Spline quasi-interpolation in the Bernstein basis on the Powell–Sabin 6-split of a type-1 triangulation

Abstract

In this paper, we provide quasi-interpolation schemes defined on a uniform triangulation of type-1 endowed with a Powell–Sabin refinement. In contrast to the usual construction of quasi interpolation splines on the 6-split, the approach described in this work does not require a set of appropriate basis functions. The approximating splines are directly defined by setting their Bézier ordinates to suitable combinations of the given data values. The resulting quasi-interpolants are $C^1$ continuous and reproduce quadratic polynomials. Some numerical tests are given to confirm the theoretical results.

Introduction

Spline interpolation and quasi-interpolation have become indispensable mathematical tools for the approximation of functions and data. They are topics widely developed in the literature, which includes many different ways of construction of approximating splines. In the uniform case, standard quasi-interpolation produces approximants in the space spanned by a family of non-negative compactly supported functions forming a partition of unity (for instance, B-splines or box splines) and no linear system resolution is required.

Following the idea used in [1], a new procedure was introduced in [2] and [3] based on the definition of the Bernstein–Bézier (BB-) coefficients of the spline on each triangle in the uniform partition. They are set directly from specific point values in a neighbourhood of the triangle so that $C^1$ continuity is achieved, in addition to the reproduction of the polynomials of a specific degree.

The aim of the method addressed in [3] is to construct $C^1$ quartic splines on a type-1 triangulation in such a way that the cubic polynomials are reproduced. Simple rules to produce the BB-coefficients of the quasi-interpolant on each triangle of the partition are provided. The values of the quasi-interpolated function at the domain points of order four relative each triangle of the triangulation are assumed to be known. For a triangle $T〈v_1,v_2,v_3〉$ with barycentric coordinates $\left({\lambda }_1,{\lambda }_2,{\lambda }_3\right)$, they are of the form $\frac{1}{3}\left(i{v}_1+j{v}_2+k{v}_3\right)$,with $i,j,k$ non negative integers such that $i+j+k=3$. The splines obtained from these rules interpolate the point values at vertices.

In [4] a general study of this problem is carried out in order to determine all possible rules for defining BB-coefficients giving $C^1$ continuity and exactness on the space of polynomials of total degree equal to three. It is shown that there exists a multi-parametric family of rules, having nineteen degrees of freedom, and then the reduction of the number of evaluations needed to compute the BB-coefficients is addressed. Moreover, it is proved that there exists a family of rules based on evaluation at vertices and midpoints of edges of triangles depending on only three parameters. The resulting quasi-interpolating splines also interpolate the point values at vertices. Both in [3] and [4] the used rules have symmetries, so the computational cost is reduced.

A similar methodology is used in [5] to construct $C^1$ cubic quasi-interpolants that reproduce quadratic polynomials when the values at the vertices and midpoints are known. In this case, quasi-interpolants do not interpolate the data values at vertices. Moreover, different rules correspond to different domain points. There are no symmetries applicable. However, there exists only one solution, i.e. a set of rules that allow the objectives to be met: $C^1$ continuity and reproduction of the quadratic polynomials.

It would be natural, therefore, to construct $C^1$ quadratic quasi-interpolants in the same way on type-1 triangulations, reproducing polynomials of degree 1 at most, but this is not possible. Only constants can be reproduced. Consequently, we propose to construct quadratic quasi-interpolants on a type-1 triangulation endowed with a Powell–Sabin refinement [6] to achieve the optimal approximation order. $C^1$ quadratic quasi-interpolation on general Powell–Sabin triangulations is studied in [7] and [8] by defining appropriate bases of B-spline-like functions. Super-convergent quadratic quasi-interpolants are defined in [9] and quadratic quasi-interpolation is also studied on the sphere [10].

The rest of this work is organized as follows: in Section 2, we give some preliminaries on the Bernstein–Bézier form of quadratic polynomials on type-1 triangulations and we introduce the Powell–Sabin split and some useful notations used throughout the paper. In Section 3, we provide the unique $C^1$ quasi-interpolating spline that reproduces quadratic polynomials. It is defined from the prescribed values at the vertices and midpoints of the edges of the triangulation. In Section 4, we propose some numerical tests that confirm the theoretical results established in Section 3. Another $C^1$ scheme exact on the space ${\mathbb{P}}_2$ of polynomials of degree less than or equal to two is provided in Section 5, but now defined only from the point values at vertices. Numerical tests supporting the theoretical results of Section 5 are also included. We conclude the paper with a conclusion section.