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Construction of 2D explicit cubic quasi-interpolating splines in Bernstein-Bézier form

Domingo Barrera, Salah Eddargani, María José Ibáñez, Sara Remogna

TL;DR

The paper addresses constructing a 2D $C^{1}$ cubic quasi-interpolant on a uniform three-direction triangulation by directly assigning Bernstein-Bézier coefficients from function values and gradients to reproduce polynomials up to degree $2$. It establishes a $5$-parameter family of $C^{1}$ quasi-interpolants exact on $\mathbb{P}_{2}$ and derives BB-coefficient masks to satisfy $C^{1}$ continuity, with extra parameters chosen to achieve edge-midpoint super-convergence. By fixing four parameters to enforce this super-convergence and selecting $\lambda=\tfrac{1}{2}$, the authors obtain a symmetric, high-quality quasi-interpolant and provide explicit masks. Numerical tests on Franke and Nielsen functions confirm the theoretical convergence: global $C^{1}$-error scales as $O(h^{3})$, with edge-midpoint errors exhibiting $O(h^{4})$ super-convergence, demonstrating the method’s accuracy and practicality for Hermite data on triangulations.

Abstract

In this paper, the construction of $C^{1}$ cubic quasi-interpolants on a three-direction mesh of $\RR^{2}$ is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-Bézier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.

Construction of 2D explicit cubic quasi-interpolating splines in Bernstein-Bézier form

TL;DR

The paper addresses constructing a 2D cubic quasi-interpolant on a uniform three-direction triangulation by directly assigning Bernstein-Bézier coefficients from function values and gradients to reproduce polynomials up to degree . It establishes a -parameter family of quasi-interpolants exact on and derives BB-coefficient masks to satisfy continuity, with extra parameters chosen to achieve edge-midpoint super-convergence. By fixing four parameters to enforce this super-convergence and selecting , the authors obtain a symmetric, high-quality quasi-interpolant and provide explicit masks. Numerical tests on Franke and Nielsen functions confirm the theoretical convergence: global -error scales as , with edge-midpoint errors exhibiting super-convergence, demonstrating the method’s accuracy and practicality for Hermite data on triangulations.

Abstract

In this paper, the construction of cubic quasi-interpolants on a three-direction mesh of is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-Bézier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.
Paper Structure (6 sections, 5 theorems, 30 equations, 5 figures, 1 table)

This paper contains 6 sections, 5 theorems, 30 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. $C^{1}$ cubic Hermite quasi-interpolation
  4. Choice of parameters
  5. Numerical tests
  6. Conclusions

Key Result

lemma 1

The polynomials $p$ and $\widetilde{p}$ given in (p) and (ptilde), respectively, join with $C^{r}$ smoothness across the edge defined by $v_{2}$ and $v_{3}$ if where $\tau$ denotes the barycentric coordinates of $v_{4}$ with respect to $T$.

Figures (5)

  • Figure 1: The triangulation $\Delta_{3}$ and the hexagon $H_{i,j}$ centered at $v_{i,j}$ defined by its six closest vertices.
  • Figure 2: The domain points in $T_{i,j}$ and $\widetilde{T}_{i,j}$.
  • Figure 3: The conditions equivalent to the $C^{1}$ smoothness of $Qf$.
  • Figure 4: Masks associated with vertices for achieving $C^{1}$ smoothness and superconvergence at midpoints of edges (and also at vertices), and ensuring exactness on ${\mathbb P}_{2}$.
  • Figure 5: Masks associated with barycenters.

Theorems & Definitions (9)

  • lemma 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • corollary 1
  • proof