Goal-oriented a posteriori error estimates in linear elastic fracture mechanics

Goal-oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the J-integral in linear elastic fracture mechanics using the finite element method. The J-integral provides a criterion for crack propagation of a pre-existing crack. Therefore, its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures. This can be achieved by goal-oriented adaptive finite element methods based on the well-established strategy of solving an auxiliary dual problem in order to control the error of a (non)linear functional, here the J-integral. In this paper, two principally different goal-oriented adaptive strategies are presented. The first strategy rests upon equilibrated residual error estimators 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.