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

Abstract

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