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Local Lagrange interpolation by quintic C 1 splines on type-6 tetrahedral partitions

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Local Lagrange interpolation by quintic C 1 splines on type-6 tetrahedral partitions

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Abstract

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.
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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.