68 research outputs found
Univariate spline quasi-interpolants and applications to numerical analysis
We describe some new univariate spline quasi-interpolants on uniform
partitions of bounded intervals. Then we give some applications to numerical
analysis: integration, differentiation and approximation of zeros
Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains
Given a non-uniform criss-cross partition of a rectangular domain ,
we analyse the error between a function defined on and two types
of -quadratic spline quasi-interpolants (QIs) obtained as linear
combinations of B-splines with discrete functionals as coefficients. The main
novelties are the facts that supports of B-splines are contained in
and that data sites also lie inside or on the boundary of . Moreover,
the infinity norms of these QIs are small and do not depend on the
triangulation: as the two QIs are exact on quadratic polynomials, they give the
optimal approximation order for smooth functions. Our analysis is done for
and its partial derivatives of the first and second orders and a particular
effort has been made in order to give the best possible error bounds in terms
of the smoothness of and of the mesh ratios of the triangulation
Weierstrass quasi-interpolants
International audienceIn this paper, the expression of Weierstrass operators as differential operators on polynomials is used for the construction of associated quasi-interpolants. Then the convergence properties of these operators are studied
Quadrature rules associated with Baskakov quasi-interpolants
Quadrature rules on the positive real half-line obtained by integrating the Baskakov quasi-interpolants described in \cite{MM, Sab7} are constructed and their asymptotic convergence orders are studied. These results are illustrated by some numerical examples. The formulas are based on series of values of the function on uniform partitions and are not comparable with Gauss quadrature rules
A quadrature formula associated with a univariate quadratic spline quasi-interpolant
International audienceWe study a new simple quadrature rule based on integrating a quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions and we compare this formula with Simpson's rule
Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants
paru dans la revue sous le titre : Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshesGiven a bivariate function defined in a rectangular domain , we approximate it by a quadratic spline quasi-interpolant (abbr. QI) and we take partial derivatives of this QI as approximations to those of . We give error estimates and asymptotic expansions for these approximations. We also propose a simple algorithm for the determination of stationary points, illustrated by a numerical example
Quadratic Spline Quasi-interpolants on Powell-Sabin Partitions
2004-16International audienceIn this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented
Quadratic spline quasi-interpolants and collocation methods
International audienceUnivariate and multivariate quadratic spline quasi-interpolants provide interesting approximation formulas for derivatives of approximated functions that can be very accurate at some points thanks to the superconvergence properties of these operators. Moreover, they also give rise to good global approximations of derivatives on the whole domain of definition. From these results, some collocation methods are deduced for the solution of ordinary or partial differential equations with boundary conditions. Their convergence properties are illustrated on some numerical examples of elliptic boundary value problems
Near-best quasi-interpolants associated with H-splines on a three-directional mesh.
International audienceSpline quasi-interpolants with best approximation orders and small norms are useful in several applications. In this paper, we construct the so-called near-best discrete and integral quasi-interpolants based on H-splines, i.e., B-splines with regular hexagonal supports on the uniform three-directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials, and minimize an upper bound of their infinite norms depending on a finite number of free parameters. We show that this problem has always a solution, but it is not unique in general. Concrete examples of these types of quasi-interpolants are given in the two last sections
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