On Chebyshev-type integral quasi-interpolation operators

Spline quasi-interpolants on the real line are approximating splines to given functions with optimal approximation orders. They are called integral quasi-interpolants if the coefficients in the spline series are linear combinations of weighted mean values of the function to be approximated. This paper is devoted to the construction of new integral quasi-interpolants with compactly supported piecewise polynomial weights. The basic idea consists of minimizing an expression appearing in an estimate for the quasi-interpolation error. It depends on how well the quasi-interpolation operator approximates the first non-reproduced monomial. Explicit solutions as well as some numerical tests in the B-spline case are given.     

A modified Kulkarni's method based on a discrete spline quasi-interpolant

We use a discrete spline quasi-interpolant (abbr. dQI) defined on a bounded interval for the numerical solution of linear Fredholm integral equations of the second kind with a smooth kernel by collocation and a modified Kulkarni's method together with its Sloan's iterated version. We study the approximation errors of these methods and we illustrate the theoretical results by a numerical example.     

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

The aim of this paper is to investigate, in a bounded domain of R³, two blending sums of univariate and bivariate C¹ 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 quasi-interpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.     

An identity for multivariate Bernstein polynomials

We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order.     

Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations

Hamiltonian PDEs have some invariant quantities such as energy and momentum, etc., which should be well conserved with the numerical integration. In this paper we concentrate on the nonlinear wave equation. We study how a space discretization by using multiquadric quasi-interpolation method makes the space discretized system also possess some invariants which are good approximation of the continuous energy. Then, appropriate symplectic scheme is employed for the integration of the semi-discretized system. Theoretical results show that the proposed method has not only high order accuracy but also good properties of long-time tracking capability. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.     

Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions

We derive $\mathit{H}$(curl)-error estimates and improved $\mathit{L}$²-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than $\frac{1}{2}$ and can be arbitrarily close to $0$ when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in $\mathit{H}$(curl)- and $\mathit{H}$(div)-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev smoothness index arbitrarily close to $0$. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.