Local Lagrange Interpolation By Super Spline Spaces On Triangulations

Bivariate spline spaces have been widely applied in finite element method, numerical approxi-mation theory, surface fitting, scattered data interpolation, numerical solution of partial di?erentialequations and computer aided geometic design (CAGD).In interpolation theory of bivariate splines, there are generally two methods: Hermite interpo-lation and Lagrange interpolation. We discuss Lagrange interpolation in this paper. First,Lagrangeinterpolation schemes are constructed based on bivariate C2 quintic super spline spaces on Wang'srefined triangulations. Firstly,a suitable coloring of the triangles in the original triangulation isused and about half of the triangles are subdivided by Wang's refined triangulation. Then,Lagrangeinterpolation points are chosen in the refined triangulation by requiring certain additional smooth-ness conditions across inserted edges. The corresponding fundamental splines have local sup-ports. Then,Lagrange interpolation scheme is constructed based on bivariate C2 nine supersplines spaces.At first,a suitable coloring of the triangles is used by black and white colors.Then,Lagrange interpolation points are chosen in the triangulations. It is shown that the corre-sponding fundamental splines have local supports.