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 QdQd, which is exact on the space PmPm of polynomials of total degree at most mm, we first propose a general method to determine a new differential quasi-interpolation operator QrD which is exact on Pm+rPm+r. QrD uses the values of the function to be approximated at the points involved in the linear functional defining QdQd as well as the partial derivatives up to the order rr at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C1C1 cubic 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−QrD[f] and f−Q̃d[f] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.