Local Lagrange Interpolation with C2 splines of degree nine on arbitrary tetrahedral partitions

Abstract

In this chapter, we construct a local Lagrange interpolation method for trivariate C2 splines of degree nine on arbitrary tetrahedral partitions. In section 5.1, we consider a decomposition of a tetrahedral partition, which forms a basis for the construction of the Lagrange interpolation method in this chapter. In the next section, we construct a refined tetrahedral partition according to the decomposition obtained in the previous section. Moreover, the spline space used for the Lagrange interpolation in this chapter, which is endowed with several additional smoothness conditions, is defined. In section 5.3, the Lagrange interpolation with the spline space and the refined partition constructed in section 5.2 is investigated. Thereby, we show that the interpolation is 11-local and stable. We also give a nodal minimal determining set for the spline space. Finally, in section 5.4, we prove that the Lagrange interpolation method constructed in this chapter yields optimal approximation order.