A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

Abstract

Abstract. A trivariate Lagrange interpolation method based on C 1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity. §1. Introductionbe a set of points in R 3 . In this paper we are interested in the following problem. Problem 1. Find a tetrahedral partition whose set of vertices includes V, an Ndimensional space S of trivariate splines defined on with a prescribed smoothness r, and a set of additional points {η i } N i=n+1 such that for every choice of the data, there is a unique spline s ∈ S satisfying (1.1)We call P :and S a Lagrange interpolation pair. It is easy to solve this problem using C 0 splines; see Remark 1. In this case no additional interpolation points are needed. However, the situation is much more complicated if we want to use C r splines with r ≥ 1. For r = 1 the problem was first solved in [29] for the special case where the points in V are the vertices of a cube partition. The method uses quintic splines, and provides full approximation power six for sufficiently smooth functions. In [23] we recently described two C 1 methods which solve the problem for arbitrary initial point sets V. The first method is based on quadratic splines, and requires that each tetrahedron be split into 24 subtetrahedra. The second method is based on cubic splines, and requires that each tetrahedron be split into 12 subtetrahedra. Both methods provide approximation order three for sufficiently smooth functions. For the quadratic spline case this is the optimal approximation order, but for the cubic case it is suboptimal.The aim of this paper is to describe a new method based on cubic C 1 splines which yields optimal approximation order four. To achieve this, we restrict ourselves

Local Lagrange interpolation using cubic C1 splines on type-4 cube partitions

“…We note that the algorithm is different from the general method in [4] and from the algorithm in [3] for Freudenthal partitions. In [4], the computation of the interpolating spline is based on chains of tetrahedra with common vertices and common edges, while in [3], a black and white coloring of the tetrahedra is used.…”

“…On the other hand, only a few results are known for this problem in the trivariate case (see [3,4,12,10]). Up to now, no Lagrange interpolation algorithms using trivariate cubic C 1 splines have been implemented.…”

“…Moreover, we also implement the method of [3] for Freudenthal partitions. The results for our method are slightly better than those for the Freudenthal partition, although fewer tetrahedra are used.…”

Lagrange interpolation set along linear piecewise algebraic curves

“…Davydov et al showed the characterization of the interpolation set on arbitrary triangulations in [3] by investigating the zero set of bivariate linear spline. Other scholars, e.g., Chui, He, Hecklin, Nürnberger, Schumaker and Wang, did much research on this topic [1,2,4,7,10]. Nürnberger and Zeilfelder gave a summary of interpolation methods on bivariate spline spaces in [8].…”

Optimal Lagrange interpolation by quartic C1 splines on triangulations

“…Local Lagrange interpolation methods with optimal approximation order were developed recently in [14,28,18,19,15] for the spaces S 1 q ( ), q 3, and S r q ( ) for r 2 and certain q, where about half of the triangles of are split into three subtriangles. The resulting triangulation has twice as much triangles as .…”