Constructing Good Coefficient Functionals for Bivariate C 1 Quadratic Spline Quasi-Interpolants

Remogna, Sara

doi:10.1007/978-3-642-11620-9_22

Cited by 20 publications

(19 citation statements)

References 12 publications

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“…This can increase the value of the infinity norm of the quasiinterpolant and eventually give some trouble in numerical computations. For that reason, we shall develop in a further paper alternative functionals having larger supports and lower infinity norms (as is done in the corresponding work by Sara Remogna [25] on quadratic spline QIs).…”

Section: Quasi-interpolants In a Rectangular Domainmentioning

confidence: 98%

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

Davydov

Sablonnière²

2010

Journal of Approximation Theory

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We study two kinds of quasi-interpolants (abbr. QI) in the space of C 2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinity norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples.

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Section: Quasi-interpolants In a Rectangular Domainmentioning

confidence: 98%

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

Davydov

Sablonnière²

2010

Journal of Approximation Theory

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“…The simplest one, S 1 , is a bivariate extension of the Schoenberg-Marsden operator (Chui and Wang, 1984;Sablonnière, 2003a,b). The other two kinds of operators, constructed in Remogna (2010a), are related to the operator S 2 , proposed in Sablonnière (2003a,b), exact on P 2 [x, y] and obtained from (6) by replacing the Laplacian ∆ by its five points discretisation. The operator S 2 has the form…”

Section: Quadratic Spline Quasi-interpolants On a Bounded Rectanglementioning

confidence: 99%

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Remogna¹,

Sablonnière²

2011

Computer Aided Geometric Design

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The aim of this paper is to investigate, in a bounded domain of R 3 , two blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants.The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm.We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasiinterpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.

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“…Using the method presented in [19], we are interested in the construction of two different types of discrete quasi-interpolants…”

Section: Fig 7 Points Of Superconvergencementioning

confidence: 99%

“…The method used in this subsection is closely related to the techniques given in [1][2][3]17] to define near-best discrete quasi-interpolants on type-1 and type-2 triangulations (see also [4,19,21]). …”

Section: Construction Of Near-best Boundary Functionalsmentioning

confidence: 99%

Bivariate C 2 cubic spline quasi-interpolants on uniform Powell–Sabin triangulations of a rectangular domain

Remogna

2011

Adv Comput Math

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In this paper we construct discrete quasi-interpolants based on C 2 cubic multi-box splines on uniform Powell-Sabin triangulations of a rectangular domain. The main problem consists in finding the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters, or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain.Finally, we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed operators.

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Constructing Good Coefficient Functionals for Bivariate C 1 Quadratic Spline Quasi-Interpolants

Cited by 20 publications

References 12 publications

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Bivariate C 2 cubic spline quasi-interpolants on uniform Powell–Sabin triangulations of a rectangular domain

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