Interpolation by spline spaces on classes of triangulations
type: Item
instance of: scholarly article
title: Interpolation by spline spaces on classes of triangulations
zbMATH Open document ID: 0966.65018
DOI: 10.1016/S0377-0427(00)00387-3
published in: Journal of Computational and Applied Mathematics
publication date: 2001-08-21T00:00:00Z
review text: This paper presents an inductive method for the construction of triangulations \(\Delta\). These triangular elements are suitable for interpolation by the space \(S^r_q(\Delta)\) of splines of degree \(q\) and smoothness \(r= 1,2\). The article is divided into seven sections. The procedures to construct triangulations and admissible sets are described in sections 2 and 3, respectively. The notion of admissible sets, which is considered here for technical reasons, is given in terms of the Bézier-Bernstein representation \[ p^{[l]}(x, y)= \sum_{i+j+k= q} a^{[l]}_{i,j,k} {q!\over i!j!k!} \Phi^i_1(x,y) \Phi^j_2(x, y)\Phi^k_3(x, y),\quad (x,y)\in T^{[l]}, \] where \(p^{[l]}= s|_{T^{[l]}}\) are the polynomials pieces of the spline \(s\in S^r_q(\Delta)\), \(T^{[l]}= \Delta(v^{[l]}_1, v^{[l]}_2, v^{[l]}_3)\) are the triangles of \(\Delta\), \(l= 1,\dots, N\), and \(\Phi_\mu\), \(\mu= 1,2,3\), are linear polynomials uniquely defined by \(\Phi_\mu(v^{[l]}_\nu)= \delta_{\mu,\nu}\), \(\nu= 1,2,3\). Some sets \({\mathcal A}_r\subset I= \{(i,j,k,l): i+ j+ k= q, l= 1,\dots, N\}\), are found to be admissible sets for \(S^r_q(\Delta)\), \(r= 1,2\). In this respect two theorems are presented in the third section for the cases \(q\geq 3\), \(r= 1\); and \(q\geq 5\), \(r= 2\). The problem of constructing Lagrange and Hermite interpolation sets for \(S^r_q(\Delta)\) is investigated in section 4. Sections 5 and 6 are devoted to prove the main theorems of the paper and contain other results of technical interest. In order to illustrate the power of the new method, the last section contains tables with numerical results on interpolation of functions and scattered data. The procedure is used to approximate the test function of Franke, by \(S^1_q(\Delta)\), \(q= 3,4\), and \(S^1_2(\Delta)\).
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1491265
zbMATH Keywords: Burnside splines
zbMATH Keywords: triangulations
zbMATH Keywords: interpolation
zbMATH Keywords: Bézier-Bernstein representation
zbMATH Keywords: numerical results
zbMATH Keywords: scattered data
author: Günther Nürnberger
author: Frank Zeilfelder
reviewed by: J. Illán
MaRDI profile type: Publication
cites work: A bivariate \(C^ 2\) Clough-Tocher scheme
cites work: An Explicit Basis for $C^1 $ Quartic Bivariate Splines
cites work: The dimension of bivariate spline spaces of smoothness r for degree \(d\geq 4r+1\)
cites work: A survey of curve and surface methods in CAGD
cites work: Approximation power of smooth bivariate pp functions
cites work: A sharp upper bound on the approximation order of smooth bivariate pp functions
cites work: Q4205895
cites work: Q3831405
cites work: Swapping Edgesof Arbitrary Triangulations to Achieve the Optimal Order of Approximation
cites work: Construction of local 𝐶¹ quartic spline elements for optimal-order approximation
cites work: Stability of Optimal-Order Approximation by Bivariate Splines over Arbitrary Triangulations
cites work: On bivariate super vertex splines
cites work: Q3839092
cites work: Approximation order of bivariate spline interpolation for arbitrary smoothness
cites work: Q4240347
cites work: Bivariate spline interpolation with optimal approximation order
cites work: Q4379943
cites work: Q3776153
cites work: A conforming finite element for plate bending
cites work: Q5289605
cites work: A \(C^ 2\) finite element and interpolation
cites work: Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree
cites work: An iterative method for computing multivariate \(C^ 1\) piecewise polynomial interpolants
cites work: Super spline spaces of smoothness \(r\) and degree \(d\geq{} 3r+2\)
cites work: Q4008655
cites work: Q4325863
cites work: Triangular finite elements of HCT type and class \(C^ \rho\)
cites work: Quadrilateral finite elements of FVS type and class \(C^ \rho\)
cites work: Scattered data interpolation and approximation using bivariate \(C^{1}\) piecewise cubic polynomials
cites work: On the approximation power of bivariate splines
cites work: On the Approximation Power of Splines on Triangulated Quadrangulations
cites work: A Nodal Basis for C 1 Piecewise Polynomials of Degree n ≥5
cites work: Approximation order of bivariate spline interpolation
cites work: Q4379954
cites work: Lagrange and hermite interpolation by bivariate splines
cites work: Bivariate spline interpolation at grid points
cites work: Error analysis in interpolation by bivariate C1-splines
cites work: Q4240375
cites work: Piecewise Quadratic Approximations on Triangles
cites work: Interpolation by bivariate quadratic splines on a four-directional mesh
cites work: Q3773161
cites work: Dual bases for spline spaces on cells
cites work: On Super Splines and Finite Elements
cites work: Q3737948
cites work: Q3728562
cites work: A \(C^ 2\)-quintic spline interpolation scheme on triangulation
description: scientific article; zbMATH DE number 1491265