The paper is concerned with a posteriori error estimation in terms of special problem-oriented quantities. In many practically interesting cases, such a quantity is represented as a linear functional that controls the behavior of a solution in certain subdomains, along some lines, or at especially interesting points. The method of estimating quantities of interest is usually based upon the analysis of the adjoint boundary-value problem, whose right hand side is formed by the considered linear functional. On this way, we propose a new effective modus operandi. It is based on two principles: (a) the original and adjoint problems are solved on non-coinciding meshes, and (b) the term presenting the product of gradients of errors of the primal and adjoint problems is estimated by using the 'gradient averaging' technique. Numerical tests confirm high effectivity of this approach.