Hermite and Bernstein style basis functions for cubic serendipity spaces on squares and cubes

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

Abstract

We introduce new Hermite style and Bernstein style geometric decompositions of the cubic serendipity finite element spaces S3(I2) and S3(I3), 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 H1 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 languageEnglish (US)
Title of host publicationApproximation Theory XIV
Subtitle of host publicationSan Antonio 2013
EditorsGregory E. Fasshauer, Larry L. Schumaker
PublisherSpringer New York LLC
Pages103-121
Number of pages19
ISBN (Electronic)9783319064031
DOIs
StatePublished - Jan 1 2014
Event14th International conference Approximation Theory XIV, AT 2013 - San Antonio, United States
Duration: Apr 7 2013Apr 10 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume83
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other14th International conference Approximation Theory XIV, AT 2013
CountryUnited States
CitySan Antonio
Period4/7/134/10/13

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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    Gillette, A. (2014). Hermite and Bernstein style basis functions for cubic serendipity spaces on squares and cubes. In G. E. Fasshauer, & L. L. Schumaker (Eds.), 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