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Bivariate spline interpolation with optimal approximation order

2001; Springer Science+Business Media; Volume: 17; Issue: 2 Linguagem: Inglês

10.1007/s003650010034

1432-0940

Oleg Davydov, Günther Nürnberger, Frank Zeilfelder,

Numerical methods in engineering

Let Δ be a triangulation of some polygonal domain Ω ⊂ R2 and let S q r (Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].