Purpose – This paper aims to focus on the local quality of outputs of interest computed by a finite element analysis in linear elasticity. Design/methodology/approach – In particular outputs of interest are studied which do not depend linearly on the solution of the problem considered such as the L2‐norm of the stress and the von Mises' stress. The method is based on the concept of error in the constitutive relation. Findings – The method is illustrated through 2D test examples and shows that the proposed error estimator leads in practice to upper bounds of the output of interest being studied. Practical implications – This tool is directly usable in the design stage. It can be used to develop efficient adaptive techniques. Originality/value – The interest of this paper is to provide an estimation of the local quality of L2‐norm of the stress and the Von Mises' stress as well as practical upper bounds for these quantities.