Point and differential C1 quasi-interpolation on three direction meshes

Abstract In this paper we construct and analyse C 1 cubic and quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data, without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions. In particular, the C 1 cubic splines are directly determined by setting their Bernstein–Bezier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain, in such a way that the C 1 -smoothness conditions are satisfied and approximation order three is guaranteed, for smooth functions. We also propose some numerical tests that confirm the theoretical results. Then, from the above C 1 cubic splines we obtain C 1 quartic splines exact on P 3 , achieving approximation order four. The associated differential quasi-interpolation operator involves the values of the first partial derivatives in its definition.