C1 Natural Neighbor Interpolant for Partial Differential Equations

Natural neighbor coordinates [20] are optimum weighted-average measures for an irregular arrangement of nodes in R. [26] used the notion of Bézier simplices in natural neighbor coordinates Φ to propose a C interpolant. The C interpolant has quadratic precision in Ω ⊂ R, and reduces to a cubic polynomial between adjacent nodes on the boundary ∂Ω. We present the C formulation and propose a computational methodology for its numerical implementation (Natural Element Method) for the solution of partial differential equations (PDEs). The approach involves the transformation of the original Bernstein basis functions B i (Φ) to new shape functions Ψ(Φ), such that the shape functions ψ3I−2(Φ), ψ3I−1(Φ), and ψ3I(Φ) for node I are directly associated with the three nodal degrees of freedom wI , θIx , and θIy , respectively. The C shape functions interpolate to nodal function and nodal gradient values, which renders the interpolant amenable to application in a Galerkin scheme for the solution of fourth-order elliptic PDEs. Results for the biharmonic equation with Dirichlet boundary conditions are presented. The generalized eigenproblem is studied to establish the ellipticity of the discrete biharmonic operator, and consequently the stability of the numerical method. c © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 417–447, 1999