C1 natural neighbor interpolant for partial differential equations

N. Sukumar*, B. Moran

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

77 Scopus citations


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 Bi3 (Φ) to new shape functions Ψ(Φ), such that the shape functions ψ3I-2 (Φ), ψ3I-2 (Φ), and ψ3I (Φ) for node I are directly associated with the three nodal degrees of freedom wI, θIx, and θIy, 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.

Original languageEnglish (US)
Pages (from-to)417-447
Number of pages31
JournalNumerical Methods for Partial Differential Equations
Issue number4
StatePublished - Jul 1999
Externally publishedYes


  • Bernstein-Bézier polynomials
  • Biharmonic equation
  • C interpolant
  • Natural element method
  • Natural neighbor coordinates
  • Plate bending

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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