Bivariate Spline Quasi-interpolants on Criss-cross Triangulations for the Approximation of Piecewise Smooth Functions

Spline quasi-interpolation is well known to be a powerful and useful tool for the approximation of bivariate functions and data, important problem in many mathematical and scientific applications. If the function to be approximated is smooth, a spline quasi-interpolant is able to well reconstruct it, but if the function has jump discontinuities, the approximatingspline presents oscillations of magnitude proportional to the jump. Therefore, the aim of this talk is to apply Weighted Essentially Non-Oscillatory (WENO) techniques to modify classical quasi-interpolants in the space of C1 quadratic and C2 quartic splines on criss-cross triangulations, in order to avoid such oscillations. Using such a nonlinear modification we are able to avoid Gibbs phenomenon near discontinuities and, at the same time, maintain the high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.