Error bounds on the approximation of functions and partial derivatives by quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of a rectangular domain

Abstract

Given a non-uniform criss-cross triangulation of a rectangular domain Ω , we consider the approximation of a function f and its partial derivatives, by general C 1 quadratic spline quasi-interpolants and their derivatives. We give error bounds in terms of the smoothness of f and the characteristics of the triangulation. Then, the preceding theoretical results are compared with similar results in the literature. Finally, several examples are proposed for illustrating various applications of the quasiinterpolants studied in the paper.

NURBS on Criss-cross Triangulations and Applications

“…We remark that the first-boundary-layer and the second-boundary-layer B-splines can be expressed as linear combination of classical B-splines with octagonal support and simple knots [3,23].…”

“…In particular, considering the subtriangulation T 2m,2n of T mn , obtained by subdividing each rectangle Furthermore, given a function ρ ∈ C(Ω 0 ), we can establish how well a spline in S 1 2 (T mn ) is able to approximate such a function. Indeed, concerning the approximation properties of bivariate C 1 quadratic spline spaces on criss-cross triangulations, we can refer to [9], where the authors give error bounds for functions and partial derivatives in terms of the smoothness of functions that are approximated and the characteristics of the triangulation. In particular, since we know that there exist quasi-interpolating operators exact on the space P 2 (see e.g [4, Sect.…”

“…In particular, since we know that there exist quasi-interpolating operators exact on the space P 2 (see e.g [4, Sect. 2.4], [7,9,19]), then the optimal approximation order is achieved for sufficiently smooth functions, i.e. if ρ ∈ C 3 (Ω 0 ) there exists a constant C > 0 such that…”

“…Since many domains of interest in engineering problems are often described by conic sections, we consider as geometry function G a quadratic NURBS surface on a criss-cross triangulation, presented in Section 3, in order to exactly reproduce them. More precisely, we consider a criss-cross triangulation T mn of Ω 0 , the space S 1 2 (T mn ), its spanning set B mn and, given opportune weights {w ij } (i,j)∈Kmn , from B mn we get N mn defined in (9). Then, we define the global geometry function…”

“…[2,3,4,5,6,7,8,9]) has been considered in the solution of several kinds of problems, like construction of approximation operators with special properties (see e.g. [10,11,12,13,14,15,16]), solution of integral equations (see e.g.…”