TY - GEN
T1 - Hermite and Bernstein style basis functions for cubic serendipity spaces on squares and cubes
AU - Gillette, Andrew
N1 - Funding Information:
Support for this work was provided in part by NSF Award 0715146 and the National Biomedical Computation Resource while the author was at the University of California, San Diego.
Publisher Copyright:
© Springer International Publishing Switzerland 2014.
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
KW - Finite elements
KW - Hermite interpolation
KW - Multivariate polynomial interpolation
KW - Serendipity elements
KW - Tensor product interpolation
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U2 - 10.1007/978-3-319-06404-8_7
DO - 10.1007/978-3-319-06404-8_7
M3 - Conference contribution
AN - SCOPUS:84927645853
T3 - Springer Proceedings in Mathematics and Statistics
SP - 103
EP - 121
BT - Approximation Theory XIV
A2 - Fasshauer, Gregory E.
A2 - Schumaker, Larry L.
PB - Springer New York LLC
T2 - 14th International conference Approximation Theory XIV, AT 2013
Y2 - 7 April 2013 through 10 April 2013
ER -