Natural neighbour Galerkin methods

Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data fitting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that natural neighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealing choice to construct meshless interpolants for the solution of partial differential equations. In Belikov et al. (Computational Mathematics and Mathematical Physics 1997; 37(1):9-15), a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the non-Sibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution of elliptic partial differential equations that arise in linear elasticity is studied. A methodology to couple finite elements to NEM is also described. Two significant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and numerically 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 of convex and non-convex domains.