Local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation

Superconvergent local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation

“…Our aim in this paper is to give a new B-spline representation of the Hermite interpolant of any piecewise polynomial s on D of class at least the class of elements of S 1;0 2 ðDÞ and of degree n P 2 in terms of the polar forms of s. Then, for any sufficiently smooth function f, we give a method for constructing superconvergent discrete quasi-interpolants Qf which approximate f and its first derivatives at the vertices of the quadrangulation AE with an order at least equal to 3. The new results that we present in this paper are a generalization and an improvement of the ones developed in [22].…”

“…It is well known that splines constructed in the Powell-Sabin space and S 1;0 2 ðDÞ reproduce the polynomials with total degree two. Therefore, they have the same optimal approximation order three (see [22,26,29]). However, we know that the number of the polynomial pieces are key factors affecting the computational efficiency.…”

“…One can use the same technique given in [22,29] to prove that kQ k 1;X is bounded. The superconvergent discrete quasiinterpolant developed in this paper does not only reproduce the quadratic polynomials, but also satisfies the following more general property Qp ¼ Ip; 8p 2 P n ðR 2 Þ:…”

“…In this subsection we recall the construction and the principal properties of the multivariate B-splines over a refined quadrangulation of a planar domain introduced in [22].…”

“…Interesting results have been obtained for these types of partitions (see [16,30,31] and references therein). Recently, Lamnii et al use the same technique in [22] to give a suitable normalized B-spline representation for a special multivariate quadratic spline space S 1;0 2 ðDÞ over a refined quadrangulation. As in [29], these B-splines have been used by the same authors for constructing some quasi-interpolants with optimal approximation order.…”