INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 86:1457–1480 Published online 24 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3119 Dual boundary-element method: Simple error estimator and adaptivity A. Portela ∗, †, ‡ Universidade de Évora, Escola de Ciências e Tecnologia, 7004-516 Évora, Portugal SUMMARY This paper is concerned with the effective numerical implementation of the adaptive dual boundary-element method (DBEM), for two-dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single-region analysis. Taking advantage on the use of non-conforming parametric boundary-elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM. Copyright  2011 John Wiley & Sons, Ltd. Received 30 June 2010; Revised 17 November 2010; Accepted 26 November 2010 KEY WORDS: DBEM; non-conforming elements; error estimator; mesh refinement; adaptive dual boundary-element method 1. INTRODUCTION The boundary-element method (BEM) is a well-established numerical technique in the engineering community, see Brebbia [1] and Brebbia et al. [2]. The BEM has been successfully applied to potential problems in domains containing no degenerate geometries. These degeneracies, either internal or edge surfaces, which include no area or volume and across which the potential field is discontinuous, are defined as mathematical cracks or slits. In a single-region analysis, the solution of general potential problems cannot be achieved with the direct application of the BEM, as first noted by Hong and Chen [3]. On the other hand, the dual boundary-element method (DBEM) is the most efficient technique devised to overcome this difficulty, as first presented by Portela [4] in Elasticity. A thorough review article of DBEMs was presented by Chen and Hong [5]. A posteriori error estimation and adaptivity is now well established for the finite element method, see Zienkiewicz and Zhu [6, 7]. Likewise the finite element method, a posteriori error estimation and adaptivity, for conforming boundary-element schemes, has been the subject of extensive investigation in the engineering and scientific community. Reviews for error estimation and adaptivity methods, for conforming boundary-element schemes, can be found in Liapis [8] and Kita and Kamiya [9, 10]. In this context, error estimation processes are usually classified into five types: residual, interpolation ∗ Correspondence to: A. Portela, Universidade de Évora, Escola de Ciências e Tecnologia, 7004-516 Évora, Portugal. † E-mail: aportela@uevora.pt ‡ Professor. Copyright  2011 John Wiley & Sons, Ltd.