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

Catterina Dagnino (1) , Paul Sablonnière (2)

1 Dipartimento di Matematica "Giuseppe Peano" [Torino]
2 IRMAR - Institut de Recherche Mathématique de Rennes

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 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 $\Omega$ 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 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.

Mots clés: Bivariate splines; Approximation by splines

Domaines: Analyse numérique [math.NA]

Catterina Dagnino, Paul Sablonnière. Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains. 2006. ⟨hal-00019042⟩