A posteriori estimation and adaptive control of the error in the quantity of interest. Part I: A posteriori estimation of the error in the von Mises stress and the stress intensity factor

In this paper we address the problem of a posteriori estimation of the error in an engineering quantity of interest which is computed from a finite element solution. As an example we consider the plane elasticity problem with the von Mises stress and the stress intensity factor, as the quantities of interest. The estimates of the error in the von Mises stress at a point are obtained by partitioning the error into two components with respect to the element which includes the point, the local and the pollution errors, and by constructing separate estimates for each component. The estimates of the error in the stress intensity factors are constructed by employing an extraction method. We demonstrate that our approach gives accurate estimates for rather coarse meshes and elements of various degrees. In Part II we will address the problem of the adaptive control of the error in the quantity of interest (the goal of the computation), and the construction of goal-adaptive meshes for one or multiple goals.