Sibson and non-Sibsonian interpolants for elliptic partial differential equations

The Natural Element Method (NEM) is a meshless Galerkin method which has shown promise in the area of computational mechanics. In earlier applications of NEM [1–3], natural neighbor (Sibson) coordinates [4] were used to construct the trial and test functions. Recently, Belikov and co-workers [5] proposed a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbors. In this paper, we present the Sibson and the non-Sibsonian interpolants, and discuss their use in a Galerkin scheme for the solution of elliptic PDEs. In particular, by choosing the non-Sibsonian interpolant, the exact imposition of essential boundary conditions in a meshless method is realized.