Approximation of piecewise smooth functions by nonlinear bivariate C2 quartic spline quasi-interpolants on criss-cross triangulations

In this paper we focus on the space of $C^2$ quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.