Quasi-interpolation by C1 quartic splines on type-1 triangulations

Abstract

In this paper we construct two new families of C1 quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results.