Natural neighbour Galerkin methods

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interp olation scheme for multivariate data fitting and smoothing. The numerical i mplementation of natural neighbour co-ordinates in a Galerkin method is kno wn as the natural element method (NEM). In the natural element method, natu ral neighbour co-ordinates are used to construct the trial and test functio ns. Recent studies on NEM have shown that natural neighbour co-ordinates, w hich are based on the Voronoi tessellation of a set of nodes, are an appeal ing choice to construct meshless interpolants for the solution of partial d ifferential equations. In Belikov et al. (Computational Mathematics and Mat hematical Physics 1997; 37(1):9-15), a new interpolation scheme (non-Sibson ian interpolation) based on natural neighbours was proposed. In the present paper, the non-Sibsonian interpolation scheme is reviewed and its performa nce in a Galerkin method for the solution of elliptic partial differential equations that arise in linear elasticity is Studied. A methodology to coup le finite elements to NEM is also described. Two significant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and num erically verified: the computational efficiency of the non-Sibson algorithm in 2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essential boundary conditions on the boundaries o f convex and non-convex domains. Copyright (C) 2000 John Wiley & Sons, Ltd.