On the Duality of Global Finite Element Discretization Error-Control in Small Strain Newtonian and Eshelbian Mechanics

Abstract

In this paper global a posteriori error estimators are presented for the error obtained during the finite element discretization of the linear elasticity problem. Thereby, the duality of global error measures is established that are chosen within the framework of traditional Newtonian mechanics as well as within the framework of Eshelbian mechanics. In Newtonian mechanics we are concerned with the (physical) Cauchy stress tensor that reflects the internal resistance of an elastic body to an applied physical force, whereas in Eshelbian mechanics the applied force is a material force acting on a defect of the elastic body with associated (material) (Newton-)Eshelby stress tensor which is the dual stress tensor of the Cauchy stress tensor. The derivation of a posteriori error estimators is based on the well-established strategy of solving an auxiliary dual problem in order to control the global error measures defined in terms of bounded non-linear functionals. In this paper, two principally different strategies are presented to estimate the error measure. The first strategy rests upon an equilibrated residual error estimator based on local Neumann problems, whereas the second one makes use of averaging techniques. The paper is concluded by a numerical example that illustrates our theoretical results.