INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2006; 65:2167–2202 Published online 12 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1534 Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods M. Arroyo ‡, § and M. Ortiz ∗, † Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A. SUMMARY We present a one-parameter family of approximation schemes, which we refer to as local maximum- entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent ) statistical inference. Local max-ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of anal- ysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regu- larization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements. Copyright  2005 John Wiley & Sons, Ltd. KEY WORDS: maximum entropy; information theory; approximation theory; meshfree methods; Delaunay triangulation 1. INTRODUCTION This paper is concerned with the formulation of approximation schemes that bridge continuously two important limits: Delaunay triangulation and maximum-entropy statistical inference. The resulting basis functions bear similarities with those obtained from the moving least squares ∗ Correspondence to: M. Ortiz, Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, U.S.A. † E-mail: ortiz@aero.caltech.edu ‡ E-mail: marino.arroyo@upc.edu § Current address: LaCàN, Universitat Politècnica de Catalunya, C/Jordi Girona 1-3, Barcelona 08034, Spain. Contract/grant sponsor: Department of Energy Contract/grant sponsor: NSF Received 17 December 2004 Revised 27 April 2005 Copyright  2005 John Wiley & Sons, Ltd. Accepted 17 August 2005