Local Lagrange Interpolation By Bivariate Splines On Refining Triangulation

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 in-terpolation and Lagrange interpolation. We discuss Lagrange interpolation in this paper. First,a Lagrange interpolation set is constructed for bivariate spline space S₃¹(△_CT) on Clough-Tocherrefining triangulation; then a C² spline space of degree five is constructed on refining triangula-tion(△|ˉ)_DCT, and a Lagrange interpolation set of this space is also given, where DCT can be gainedthrough using Double Clough-Tocher method to refine some triangles in an arbitrary triangulation. The two interpolation sets constructed here are local, in other words, corresponding fundamen-tal splines have local supports.