Approximation Order of Bivariate Spline Interpolation

AbstractIn [G. Nürnberger and Th. Riessinger,Numer. Math.71(1995), 91–119], we developed an algorithm for constructing point sets at which unique Lagrange interpolation by spaces of bivariate splines of arbitrary degree and smoothness on uniform type triangulations is possible. Here, we show that similar Hermite interpolation sets yield (nearly) optimal approximation order. This is shown for differentiable splines of degree at least four defined on non-rectangular domains subdivided in uniform type triangles. Therefore, in practice we use Lagrange configurations which are “close” to these Hermite configurations. Applications to data fitting problems and numerical examples are given