Paul Sablonnière

Quadratic spline quasi-interpolants on Powell-Sabin partitions

In 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.

Recent Progress on Univariate and Multivariate Polynomial and Spline Quasi-interpolants

Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral We shall also present some applications of QIs to numerical methods.

C r -finite elements of Powell-Sabin type on the three direction mesh

Letτ be the triangulation generated by a uniform three direction mesh of the plane. Letτ6 be the Powell-Sabin subtriangulation obtained by subdividing each triangleT ∈τ by connecting each vertex to the midpoint of the opposite side.Given a smooth functionu, we construct a piecewise polynomial functionυ ∈Cr (ℝ2) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle ofτ6, interpolating derivatives ofu up to orderr at the vertices ofτ.

Representation of quasi-interpolants as differential operators and applications

Most of the best known positive linear operators are isomorphisms of the maximal subspace of polynomials that they preserve. We give here the differential forms of these isomorphisms and of their inverses for Bernstein and Szasz-Mirakyan operators, and their Durrmeyer and Kantorovitch extensions. They allow to define families of intermediate left and right quasiinterpolants of which we study some properties like Voronowskaya type asymptotic error estimates and uniform boundedness of norms. In the Durrmeyer case, the polynomial coefficients of the associated linear differential operators are nicely connected with Jacobi or Laguerre orthogonal polynomials.

Triangular finite elements of HCT type and class C rho .

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈Cρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ3 of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC1 cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.

Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes

Given a bivariate function f defined in a rectangular domain @W, we approximate it by a C^1 quadratic spline quasi-interpolant (QI) and we take partial derivatives of this QI as approximations to those of f. 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.

Positive spline operators and orthogonal splines

We study some convergence (in Lp) and spectral properties of the positive spline operators Un,kƒ(x) = ∑i (∝01 Mi,k(t)ƒ(t) dt) Ni,k(x), where ∑i Ni,k = 1 and ∝01 Mi,k = 1, Ni,k and Mi,k being the B-splines of degree k and class Ck − 1 associated with some partition of I = [0, 1] into n subintervals. Their eigenfunctions are orthogonal splines generalizing in some sense the Legendre polynomials with which they share many properties.

Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants

We study two methods for solving a univariate Fredholm integral equation of the second kind, based on (left and right) partial approximations of the kernel K by a discrete quartic spline quasi-interpolant. The principle of each method is to approximate the kernel with respect to one variable, the other remaining free. This leads to an approximation of K by a degenerate kernel. We give error estimates for smooth functions, and we show that the method based on the left (resp. right) approximation of the kernel has an approximation order O(h5) (resp. O(h6)). We also compare the obtained formulae with projection methods.

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

Quadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels.

Quadratic spline quasi-interpolants and collocation methods

Univariate 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 and compared with finite difference methods on some numerical examples of elliptic boundary value problems.