INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 76:1782–1818 Published online 10 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2380 Bounds for quantities of interest and adaptivity in the element-free Galerkin method Yolanda Vidal, N´ uria Par´ es, Pedro D´ ıez ∗, † and Antonio Huerta Departament de Matem` atica Aplicada III, Laboratori de C` alcul Num` eric (LaC` aN), Universitat Polit` ecnica de Catalunya, Jordi Girona 1, E-08034 Barcelona, Spain SUMMARY A novel approach to implicit residual-type error estimation in mesh-free methods and an adaptive refinement strategy are presented. This allows computing upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual-type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods. The adaptive strategy proposed leads to a fast convergence of the bounds to the desired precision. Copyright 2008 John Wiley & Sons, Ltd. Received 26 November 2007; Revised 22 December 2007; Accepted 4 April 2008 KEY WORDS: element-free Galerkin; mesh-free; error estimation; engineering outputs; functional outputs; goal-oriented error estimation; residual-based estimators; adaptivity 1. INTRODUCTION Assessment of functional outputs of the solution (goal-oriented error estimation) in computational mechanics problems is a real need in standard engineering practice. End-users of finite elements, finite differences or mesh-free codes are interested in obtaining bounds for quantities of engineering interest. Using a proper error representation, the bounds in the quantities of interest are obtained from bounds of the energy norm. Actually, these bounds are recovered combining upper and lower bounds of the energy error for both the original problem (primal) and an adjoint problem (associated with the selected functional output) [1–4]. It is also important to note that bounds for the energy and for quantities of interest are usually obtained with respect to a reference solution (associated with a much larger space of approximation ∗ Correspondence to: Pedro D´ ıez, Departament de Matem` atica Aplicada III, Laboratori de C` alcul Num` eric (LaC` aN), Universitat Polit` ecnica de Catalunya, Jordi Girona 1, E-08034 Barcelona, Spain. † E-mail: pedro.diez@upc.es Copyright 2008 John Wiley & Sons, Ltd.