaranteed upper error bound.

In [@DolRoskovec_AM17], we employed an approach similar to [@Meidner2009Goal] and studied the convergence of the error estimator with respect to algebraic iterations of the primal and dual problems. We observed a delay in the convergence for insufficiently accurate resolution of the primal or the dual problem. We arrived to the conclusion that it is difficult to control the efficiency of the computational process when the primal and dual problems are solved alternatively.

Therefore, in this paper, we develop a technique to solve the primal and dual problems simultaneously using the [*bi-conjugate gradient*]{} (BiCG) method. At each BiCG iteration, the approximations of $\uh$ and $\zh$ are available and the estimate of the algebraic error can be computed and the accuracy and efficiency can be controlled. Motivated by [@StTi2011], we proposed a stopping criterion for the BiCG solver which is very cheap for evaluation and in contrary to [@Meidner2009Goal; @DolRoskovec_AM17], it does not require the 