Bivariate quartic spline spaces and quasi-interpolation operators

In this paper, we study two bivariate quartic spline spaces and , and present two classes of quasi-interpolation operators in the two spaces, respectively. Some results on the operators are given.     

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.     

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.     

Trivariate spline approximations of 3D Navier-Stokes equations

We present numerical approximations of the 3D steady state Navier-Stokes equations in velocity-pressure formulation using trivariate splines of arbitrary degree and arbitrary smoothness with . Using functional arguments, we derive the discrete Navier-Stokes equations in terms of -coefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergence-free condition are enforced through Lagrange multipliers. The pressure is computed by solving a Poisson equation with Neumann boundary conditions. We have implemented this approach in MATLAB and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.

     

Quadrature formulas descending from BS Hermite spline quasi-interpolation

Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy.     

Quasi-interpolants from spline interpolation operators

For bi-infinite Toeplitz matrices, it is easy to see that thekth partial sum of the Neumann series reproduces polynomials of orderk There is no guarantee, however, that the spectral radius is less than 1. A principal result of this paper is to show that for the spline interpolation Toeplitz case the spectral radius is less than 1 whenA is invertible and the main diagonal is the central diagonal. This is not true for all totally positive Toeplitz matrices as shown by an example in Section 2.Communicated by Charles A. Micchelli.     

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept “generator”. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.     

Stability of spline approximation methods for multidimensional pseudodifferential operators

The present paper provides stability considerations of spline approximation methods for multidimensional singular operators. This paper should be regarded as a first step in establishing spline approximation methods for pseudodifferential operators on manifolds.     

Approximate Hermite quasi-interpolation

In this article, we derive approximate quasi-interpolants when the values of a function u and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants, which provide high-order approximations for solutions to elliptic differential equations with constant coefficients.     

Cubature rule associated with a discrete blending sum of quadratic spline quasi-interpolants

A new cubature rule for a parallelepiped domain is defined by integrating a discrete blending sum of C¹ quadratic spline quasi-interpolants in one and two variables. We give the weights and the nodes of this cubature rule and we study the associated error estimates for smooth functions. We compare our method with cubature rules based on the tensor products of spline quadratures and classical composite Simpson’s rules.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

Uniform bounds for cardinal Hermite spline operators with double knots

A method is given for computing the uniform norm of the cardinal Hermite spline operator. This is the operator that takes two bounded biinfinite sequences of numbers into the unique bounded spline of degree 2k − 1(k 2) with knots of multiplicity two at the integers and that interpolates the two given sequences for both functional and first derivative values at the integers. The computational schema relies on knowledge of the Bernoulli splines, while the theoretical aspects make use of some properties of zeros of periodic splines.     

Computing near-best fixed pole rational interpolants

We study rational interpolation formulas on the interval [−1,1] for a given set of real or complex conjugate poles outside this interval. Interpolation points which are near-best in a Chebyshev sense were derived in earlier work. The present paper discusses several computation aspects of the interpolation points and the corresponding interpolants. We also study a related set of points (that includes the end points), which is more suitable for applications in rational spectral methods. Some examples are given at the end of this paper.     

Lebesgue constants for periodic Hermite spline interpolation operators on uniform lattices

We derive a complex line integral representation for the ebyshev norm of periodic spline interpolation operators of odd degree on uniform lattices. Several generalizations are indicated.     

On the cardinal spline interpolation corresponding to infinite order differential operators

This paper discusses some problems on the cardinal spline interpolation corresponding to infinite order differential operators. The remainder formulas and a dual theorem are established for some convolution classes, where the kernels arePF densities. Moreover, the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of averagen-Kolmogorov widths are obtained for the convolution class. Supported in part by NSFC.     

A Trivariate Box Macroelement

Given a rectangular box which has been split into 24 tetrahedra, we show how to construct a C¹ macroelement using polynomial pieces of degree 6.