Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains

摘要

Given a non-uniform criss-cross partition of a rectangular domain $\Omega$,we analyse the error between a function $f$ defined on $\Omega$ and two typesof $C^1$-quadratic spline quasi-interpolants (QIs) obtained as linearcombinations of B-splines with discrete functionals as coefficients. The mainnovelties are the facts that supports of B-splines are contained in $\Omega$and that data sites also lie inside or on the boundary of $\Omega$. Moreover,the infinity norms of these QIs are small and do not depend on thetriangulation: as the two QIs are exact on quadratic polynomials, they give theoptimal approximation order for smooth functions. Our analysis is done for $f$and its partial derivatives of the first and second orders and a particulareffort has been made in order to give the best possible error bounds in termsof the smoothness of $f$ and of the mesh ratios of the triangulation.