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We develop a Lagrange interpolation method for quintic C1 splines on cube partitions with 24 tetrahedra in each cube. The construction of the interpolation points is based on a new priority principle by decomposing the tetrahedral partition into special classes of octahedra such that no tetrahedron has to be refined. It follows that the interpolation method is local and stable, and has optimal approximation order six and linear complexity. The interpolating splines are uniquely determined by data values, but no derivatives are needed.

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In this work, we construct two trivariate local Lagrange interpolation methods which yield optimal approximation order and Cr macro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition. The first interpolation method is based on cubic C1 splines over type-4 cube partitions, for which numerical tests are given. The other one is the first trivariate Lagrange interpolation method using C2 splines. It is based on arbitrary tetrahedral partitions using splines of degree nine. In order to obtain this method, several new results on C2 splines over partial Worsey- Farin splits are required. We construct trivariate macro-elements based on the Alfeld, where each tetrahedron is divided into four subtetrahedra, and theWorsey-Farin split, where each tetrahedron is divided into twelve subtetrahedra, of a tetrahedral partition. In order to obtain the macro-elements based on theWorsey-Farin split we construct minimal determining sets for Cr macro-elements over the Clough-Tocher split of a triangle, which are more variable than those in the literature. © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012. All rights are reserved.

We use C 1 quintic superspline functions to interpolate any given scattered data. The space of C 1 quintic superspline functions is introduced in [Lai and LeMehaute'99] and is an improvement of the Alfeld scheme of 3D scattered data interpolation. We have implemented the spline space in MATLAB and tested for the accuracy of reproduction of all quintic polynomials. We present some numerical evidences that when the partition is refined, the spline interpolant converges to the function to be approximated. x1.

We propose a construction of a trivariate C1 macro-element over a special tetrahedral partition and compare our construction with known C 1 macro-elements which are summarized in this paper. Also, we propose an improvement of the Alfeld construction of a C1 quintic macro-element such that the new scheme is able to reproduce all polynomials of total degree ≤5.

Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type-4) partition and associated trivariate C
1 quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.

We describe a local Lagrange interpolation method using cubic (i.e. non-tensor product) C1 splines on cube partitions with five tetrahedra in each cube. We show, by applying a complex proof, that the interpolation method is local, stable, has optimal approximation order and linear complexity. Since no numerical results on trivariate cubic C1 spline interpolation are known from the literature, the steps of the algorithm, which are different from those of the known methods, are focused on its implementation. In this way, we are able to describe the first implementation of a trivariate C1 spline interpolation method, run numerical tests and visualize the corresponding isosurfaces. These tests with up to 5.5×1011 data confirm the efficiency of the algorithm.

We describe an algorithm for constructing a Lagrange interpolation pair based on C1 cubic splines defined on tetrahedral partitions. In particular, given a set of points V∈R3, we construct a set P containing V and a spline space S31(▵) based on a tetrahedral partition ▵ whose set of vertices include V such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets V, and (2) the method provides optimal approximation order of smooth functions.

A trivariate Lagrange interpolation method based on C1 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 in- terpolation method is local and stable, provides optimal order approximation, and has linear complexity.

We consider a linear space of piecewise polynomials in three variables which are globally smooth, i.e. trivariate C1-splines of arbitrary polynomial degree. The splines are defined on type-6 tetrahedral partitions, which are natural generalizations of the fourdirectional mesh. By using Bernstein-B´ezier techniques, we analyze the structure of the spaces and establish formulae for the dimension of the smooth splines on such uniform type partitions.

C 1 quintic splines on type-4 tetrahedral partitions

- L L Schumaker
- T Sorokina

L.L. Schumaker, T. Sorokina, C 1 quintic splines on type-4 tetrahedral partitions, Adv. Comput. Math. 21 (2004) 421–444.