Nonlinear 2D C¹ Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

The aim of this paper is to present and study nonlinear bivariate C-1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain 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.