Optimal bivariate C1C1 cubic quasi-interpolation on a type-2 triangulation

In [A. Guessab, O. Nouisser, G. Schmeisser, Multivariate approximation by a combination of modified Taylor polynomials, J. Comput. Appl. Math. 196 (2006) 162–179], a general method is proposed to increase the approximation order of approximation operators. In this work, by using these enhancement techniques, we introduce and study new schemes based on a C1C1-spline quasi-interpolant on a type-2 triangulation. They are designed for approximating real-valued functions defined on R2R2. The proposed method is based on the following idea: from a discrete quasi-interpolant defined by the quadratic box-spline exact on P2P2 and by judiciously choosing the first-order Taylor coefficients, we derive a cubic differential quasi-interpolant yielding optimal approximation order. In addition, when the derivatives are not available or are extremely expensive to compute, we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi-interpolants. As an essential difference to some of the existing methods, we only use the given data values and, then, we do not modify the original triangulation. Finally, we present some numerical tests which confirm the efficiency of the newly quasi-interpolant and demonstrate good visual quality. In particular, we compare it with a differential quasi-interpolant done by Lai [M.-J. Lai, Approximation order from bivariate C1C1-cubics on a four-directional mesh is full, Comput. Aided Geom. Design 11 (2) (1994) 215–223] which is also exact on P3P3 but uses third order partial derivatives.