Scattered data interpolation by bivariate splines with higher approximation order

Given a set of scattered data, we usually use a minimal energy method to find a Lagrange interpolation in a bivariate spline space over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in this paper. Mainly we follow the ideas of clamped cubic interpolatory splines and not-a-knot interpolatory splines in the univariate setting and extend them to the bivariate setting. In addition, instead of the energy functional of the second order, we propose to use higher order versions. We shall present some theoretical analysis as well as many numerical results to demonstrate that our new interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory spline.