Increasing the approximation order of spline quasi-interpolants

In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants, which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Qd, which is exact on the space Pm of polynomials of total degree at most m, we first propose a general method to determine a new differential quasi-interpolation operator Qr which is exact on Pm+r.Q D r uses the values of the function to be approximated at the points involved in the linear functional defining Qd as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C1 B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolants Q̃d. We estimate with small constants the quasi-interpolation errors f−Qr [f ] and f−Q̃d[f ] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.