Minimizing the quasi-interpolation error for bivariate discrete quasi-interpolants

Error estimates for some quasi-interpolation operators

‖u− Ihu‖L2(T ) ≤cThT ‖∇ku‖L2(ω̃T ), ‖u− Ihu‖L2(E) ≤cEh E ‖∇ku‖L2(ω̃E). Here, k ∈ {1, 2}, Ih is some quasi-interpolation operator, T and E are a simplex and a face thereof, hT and hE measure the size of T and E, and ω̃T and ω̃E are neighbourhoods of T and E which should be as small as possible. Note that the interpolate Ihu never needs to be computed explicitely. Moreover, for problems in two and th...

Quasi-interpolants and Quasi-interpolation: Important Properties and Propositions of Terminology

Weierstrass quasi-interpolants

In 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. AMS Classification: 41A35.

Numerical Integration Based on Bivariate Quartic Quasi-Interpolation Operators

In this paper, we propose a method to deal with numerical integral by using two kinds of C quasi-interpolation operators on the bivariate spline space, and also discuss the convergence properties and error estimates. Moreover, the proposed method is applied to the numerical evaluation of 2-D singular integrals. Numerical experiments will be carried out and the results will be compared with some...

Approximation Order without Quasi-Interpolants

In the study of approximation order, particularly in a multivariable setting, quasi-interpolants have played a major role. This report points out some limitations of quasi-interpolants and describes some recent results on approximation order obtained without the benefit of the quasi-interpolant idea. §1. Approximation Order In most general terms, “approximation order” is defined as follows. Def...