evaluation of the discretization error estimator. The use of the BiCG solver for the primal and dual problems with the proposed criterion in a combination the mesh adaptation leads to an efficient numerical method for the solution of .

Since the presented approach can be applied to with a general linear operator $\krl$ discretized by any numerical scheme based on a variational formulation, we express the problem considered and its numerical approximation only in the abstract form. However, in order to demonstrate the applicability of this technique, we present the numerical solution of purely elliptic and convection-diffusion problems by the $hp$-adaptive discontinuous Galerkin method (DGM) on possibly anisotropic meshes. The content of the rest of the paper is the following. In Section \[sec:GO\], we shortly summarize the goal-oriented error estimate technique including two variants of the expression of the algebraic errors. In Section \[sec:algeb\], we introduce the algebraic representation of the discrete problems and discuss several ways for the evaluation of the quantity of interest and their corresponding error estimates. These possibilities are numerically tested in Section \[sec:numerF\], where we solve the Laplace problem on fixed meshes. Moreover, in Section \[sec:adapt\], we describe the mesh adaptation process and proposed new 