### Abstract

We introduce new Hermite style and Bernstein style geometric decompositions of the cubic serendipity finite element spaces S_{3}(I^{2}) and S_{3}(I^{3}), as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337–344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces—12 instead of 16 for the square and 32 instead of 64 for the cube—yet are still guaranteed to obtain cubic order a priori error estimates in H^{1} norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.

Original language | English (US) |
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Title of host publication | Approximation Theory XIV |

Subtitle of host publication | San Antonio 2013 |

Editors | Gregory E. Fasshauer, Larry L. Schumaker |

Publisher | Springer New York LLC |

Pages | 103-121 |

Number of pages | 19 |

ISBN (Electronic) | 9783319064031 |

DOIs | |

State | Published - Jan 1 2014 |

Event | 14th International conference Approximation Theory XIV, AT 2013 - San Antonio, United States Duration: Apr 7 2013 → Apr 10 2013 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 83 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | 14th International conference Approximation Theory XIV, AT 2013 |
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Country | United States |

City | San Antonio |

Period | 4/7/13 → 4/10/13 |

### Keywords

- Finite elements
- Hermite interpolation
- Multivariate polynomial interpolation
- Serendipity elements
- Tensor product interpolation

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Approximation Theory XIV: San Antonio 2013*(pp. 103-121). (Springer Proceedings in Mathematics and Statistics; Vol. 83). Springer New York LLC. https://doi.org/10.1007/978-3-319-06404-8_7