Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

Abstract

AbstractThis paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.