A posteriori error estimation for elasto-plastic problems based on duality theory

In this paper we introduce a new approach to a posteriori error estimation for elasto-plastic problems based on the duality theory of the calculus of variations. We show that, in spite of the prevailing view, duality methods provide a viable way for obtaining computable a posteriori error estimates for nonlinear boundary value problems without directly solving the dual problem. Rigorous mathematical analysis leads to what we call duality error estimators consisting of two parts: the error in the constitutive law and the error in the equilibrium equations. This representation is both physically meaningful and computationally important. The duality error estimators hold for any conforming approximation of the exact solution regardless of whether or not they satisfy the Galerkin orthogonality condition. In particular, they encompass the familiar smoothening or gradient averaging techniques commonly used in practice. We prove that the duality error estimators are ‘equivalent to the error’ when the approximate solution is the exact solution of the discretised problem. Moreover, in the case of linear elasticity, the known residual type error estimators can be obtained from the duality error estimators. Numerical results for a model elasto-plastic problem show the accuracy of the duality error estimators.