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A \(C^r\) trivariate macro-element based on the Worsey–Farin split of a tetrahedron

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Abstract

We construct trivariate \(C^r\) macro-elements for any \(r\ge 1\) over the Worsey–Farin refinement of any tetrahedral partition. This extends the construction of \(C^1\) cubic Worsey–Farin elements and \(C^2\) elements of degree nine to the \(C^r\) situation with \(r>2\). We also show that the degree of polynomials used for our macro-elements is the lowest.

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Acknowledgments

I would like to thank Ming-Jun Lai for useful discussions and helpful suggestions for this paper. Moreover, I would like to thank Peter Alfeld for writing a great JAVA program [1] which can be used to explore trivariate spline spaces. I made extensive use of the program to test the results for \(r=1,\ldots ,6\).

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  1. Institute for Mathematics, University of Mannheim, 68131, Mannheim, Germany

    Michael A. Matt

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  1. Michael A. Matt
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Correspondence to Michael A. Matt.

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Matt, M.A. A \(C^r\) trivariate macro-element based on the Worsey–Farin split of a tetrahedron. Numer. Math. 123, 121–144 (2013). https://doi.org/10.1007/s00211-012-0478-4

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  • DOI: https://doi.org/10.1007/s00211-012-0478-4

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