C¹ natural neighbor interpolant for partial differential equations

Natural neighbor coordinates [20] are optimum weighted-average measures for an irregular arrangement of nodes in ℝⁿ. [26] used the notion of Bézier simplices in natural neighbor coordinates Φ to propose a C¹ interpolant. The C¹ interpolant has quadratic precision in Ω ⊂ ℝ², and reduces to a cubic polynomial between adjacent nodes on the boundary ∂Ω. We present the C¹ 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_i³ (Φ) to new shape functions Ψ(Φ), such that the shape functions ψ_3I-2 (Φ), ψ_3I-2 (Φ), and ψ_3I (Φ) for node I are directly associated with the three nodal degrees of freedom w_I, θ_Ix, and θ_Iy, respectively. The C¹ 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.